Recent zbMATH articles in MSC 30https://zbmath.org/atom/cc/302021-01-08T12:24:00+00:00WerkzeugUniqueness on the polynomial of meromorphic function sharing a small function with its derivative.https://zbmath.org/1449.300062021-01-08T12:24:00+00:00"Lin, Shanhua"https://zbmath.org/authors/?q=ai:lin.shanhua"Lin, Weichuan"https://zbmath.org/authors/?q=ai:lin.weichuanSummary: In this paper, we discuss the problem on a polynomial of meromorphic function and its derivative sharing a small function. Our results improve or generalize the recent results of some previous authors.Further results about iterated order of meromorphic functions with their derivatives.https://zbmath.org/1449.300562021-01-08T12:24:00+00:00"Ma, Hongwei"https://zbmath.org/authors/?q=ai:ma.hongwei"Qi, Jianming"https://zbmath.org/authors/?q=ai:qi.jianming"Chen, Qiaoyu"https://zbmath.org/authors/?q=ai:chen.qiaoyuSummary: The growth of some meromorphic functions that share functions with their derivative is investigated. Combining the definition of iterated order with the normal family theory, we improve the previous results.Growth of meromorphic solutions of some delay differential equations.https://zbmath.org/1449.343172021-01-08T12:24:00+00:00"Long, Fang"https://zbmath.org/authors/?q=ai:long.fang"Wang, Jun"https://zbmath.org/authors/?q=ai:wang.jun.2Summary: The growth property of meromorphic solutions of delay differential equations with rational coefficients is studied. The fact that every transcendental meromorphic solution is of order no less than one is proved under some conditions.The relation between solutions of second order linear differential equation with fixed points.https://zbmath.org/1449.300472021-01-08T12:24:00+00:00"Gong, Pan"https://zbmath.org/authors/?q=ai:gong.pan"Shi, Huangping"https://zbmath.org/authors/?q=ai:shi.huangping"Cheng, Guofei"https://zbmath.org/authors/?q=ai:cheng.guofeiSummary: This paper investigated the relations between solutions of second order linear differential equations and their derivatives with fixed point by by using the theory and the method of Nevanlinna value distribution. The precision result was obtained that convergence exponents of various points of the fixed points of equation solutions and their derivatives are infinite and the second order convergence exponent with the hyper order of solution is equal.The growth of the generalized Hadamard quotient of Dirichlet series.https://zbmath.org/1449.300022021-01-08T12:24:00+00:00"Cui, Yongqin"https://zbmath.org/authors/?q=ai:cui.yongqin"Xu, Huiqing"https://zbmath.org/authors/?q=ai:xu.huiqing"Xu, Hongyan"https://zbmath.org/authors/?q=ai:xu.hongyanSummary: The main purpose of this article is to study the growth of the Hadamard quotient of Dirichlet series which converge in the whole complex plane. By constructing the form of Hadamard quotient of Dirichlet series, we obtain some results about (lower)order and (lower)type, which are extension and improvement of the previous theorems given in literatures.Normal families of meromorphic functions.https://zbmath.org/1449.300782021-01-08T12:24:00+00:00"Lv, Fengjiao"https://zbmath.org/authors/?q=ai:lv.fengjiao"Liu, Zhixiu"https://zbmath.org/authors/?q=ai:liu.zhixiuSummary: The normal family of meromorphic functions is an important subject of the value distribution theory of meromorphic functions. In this paper, we discuss the meromorphic function family concerning shared values and the normality of holomorphic function family, and get some results which improve and supplement some previous results. The related theories of normal families have important applications in complex dynamical systems and complex differential equations.Majorization properties of two classes of analytic functions.https://zbmath.org/1449.300322021-01-08T12:24:00+00:00"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huo"Deng, Guantie"https://zbmath.org/authors/?q=ai:deng.guantieSummary: In this paper, we introduce \({M_n}\) and \({N_n}\) analytic functions associated with lemniscate of Bernoulli and exponential function, as defined by Salagean operator, and discuss the majorization properties of the functions. Our results generalize the theory of majorization in geometric function theory of one complex variable.Sharp distortion theorems for quasi-convex mapping of order \(\alpha\) on the unit ball.https://zbmath.org/1449.300202021-01-08T12:24:00+00:00"Guo, Lijuan"https://zbmath.org/authors/?q=ai:guo.lijuan"Zhang, Xiaofei"https://zbmath.org/authors/?q=ai:zhang.xiaofei"Zhang, Xinhong"https://zbmath.org/authors/?q=ai:zhang.xinhongSummary: In this article, we obtained the sharp distortion theorems of determinant and sharp distortion theorems of matrix at the extreme points for quasi-convex mapping of order \(\alpha\) using the Schwarz lemma at the boundary of unit ball in Euclidean space.The growth of meromorphic solutions of homogeneous and non-homogeneous complex linear equations for composite functions.https://zbmath.org/1449.343152021-01-08T12:24:00+00:00"Chen, Haiying"https://zbmath.org/authors/?q=ai:chen.haiying"Zheng, Xiumin"https://zbmath.org/authors/?q=ai:zheng.xiuminSummary: The growth of meromorphic solutions of a class of homogenous and non-homogeneous complex linear equations for composite functions with meromorphic coefficients is investigated by the Nevanlinna's value distribution of meromorphic function, which is generalized to the more general case of complex linear differential equations for composite functions. When more than one coefficient of the involved equations have the maximal order or the maximal lower order, some estimates on the lower bound of the order or the lower order of non-zero meromorphic solutions of involved equations are obtained under some conditions.The comparison on the growth of the maximum moduli \(M (r,f)\) and its derivative function \(M' (r,f)\) of entire function and analytic function.https://zbmath.org/1449.300512021-01-08T12:24:00+00:00"Tu, Jin"https://zbmath.org/authors/?q=ai:tu.jin"Lv, Fengheng"https://zbmath.org/authors/?q=ai:lv.fengheng"Jiang, Jie"https://zbmath.org/authors/?q=ai:jiang.jieSummary: In this paper, the growth relationship between the maximum moduli \(M (r, f)\) and its derivative function \(M' (r, f)\) is investigated by the order and type of entire function and analytic function in the unit disc, where \(f (z)\) is an entire function or analytic function in the unit disc. The existence of proximate type of analytic function in the unit disc is studied under some conditions.Uniqueness of meromorphic solutions of nonlinear differential equations.https://zbmath.org/1449.300482021-01-08T12:24:00+00:00"Wang, Lingshuang"https://zbmath.org/authors/?q=ai:wang.lingshuang"Huang, Zhigang"https://zbmath.org/authors/?q=ai:huang.zhigangSummary: This paper is devoted to studying the uniqueness of meromorphic functions by Nevanlinna value distribution theory. It is obtained that if \(f (z)\) is a finite order meromorphic solution of the equation \(f'f = F (z)\) and shares \(0,1,\infty\) with finite order meromorphic function \(g (z)\) where \(F (z)\) is an entire function and \(\lambda (F (z)) < 1\), then \(f (z)=g (z)\). If \(f (z)\) is a finite order meromorphic solution of the equation \(f'+ A (z){f^n} = F (z)\) and shares \(0,1,\infty\) with finite order meromorphic function \(g (z)\), where \(A (z)\) is a nonzero polynomial and \(F (z)\) is an entire function, \(\lambda (F (z)) < 1\), \(A (z)\ne F (z)\), then \(f (z)=g (z)\).On the order of growth of entire function and its maximum term and central index.https://zbmath.org/1449.300522021-01-08T12:24:00+00:00"Wu, Xingqun"https://zbmath.org/authors/?q=ai:wu.xingqun"Long, Jianren"https://zbmath.org/authors/?q=ai:long.jianrenSummary: The relationship among the order of growth and the maximum term and the central index of entire function is discussed by using the value distribution theory of meromorphic function. Some characterizations of the order of growth of entire function are obtained in this paper.Univalent criteria for analytic functions involving Schwarzian derivative.https://zbmath.org/1449.300412021-01-08T12:24:00+00:00"Hu, Zhenyong"https://zbmath.org/authors/?q=ai:hu.zhenyong"Wang, Qihan"https://zbmath.org/authors/?q=ai:wang.qihan"He, Liangmiao"https://zbmath.org/authors/?q=ai:he.liangmiao"Long, Boyong"https://zbmath.org/authors/?q=ai:long.boyongSummary: In this paper, some new criteria for univalence of analytic functions defined in the unit disk in terms of two parameters are presented. Moreover, the related result in literature is generalized.Third Hankel determinant for the inverse of starlike and convex functions.https://zbmath.org/1449.300172021-01-08T12:24:00+00:00"Guo, Dong"https://zbmath.org/authors/?q=ai:guo.dong"Ao, En"https://zbmath.org/authors/?q=ai:ao.en"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huo"Xiong, Liangpeng"https://zbmath.org/authors/?q=ai:xiong.liangpengSummary: Let \(S\) denote the class of functions which are analytic, normalized and univalent in the open unit disk \(\mathbb{U} = \{z:| z | < 1\}\). The important subclasses of \(S\) are the class of starlike and convex functions, which are denoted by \({S^*}\) and \(C\). In this paper, we obtain the third Hankel determinant for the inverse of functions \(f (z) = z + \sum\limits_{n = 2}^\infty {a_n}{z^n}\) belonging to \({S^*}\) and \(C\).Coefficient estimates on general compact sets.https://zbmath.org/1449.300082021-01-08T12:24:00+00:00"Totik, Vilmos"https://zbmath.org/authors/?q=ai:totik.vilmosThe article studies coefficient estimates for complex polynomials. The main result is the following: Let \(K\) be a planar compact set of positive logarithmic capacity cap\((K)\). If \(p_n(z)=\sum_{k=0}^n a_kz^k\) is a polynomial of degree \(n\), then \[|a_k|\leq \binom{n}{k} \sup_{z\in K}|z|^{n-k}\sup_{z\in K}|p_n(z)|\frac{1}{{\text{cap}}(K)^n},\;\;\;k=0,1,\dots, n.\] The proof uses potential theoretic tools. Some related results are also proved.
Reviewer: Dimitrios Betsakos (Thessaloniki)Quasimöbius maps and the connectedness properties of quasi-metric spaces.https://zbmath.org/1449.300432021-01-08T12:24:00+00:00"Liu, Hongjun"https://zbmath.org/authors/?q=ai:liu.hongjun"Huang, Xiaojun"https://zbmath.org/authors/?q=ai:huang.xiaojun.1Summary: This paper investigates the connectedness properties of quasi-metric space, and shows that connectedness properties of quasi-metric space are preserved under quasimöbius maps.Commutators of bilinear \(\theta\)-type Calderón-Zygmund operators on Morrey spaces over non-homogeneous spaces.https://zbmath.org/1449.420162021-01-08T12:24:00+00:00"Lu, G.-H."https://zbmath.org/authors/?q=ai:lu.gui-hua|lu.guohao|lu.genghong|lu.guanhua|lu.guanghui|lu.guang-hongThe author proves some boundedness properties for commutators which are generated by the bilinear \(\theta\)-type Calderón-Zygmund operators \(T_\theta\) and two functions \(b_1\), \(b_2\) belonging to the space that is a variant of the bounded mean oscillation class, firstly defined by \textit{F. John} and \textit{L. Nirenberg} [Commun. Pure Appl. Math. 14, 415--426 (1961; Zbl 0102.04302)].
Precisely, \([b_1,b_2,T_\theta]\) is bounded from the Lebesgue space \(L^p(\mu)\) into the product of Lebesgue spaces \[L^{p_1}(\mu) \times L^{p_2}(\mu), \quad \frac{1}{p}\,=\, \frac{1}{p_1}\,+\, \frac{1}{p_2}, \,\,(1 < p, p_1, p_2 < \infty)\] being \(\mu\) a Borel measure.
Moreover the boundedness of the commutator \([b_1, b_2, T_\theta]\) on the Morrey space \(M^q_p(\mu)\), \(1\,<\,q\,<\,p\,<\,\infty\) is obtained. Main tools are the definitions of geometrically doubling metric space and upper doubling metric measure space.
Reviewer: Maria Alessandra Ragusa (Catania)Some inequalities of meromorphic function on an angular domain concerning small functions.https://zbmath.org/1449.300722021-01-08T12:24:00+00:00"Tao, Sijun"https://zbmath.org/authors/?q=ai:tao.sijun"Chen, Yuxian"https://zbmath.org/authors/?q=ai:chen.yuxianSummary: By using the properties of the Nevanlinna characteristic function for meromorphic functions in angular domain, this paper investigates the value distribution of meromorphic functions and their derivatives in angular domain. Some inequalities are obtained for meromorphic functions and their derivatives involving a mall function, and they complement the previous results.No entire inner functions.https://zbmath.org/1449.300892021-01-08T12:24:00+00:00"Cobos, A."https://zbmath.org/authors/?q=ai:cobos.a"Seco, D."https://zbmath.org/authors/?q=ai:seco.daniel|seco.diegoIt is well known that the only entire inner functions in the classical Hardy space \(H^2(\mathbb{D})\) are normalized monomials. The authors discuss the similar problem for other reproducing kernel Hilbert spaces of analytic functions on the unit disk. In particular, they consider the weighted Hardy spaces \(H^2_\omega(\mathbb{D})\) with \(\omega=(\omega_n)_{n\ge0}\), \(\omega_0=1\), and \(\omega_n>0\), \(n=1,2,\ldots\), satisfying \[ \lim_{n\to\infty} \frac{\omega_n}{\omega_{n+1}}=1, \qquad \sup_{n\le k\le 2n}\omega_k\le C\omega_n, \ \ n=1,2,\ldots, \] and the standard inner product \(\langle,\rangle_\omega\) with weights \(\omega_n\). A function \(f\in H^2_\omega(\mathbb{D})\) is called inner, if \(\langle z^n f, f\rangle_\omega=\delta_{n,0}\) for all positive integers \(n\).
A Shapiro-Shields function in \(H^2_\omega(\mathbb{D})\) is a function which is a proper analogue of the finite Blaschke products in \(H^2(\mathbb{D})\). The space \(H^2_\omega(\mathbb{D})\) has no extraneous zeros if all zeros of all Shapiro-Shields functions in the space are regular. The main result of the paper claims that the only entire inner functions for such weighted Hardy spaces are normalized monomials.
Reviewer: Leonid Golinskii (Kharkov)Connections between various subclasses of planar harmonic mappings involving Poisson distribution series.https://zbmath.org/1449.300242021-01-08T12:24:00+00:00"Porwal, Saurabh"https://zbmath.org/authors/?q=ai:porwal.saurabhSummary: The purpose of the present paper is to establish connections between various subclasses of harmonic univalent functions by applying certain convolution operator involving Poisson distribution series. To be more precise, we investigate such connections with Goodman-Ronning-type harmonic univalent functions in the open unit disc \(U\).Schwarz-Pick lemma under generalized complex gradient.https://zbmath.org/1449.300452021-01-08T12:24:00+00:00"Wang, Gen"https://zbmath.org/authors/?q=ai:wang.gen"Liu, Yang"https://zbmath.org/authors/?q=ai:liu.yang.8|liu.yang|liu.yang.20|liu.yang.14|liu.yang.16|liu.yang.6|liu.yang.3|liu.yang.7|liu.yang.22|liu.yang.18|liu.yang.11|liu.yang.13|liu.yang.5|liu.yang.1|liu.yang.9|liu.yang.12|liu.yang.2|liu.yang.19|liu.yang.10|liu.yang.23|liu.yang.15|liu.yang.4|liu.yang.17|liu.yang.21Summary: The transformation form of Schwarz-Pick lemma was investigated by using the method of structural transformation of complex functions. The expressions of structural complex differential and generalized complex gradient were obtained. The generalized Schwarz-Pick lemma related to structural functions was derived. It greatly enlarged the scope of Schwarz-Pick lemma research and the universal applicability of relevant conclusions.On the third Hankel determinant for a subclass of close-to-convex functions.https://zbmath.org/1449.300272021-01-08T12:24:00+00:00"Sahoo, Pravati"https://zbmath.org/authors/?q=ai:sahoo.pravatiSummary: Let \(\mathcal{A}\) denote the class of all normalized analytic function \(f\) in the
unit disc \(\mathbb{U}\) of the form \(f(z)=z+\sum_{n=2}^\infty a_n z^n\). The object of this paper is to obtain a bound to the third Hankel determinant denoted by \(H_3(1)\) for a subclass of close-to-convex functions.The inequality and its application of algebroid functions on annulus concerning some polynomials.https://zbmath.org/1449.300742021-01-08T12:24:00+00:00"Xu, Hong Yan"https://zbmath.org/authors/?q=ai:xu.hongyan"Wu, Zhao Jun"https://zbmath.org/authors/?q=ai:wu.zhaojunThe authors extend work of \textit{Y. Tan} [Acta Math. Sci., Ser. B, Engl. Ed. 36, No. 1, 295--316 (2016; Zbl 1363.30075)] who obtains in Lemma 3.5 a second fundamental theorem of Nevanlinna theory for algebroid functions on annuli. If \(W (z)\) is a k-valued algebroid function determined by \[A_k(z)W^k+A_{k-1}(z)W^{k-1}+\cdots+A_0(z)=0,\] where the \(A_k (z), \dots A_0 (z)\) are analytic functions on \(\{\frac{1}{R_0} < | z | < R_0 \}\) \(( 1 < R_0 \leq +\infty)\) and \(Q_j (z) (j = 1, 2, \dots, q )\) are \(q\) distinct polynomials of degree \(< d\), the current authors derive a relation of the form \[[q-2k-(4k-3)d]~T_0(r,W)<\sum_{j=1}^q N_0 (r,\frac{1}{W(z)-Q_j(z))} )+S_0(r,W),\] where \(T_0\), \( N_0\), and \( S_0\) are the Nevanlinna theory functions as defined in Tan's work. Also shown are related results for \(W (z )\) concerning its derivatives.
Reviewer: Linda R. Sons (DeKalb)Angular derivatives and compactness of composition operators on Hardy spaces.https://zbmath.org/1449.470512021-01-08T12:24:00+00:00"Betsakos, Dimitrios"https://zbmath.org/authors/?q=ai:betsakos.dimitriosLet \(D_0\) be a simply connected domain included in the unit disc \(\mathbb{D}\) of the complex plane. Let \(D \subset D_0\) be a domain such that \(D_0 \setminus D\) is a compact subset of \(D_0\). Let \(\phi\) be a universal covering map of \(\mathbb{D}\) onto \(D\) and let \(\psi\) be a Riemann map of \(\mathbb{D}\) onto \(D_0\). The author proves that the following statements are equivalent: (1) the composition operator \(C_{\phi}\) is compact on the Hardy space \(H^p\), \(0<p \leq \infty\), (2) the composition operator \(C_{\psi}\) is compact on the Hardy space \(H^p\), \(0<p \leq \infty\), (3) \(\phi\) does not have an angular derivative at any point of the unit circle, (4) \(\psi\) does not have an angular derivative at any point of the unit circle. This result improves recent work by \textit{M. M. Jones} [J. Funct. Anal. 268, No. 4, 887--901 (2015; Zbl 1308.47030); Ill. J. Math. 59, No. 3, 707--715 (2015; Zbl 1353.47052)]. Different tools, such as Green functions, subordination, and prime ends, are used to prove the stronger result.
Reviewer: José Bonet (Valencia)Non-linear complex differential-difference equations admit meromorphic solutions.https://zbmath.org/1449.300702021-01-08T12:24:00+00:00"Liu, K."https://zbmath.org/authors/?q=ai:liu.kangping|liu.kuikui|liu.keliang|liu.keqing|liu.kangsheng|liu.kejing|liu.kaiqing|liu.kexuan|liu.kai.4|liu.keying|liu.kaiyu|liu.kejian|liu.kaizhen|liu.kefu|liu.kaifeng|liu.kaidi|liu.kehfei|liu.kexiu|liu.kaiyuan|liu.kexin|liu.keqin|liu.kaixin|liu.kaituo|liu.kanglin|liu.keji|liu.kan|liu.kenneth|liu.kaihua|liu.keping|liu.kexiao|liu.kepan|liu.kongjie|liu.kefeng|liu.ketao|liu.kaiying|liu.kai.3|liu.kai.5|liu.kuang|liu.kai|liu.kexi|liu.kunlun|liu.kefei|liu.kaisheng|liu.kebin|liu.kairan|liu.kehui|liu.kunlin|liu.keguang|liu.kui|liu.kuangyu|liu.kairong|liu.kuan|liu.kai.1|liu.kang|liu.kaifu|liu.kai.2|liu.kimfung|liu.kunkun|liu.kaijun|liu.keqiang|liu.kunlong|liu.kunqi|liu.kecheng|liu.kaizhou|liu.kejia|liu.kaifang|liu.kangyan|liu.kaihui|liu.kean|liu.kaien|liu.kening|liu.kangqi|liu.kaihe|liu.ke|liu.kangjie|liu.kun|liu.kunhong|liu.kaishin|liu.kunhui|liu.kewei|liu.kelan|liu.kuo|liu.kangze|liu.kangling|liu.kangman|liu.kesheng|liu.kaiming|liu.kaile|liu.keyu|liu.kunming|liu.kangni|liu.kiang|liu.keke"Song, C. J."https://zbmath.org/authors/?q=ai:song.chong-jae|song.changjiang|song.chuanjing|song.changjin|song.chengjun|song.chengjuSummary: We obtain necessary conditions for the non-linear complex differential-difference equations \[w(z + 1)w(z - 1) + a(z)\frac{w'(z)}{w(z)}= R(z,w(z))\] to admit transcendental meromorphic solutions \(w(z)\) such that \(\rho_2(w) < 1\), where \(R(z,w(z))\) is rational in \(w(z)\) with rational coefficients, \(a(z)\) is a rational function and \(\rho_2(w)\) is the hyper-order of \(w(z)\). Our results can be seen as the product versions on an equation of another type investigated by \textit{R. Halburd} and \textit{R. Korhonen} [Proc. Am. Math. Soc. 145, No. 6, 2513--2526 (2017; Zbl 1361.30049)]. We also provide an idea which implies that the case of \(\deg_w(R(z,w)) = 4\) in the original proof of Theorem 1.1 of the above mentioned paper can be organized in a short way.Coefficient estimates of certain subclasses of meromorphic bi-univalent functions.https://zbmath.org/1449.300262021-01-08T12:24:00+00:00"Qin, Chuan"https://zbmath.org/authors/?q=ai:qin.chuan"Li, Xiaofei"https://zbmath.org/authors/?q=ai:li.xiaofei"Feng, Jianzhong"https://zbmath.org/authors/?q=ai:feng.jianzhongSummary: This paper defines certain subclasses of meromorphic univalent functions \({\Omega_s} (\alpha, \beta, \lambda)\) and meromorphic bi-univalent functions \({\Omega_{s,\sigma}} (\alpha, \beta, \lambda)\) in the region \(V =\{z \in \mathbb{C}, 1 < |z| < +\infty\}\). Using the definition and property of subordination, the authors study the estimates of coefficients \(|a_0|, |a_n| (n \in \mathbb{N})\). The Fekete-Szegö inequality of the above class \({\Omega_s} (\alpha, \beta, \lambda)\) is also obtained.Some results on sum and product of relative growth factors of composite entire functions.https://zbmath.org/1449.300662021-01-08T12:24:00+00:00"Datta, Sanjib Kumar"https://zbmath.org/authors/?q=ai:kumar-datta.sanjib"Dutta, Banani"https://zbmath.org/authors/?q=ai:dutta.banani"Biswas, Nityagopal"https://zbmath.org/authors/?q=ai:biswas.nityagopalSummary: In this paper, we study about the sum and product of relative \((p,q,t)L\)-th type and relative \((p,q,t)L\)-th lower type of an entire function with respect to another entire function in the light of a special type of non-decreasing, unbounded function \(\Psi\).On the value sharing of shift monomial of meromorphic functions.https://zbmath.org/1449.300642021-01-08T12:24:00+00:00"Banerjee, Abhijit"https://zbmath.org/authors/?q=ai:banerjee.abhijit.1|banerjee.abhijit-vinayak|banerjee.abhijit"Biswas, Tania"https://zbmath.org/authors/?q=ai:biswas.taniaSummary: We employ the notion of weighted and truncated sharing to study the uniqueness problems of generalized shift monomial sharing the same 1-points. The corollary deducted from our main results will improve a number of results of recent time. As an application of the main result we will also improve a recent result under the periphery of a more generalized shift operator. Some examples have been exhibited by us relevant to the content of the paper.On some generalizations of Eneström-Kakeya theorem.https://zbmath.org/1449.260172021-01-08T12:24:00+00:00"Rather, Nisar A."https://zbmath.org/authors/?q=ai:rather.nisar-ahmed|rather.nisar-ahmad|rather.nisar-ahemadSummary: In this paper, we obtain some generalizations of a well-known result of Eneström-Kakeya concerning the bounds for the moduli of the zeros of polynomials with complex coefficients which improve some known results.An application of the distribution series for certain analytic function classes.https://zbmath.org/1449.300132021-01-08T12:24:00+00:00"Çakmak, Serkan"https://zbmath.org/authors/?q=ai:cakmak.serkan"Yalçım, Sibel"https://zbmath.org/authors/?q=ai:yalcim.sibel"Altınkaya, Şahsene"https://zbmath.org/authors/?q=ai:altinkaya.sahseneSummary: For the generalized distribution with the Pascal model defined by \[P(\mathcal{X}=j)=\begin{pmatrix}j+t-1\\t-1\end{pmatrix}p^j(1-p)^t\ j\in\{0,1,2,3,\dots\},\] let \(\mathcal{U}P(\lambda,\alpha,\mu)\) and \(\mathcal{H}P(\lambda,\alpha)\) represent the analytic function classes in the open unit disk \(\mathcal{D}=\{z:z\in\mathbb{C}\) and \(|z|<1\}\). The main aim of this paper is to derive the sufficient conditions for functions in these classes.The Dirichlet problem for the Poisson type equations in the plane.https://zbmath.org/1449.300852021-01-08T12:24:00+00:00"Gutlyanskiĭ, V.Ya."https://zbmath.org/authors/?q=ai:gutlyanskii.vladimir-ya"Nesmelova, O. V."https://zbmath.org/authors/?q=ai:nesmelova.o-v"Ryazanov, V. I."https://zbmath.org/authors/?q=ai:ryazanov.vladimir-iSummary: We present a new approach to the study of semilinear equations of the form \(\text{div} [A(z)\Delta u] = f (u)\), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions \(A(z)\), whereas its reaction term \(f (u)\) is a continuous non-linear function. We establish a theorem on the existence of weak \(C(\overline{D})\cap W^{1,2}_{\text{loc}} (D)\) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains \(D\) without degenerate boundary components and give applications to equations of mathematical physics in anisotropic media.Growth, zeros and fixed points of differences of meromorphic solutions of difference equations.https://zbmath.org/1449.300682021-01-08T12:24:00+00:00"Lan, Shuang-ting"https://zbmath.org/authors/?q=ai:lan.shuangting"Chen, Zong-xuan"https://zbmath.org/authors/?q=ai:chen.zongxuanSummary: In this paper, we study the difference equation
\[
a_1 (z) f(z+1) + a_0(z) f(z)=0,
\]
where \(a_1(z)\) and \(a_0(z)\) are entire functions of finite order. Under some conditions, we obtain some properties, such as fixed points, zeros etc., of the differences and forward differences of meromorphic solutions of the above equation.Estimates for some convolution operators with singularities of their kernels on spheres.https://zbmath.org/1449.470692021-01-08T12:24:00+00:00"Gil', Alekseĭ Viktorovich"https://zbmath.org/authors/?q=ai:gil.aleksei-viktorovich"Zadorozhniĭ, Anatoliĭ Ivanovich"https://zbmath.org/authors/?q=ai:zadorozhnii.anatolii-ivanovich"Nogin, Vladimir Aleksandrovich"https://zbmath.org/authors/?q=ai:nogin.vladimir-aleksandrovichSummary: In the framework of Hardy spaces \(H^p\), we study multidimensional convolution operators whose kernels have power-type singularities on a finite union of spheres in \(\mathbb{R}^n\). Necessary and sufficient conditions are obtained for such operators to be bounded from \(H^p\) to \(H^q\), \(0<p\leq q<\infty\), from \(H^p\) to BMO, and from BMO to BMO.Growth and fixed point of solutions for second-order differential equations in unit disc.https://zbmath.org/1449.343162021-01-08T12:24:00+00:00"Chen, Yu"https://zbmath.org/authors/?q=ai:chen.yu.2|chen.yu.1|chen.yu.6|chen.yu.8|chen.yu.4|chen.yu.3|chen.yu.5|chen.yu.7"Deng, Guantie"https://zbmath.org/authors/?q=ai:deng.guantie(no abstract)On analytic functions involving Noor-Salagean operator.https://zbmath.org/1449.300152021-01-08T12:24:00+00:00"Fayyaz, Rabia"https://zbmath.org/authors/?q=ai:fayyaz.rabia"Noor, Khalida Inayat"https://zbmath.org/authors/?q=ai:noor.khalida-inayatSummary: In the present paper, we define a new operator using Noor integral operator and generalized Salagean operator. We introduce certain new subclasses of analytic function in open unit disk by using this new operator. We investigate some interesting and significant results like inclusion relations and integral preserving properties for these classes.On the quasilinear Poisson equations in the complex plane.https://zbmath.org/1449.352232021-01-08T12:24:00+00:00"Gutlyanskii, V.Ya."https://zbmath.org/authors/?q=ai:gutlyanskii.vladimir-ya"Nesmelova, O. V."https://zbmath.org/authors/?q=ai:nesmelova.o-v"Ryazanov, V. I."https://zbmath.org/authors/?q=ai:ryazanov.vladimir-iSummary: First, we study the existence and regularity of solutions for the linear Poisson equations \(\Delta U(z) = g(z)\) in bounded domains \(D\) of the complex plane \(\mathbb{C}\) with charges g in the classes \(L^1(D)\cap L_{loc}^p(D), p > 1\). Then, applying the Leray-Schauder approach, we prove the existence of Hölder continuous solutions \(U\) in the class \(W_{loc}^{2,p}(D)\) for the quasilinear Poisson equations of the form \(\Delta U(z) = h(z) f (U(z))\) with \(h\) in the same classes as \(g\) and continuous functions \(f:\mathbb{R} \to\mathbb{R}\) such that \(f (t)/t\to 0\) as \( t\to\infty\). These results can be applied to various problems of mathematical physics.Certain subclasses of harmonic univalent functions defined by subordination.https://zbmath.org/1449.300372021-01-08T12:24:00+00:00"Cakmak, S."https://zbmath.org/authors/?q=ai:cakmak.serkan"Yalcin, S."https://zbmath.org/authors/?q=ai:yalcin.sibel"Altinkaya, S."https://zbmath.org/authors/?q=ai:altinkaya.sahseneSummary: In this article, we consider some results involving the modified Salagean operator. We give coefficient bounds for these subclasses. Moreover, we discuss necessary and sufficient convolution conditions, distortion bounds, compactness and extreme points for these subclasses of functions.Convolution properties of some slanted half-plane harmonic mappings.https://zbmath.org/1449.300282021-01-08T12:24:00+00:00"Sharma, Poonam"https://zbmath.org/authors/?q=ai:sharma.poonam-kumar"Porwal, Saurabh"https://zbmath.org/authors/?q=ai:porwal.saurabh"Mishra, Omendra"https://zbmath.org/authors/?q=ai:mishra.omendraSummary: In this paper, we consider two classes \(S (H_\alpha^a)\) and \(S (H_{c,\alpha})\) of all univalent, harmonic, sense-preserving and normalized mappings of the unit disc \(\mathbb{D}\) onto the slanted half-planes \(H_\alpha^a\) and \(H_{c,\alpha}\), respectively. We prove that under certain conditions on \(a\) and \(c\), the convolutions of \(f_{a,\alpha} \in S (H_\alpha^a)\) and \(L_{c,\alpha} \in S (H_{c,\alpha})\) having prescribed dilatations with some slanted half-plane harmonic mappings of dilatation \({e^{i\theta}}{z^n} (n \in \mathbb{N}, \theta \in \mathbb{R})\) are still convex in particular direction.Unicity of mermorphic functions concerning their derivatives and difference.https://zbmath.org/1449.300672021-01-08T12:24:00+00:00"Deng, Bingmao"https://zbmath.org/authors/?q=ai:deng.bingmao"Zeng, Cuiping"https://zbmath.org/authors/?q=ai:zeng.cuiping"Liu, Dan"https://zbmath.org/authors/?q=ai:liu.dan"Fang, Mingliang"https://zbmath.org/authors/?q=ai:fang.mingliangSummary: In this paper, we investigate a uniqueness of meromorphic functions with finite order concerning their derivative and difference and obtain one result that if \({f'}\) and \({\Delta_c}f\) share \(a, b, \infty \) CM, then \(f' \equiv {\Delta_c}f\). This result confirms the previous one in a literature.Nonuniqueness properties on asymptotic Teichmüller space.https://zbmath.org/1449.300862021-01-08T12:24:00+00:00"Huang, Zhiyong"https://zbmath.org/authors/?q=ai:huang.zhiyong"Zhou, Zemin"https://zbmath.org/authors/?q=ai:zhou.zeminSummary: Let \(AT (\Delta)\) be the asymptotic Teichmüller space on the unit disk \(\Delta\), viewed as the space of all asymptotic Teichmüller equivalence classes \([\kern-0.15em[\mu]\kern-0.15em]\) or \([\kern-0.15em[f^\mu]\kern-0.15em]\). It is shown that, for each asymptotically extremal \([\kern-0.15em[f^\mu]\kern-0.15em]\) in \(AT (\Delta)\), there exists an asymptotically extremal \({g^v}\) in \([\kern-0.15em[f^\mu]\kern-0.15em]\) such that the boundary dilatation \({h^*} (\mu_{f\circ{g^{-1}} (g (z))})\ne 0\). A parallel result in the tangent space to \(AT (\Delta)\) at the basepoint is also obtained.Uniqueness of difference operators of entire functions.https://zbmath.org/1449.300542021-01-08T12:24:00+00:00"Li, Qian"https://zbmath.org/authors/?q=ai:li.qian"Wu, Xiaoying"https://zbmath.org/authors/?q=ai:wu.xiaoyingSummary: Under the assumption that a given entire equation has positive deficiency and with the method of complex difference equations, some results about the uniqueness of difference operators of finite order entire functions sharing values with the same multiplicities are presented. These findings can be seen as the difference analogue of differential cases.Fixed points of meromorphic functions and their differences.https://zbmath.org/1449.300622021-01-08T12:24:00+00:00"Wu, Zhaojun"https://zbmath.org/authors/?q=ai:wu.zhaojunSummary: Let \(f\) be a transcendental meromorphic function in the complex plane \(\mathbb{C}\), \(k\) is a positive integer, \(\Delta f = f (z+1) - f (z)\), \(\Delta^{k+1}f = \Delta^kf (z+1) - \Delta^kf\), \(k = 1, 2, \cdots\). The author proved some results concerning the fixed points of the differences \(\Delta^kf\). The results obtained in this paper generalize some relative results.Borel direction of finite order algebroid function.https://zbmath.org/1449.300762021-01-08T12:24:00+00:00"Zhang, Xiaomei"https://zbmath.org/authors/?q=ai:zhang.xiaomei"Sun, Daochun"https://zbmath.org/authors/?q=ai:sun.daochun"Chai, Fujie"https://zbmath.org/authors/?q=ai:chai.fujieSummary: The relationship between the Borel directions of irreducible finite positive order algebroid functions and the Borel directions of their coefficient functions on the complex plane and in the unit circle was investigated. By constructing conformal transformation between the sector and the unit circle and using Nevanlinna's theory and Valiron characteristic of coefficient functions, two interesting theorems of Borel directions of algebroid functions were proved on the basis of investigating the growth of algebroid function and its coefficient function. It is proved that a \(p\)-order Borel direction of the coefficient functions of a finite positive order integral algebroid function on the complex plane must be a Borel direction of the algebroid function of at least order \(p\). In the unit circle, a \(p\)-order Borel point of a finite positive order algebroid function must be a Borel direction of a coefficient function of at least order \(p\).Further results on the periodicity of meromorphic functions.https://zbmath.org/1449.300552021-01-08T12:24:00+00:00"Lian, Gui"https://zbmath.org/authors/?q=ai:lian.gui"Chen, Junfan"https://zbmath.org/authors/?q=ai:chen.junfanSummary: In this paper we study the periodicity of meromorphic functions and their shift operators with sharing values and truncated sharing values. The theorems obtained in the paper generalize the results of literatures. Moreover, examples are given to show that the conditions are necessary.Extensions of Vieira's theorems on the zeros of self-inversive polynomials.https://zbmath.org/1449.300102021-01-08T12:24:00+00:00"Losonczi, László"https://zbmath.org/authors/?q=ai:losonczi.laszloSummary: Recently \textit{R. S. Vieira} [Ramanujan J. 42, No. 2, 363--369 (2017; Zbl 1422.30013) ] found sufficient conditions for self-inversive polynomials to have some of their zeros on the unit circle. We extend his results by giving the location of those zeros. In case of fourth degree real reciprocal polynomials we compare Vieira's sufficient conditions with the necessary and sufficient conditions obtained by help of Chebyshev transformation.A new subclass of harmonic mappings with positive coefficients.https://zbmath.org/1449.300212021-01-08T12:24:00+00:00"Haghighi, A. R."https://zbmath.org/authors/?q=ai:haghighi.ahmad-reza"Asghary, N."https://zbmath.org/authors/?q=ai:asghary.nasim"Sedghi, A."https://zbmath.org/authors/?q=ai:sedghi.aSummary: Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disk \(U\) can be written as form \(f =h+\bar{g}\), where \(h\) and \(g\) are analytic in \(U\). In this paper, we introduce the class \(S_H^1(\beta)\), where \(1<\beta\leq 2\), and consisting of harmonic univalent function \(f = h+\bar{g}\), where \(h\) and \(g\) are in the form \(h(z) = z+\sum_{n=2}^\infty |a_n|z^n\) and \(g(z) = \sum_{n=2}^\infty |b_n|\bar{z}^n\) for which \[\mathrm{Re}\,\{z^2(h''(z)+g''(z))+2z(h'(z)+g'(z))-(h(z)+g(z))-(z-1)\}<\beta.\] It is shown that the members of this class are convex and starlike. We obtain distortions bounds extreme point for functions belonging to this class, and we also show this class is closed under convolution and convex combinations.Analytic and harmonic univalent functions.https://zbmath.org/1449.000062021-01-08T12:24:00+00:00"Ravichandran, V. (ed.)"https://zbmath.org/authors/?q=ai:ravichandran.v"Ahuja, Om P. (ed.)"https://zbmath.org/authors/?q=ai:ahuja.om-prakash"Ali, Rosihan M. (ed.)"https://zbmath.org/authors/?q=ai:ali.rosihan-mohamedFrom the text: This special issue aims to disseminate recent advances in the studies of complex function theory, harmonic univalent functions, and their connections to produce deeper insights and better understanding. These are crystallized in the form of original research articles or expository survey papers.Some geometric properties for a class of Janowski functions defined by a derivative operator.https://zbmath.org/1449.300292021-01-08T12:24:00+00:00"Song, Lili"https://zbmath.org/authors/?q=ai:song.liliSummary: In this paper, a new general class of Janowski functions is introduced by using a known derivative operator and subordination. Some properties such as the radius with inclusion relations, sharp coefficient bounds and sharp distortion theorems are studied. Furthermore, a real part inequality of the square root of the functions related to the main class is presented. Several useful consequences and earlier results associated with this main works are pointed out by choosing special parameters.Third Hankel determinant for a class of generalized analytic functions related with lemniscate of Bernoulli and symmetric points.https://zbmath.org/1449.300352021-01-08T12:24:00+00:00"Zhang, Haiyan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huo"Ma, Li'na"https://zbmath.org/authors/?q=ai:ma.linaSummary: In this paper, we introduce a class of generalized analytic functions, denoted by \(SL_s^* (\alpha, \mu)\), which are associated with lemniscate of Bernoulli and symmetric points. We investigate the Hankel determinant \({H_3} (1)\) for these functions and the upper bound of the above determinant is obtained.Several theorems of algebroid functions.https://zbmath.org/1449.300592021-01-08T12:24:00+00:00"Tan, Yang"https://zbmath.org/authors/?q=ai:tan.yangSummary: Firstly, the Nevanlinna theory is used to investigate the uniqueness of algebroid functions, the strengthening five-valued theorem of meromorphic function is extended to algebroid functions, and the strengthening \(3v + 1\)-valued theorem of algebroid functions is obtained. Secondly, we study the relation between the characteristic functions of two algebroid functions when they share less values and extend the four-valued theorem of meromorphic function.Coefficient estimates for certain subclass of bi-univalent functions involving a general differential operator.https://zbmath.org/1449.300392021-01-08T12:24:00+00:00"Zhao, Wei"https://zbmath.org/authors/?q=ai:zhao.wei.5|zhao.wei.4|zhao.wei.2|zhao.wei.1|zhao.wei.3|zhao.wei.6"Qin, Chuan"https://zbmath.org/authors/?q=ai:qin.chuan"Li, Xiaofei"https://zbmath.org/authors/?q=ai:li.xiaofeiSummary: In this paper, a familiar subclass \(\mathcal{MN}_\Sigma^{h, p} (\lambda, \mu; m, \delta)\) of analytic and bi-univalent functions in the open unit disk \(\mathcal{U}\) defined by a general differential operator are introduced and investigated. Estimates for the second and third coefficients of their Taylor expansion are obtained. Some relevant connections of the result presented here with various well-known results are briefly indicated.Chebyshev polynomials on circular arcs.https://zbmath.org/1449.300802021-01-08T12:24:00+00:00"Schiefermayr, Klaus"https://zbmath.org/authors/?q=ai:schiefermayr.klausThe Chebyshev polynomial of degree \(N\), \(N\in\mathbb{N}\), on a compact set \(K\subset\mathbb{C}\) in the complex plane is that monic polynomial \(\hat{\mathcal{P}}_N\in\hat{\mathbb{P}}_N\) which is minimal with respect to the supremum norm on \(K\) within the set of all monic polynomials, i.e. \[\hat{\mathcal{P}}_N:=\min\{\|\hat{P}_N\|_K:\hat{P}_N\in\hat{\mathbb{P}}_N\}\,, \] where \(\|\cdot\|\) denotes the supremum norm on \(K\) and \(\hat{\mathbb{P}}_N\) denotes the set of all monic polynomials of degree \(N\). In this paper, an explicit parametric representation of the complex Chebyshev polynomials \(\hat{P}_N(z)\) on a given circular arc \(A_\alpha\), defined by \[ A_\alpha:=\{z\in\mathbb{C}: |z|=1,-\alpha\leq\arg(z)\leq\alpha\},\quad 0<\alpha\leq\pi\,,\] of the unit circle (in the complex plane) in terms of real Chebyshev polynomials \(\hat{\mathcal{T}}_{N'}(x)\) on two symmetric intervals \([-1,-a]\cup[a,1]\) (on the real line) is given. For example, let \(0<\alpha<\frac{2n\pi}{2n+1}\), \(0<c<\frac{n\pi}{2n+1}\) be fixed and \(a:=\cos(\alpha/2)\). Let \(\mathcal{T}_{2n+1}\in\mathbb{P}_{2n+1}\) and \(\mathcal{U}_{2n-2}\in\mathbb{P}_{2n-2}\) be uniquely determined by \[ \mathcal{T}_{2n+1}^2(x)+(1-x^2)(x^2-a^2)(x^2-c^2)\mathcal{U}_{2n-2}^2(x)=1\,. \] Then \[\hat{P}_{2n}(z)=L_{2n}z^{n-1/2}\left(\mathcal{T}_{2n+1}(x)+i\sqrt{1-x^2}(x^2-a^2)\mathcal{U}_{2n-2}(x)\right)\,,\] is a monic polynomial of degree \(2n\) in \(z\) with real coefficients, where \(x\) and \(z\) are connected by \[ z\mapsto\frac12\left(\sqrt{z}+\frac1{\sqrt{z}}\right)=:x\,. \] Moreover, \(\hat{P}_{2n}(z)\) is the Chebyshev polynomial of degree \(2n\) on \(A_\alpha\) with minimum deviation. The case \(N=2n-1\) is also considered. It is also considered representation of Chebyshev polynomials on \([-1,-a]\cup[a,1]\) with the help of Jacobian elliptic and theta functions, which goes back to the work of Akhiezer in the 1930's.
Reviewer: Konstantin Malyutin (Kursk)Product of Volterra type integral operator and composition operators between generalized Fock spaces.https://zbmath.org/1449.470672021-01-08T12:24:00+00:00"Luo, Xiaojuan"https://zbmath.org/authors/?q=ai:luo.xiaojuan"Wang, Xiaofeng"https://zbmath.org/authors/?q=ai:wang.xiaofeng.1"Xia, Jin"https://zbmath.org/authors/?q=ai:xia.jinSummary: In this paper, equivalent characterizations for the boundedness, compactness, and Schatten-\(p\) class properties of the product of a Volterra type integral operator and a composition operator between generalized Fock spaces \(F_\phi^p\) and generalized Fock spaces \(F_\phi^q\) are proposed in terms of certain Berezin integral transformations on the complex plane \(\textbf{C}\), where \(0 < p, q < \infty \). We also obtain some estimates on the essential norms of these operators.Some applications of generalized Srivastava-Attiya operator to the bi-concave functions.https://zbmath.org/1449.300112021-01-08T12:24:00+00:00"Altınkaya, Şahsene"https://zbmath.org/authors/?q=ai:altinkaya.sahsene"Yalçın, Sibel"https://zbmath.org/authors/?q=ai:yalcin.sibelSummary: In this present investigation, we are concerned with the class \(\Omega_{\Sigma;\mu ,b}^{m,k}C_{0}(\alpha)\) of bi-concave functions defined by using the generalized Srivastava-Attiya operator. Moreover, we derive some coefficient inequalities for functions in this class.Spectra of composition operators on weighted Bergman spaces.https://zbmath.org/1449.470552021-01-08T12:24:00+00:00"Pons, Matthew A."https://zbmath.org/authors/?q=ai:pons.matthew-aSummary: We extend known results on the spectra of composition operators to the weighted Bergman spaces. Our results include a study of the essential spectral radius, a determination of the spectrum when the symbol of the composition operator is univalent and non-automorphic with a fixed point in the disk, and an affirmative answer to a conjecture of \textit{B. MacCluer} and \textit{K. Saxe} [Isr. J. Math. 128, 325--354 (2002; Zbl 1024.47009)].The Fekete-Szegö functional for a subclass of analytic functions associated with quasi-subordination.https://zbmath.org/1449.300302021-01-08T12:24:00+00:00"Srivastava, H. M."https://zbmath.org/authors/?q=ai:srivastava.hari-mohan"Hussain, Saqib"https://zbmath.org/authors/?q=ai:hussain.saqib"Raziq, Alia"https://zbmath.org/authors/?q=ai:raziq.alia"Raza, Mohsan"https://zbmath.org/authors/?q=ai:raza.mohsanSummary: In the present paper, we introduce and investigate the Fekete-Szegö functional associated with a new subclass of analytic functions, which we have defined here by using the principle of quasi-subordination between analytic functions. Some sufficient conditions for functions belonging to this class are also derived. The results presented here improve and generalize several known results.Degenerate Lambert quadrilaterals and Möbius transformations.https://zbmath.org/1449.510052021-01-08T12:24:00+00:00"Demirel, Oğuzhan"https://zbmath.org/authors/?q=ai:demirel.oguzhanAn \(\varepsilon\)-Lambert quadrilateral is a quadrilateral in the hyperbolic plane \(B^2=\{z\in {\mathbb{C}}:\vert z\vert<1\}\) with angles \(\frac{\pi}{2}+\varepsilon\), \(\frac{\pi}{2}\), \(\frac{\pi}{2}-\varepsilon\), and \(\theta\), where \(0<\theta<\frac {\pi}{2}\) and \(0<\varepsilon<\frac{\pi}{2}-\frac{\theta}{2}\). The main result of this paper belongs to characterizations of geometric transformations under mild hypotheses. It states that a surjective transformation \(f : B^2\rightarrow B^2\) is a Möbius transformation or a conjugate Möbius transformation if and only if \(f\) preserves all \(\epsilon\)-Lambert quadrilaterals where \(0<\varepsilon<\frac{\pi}{2}\). The reasoning is geometric.
Reviewer: Victor V. Pambuccian (Glendale)Third-order Hankel determinant for analytic functions based on the left-half of lemniscate of Bernoulli.https://zbmath.org/1449.300362021-01-08T12:24:00+00:00"Zhang, Haiyan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huo"Niu, Xiaomeng"https://zbmath.org/authors/?q=ai:niu.xiaomengSummary: In this paper, we introduce a class of analytic functions \(B{L_\alpha} (0 \le \alpha \le 1)\) in the left-half bounded domain of lemniscate of Bernoulli, which are defined by using subordination relationship. We then investigate the third Hankel determinant \(\boldsymbol{H}_3 (1)\) of \(B{L_\alpha}\), and obtain its upper bound. In addition, some special cases of the results are given.Integral representation of the viscoelastic relaxation function.https://zbmath.org/1449.300842021-01-08T12:24:00+00:00"Xiu, Guozhong"https://zbmath.org/authors/?q=ai:xiu.guozhong"Wang, Liying"https://zbmath.org/authors/?q=ai:wang.liying"Shi, Bao"https://zbmath.org/authors/?q=ai:shi.bao"He, Yingzheng"https://zbmath.org/authors/?q=ai:he.yingzhengSummary: In this paper, the integral representation of relaxation function is discussed. The stress-strain relationship equation of Maxwell model is obtained by Laplace transformation. The relaxation function in the equation can be expressed by Mittag-Leffler function. Because of the existence of large negative arguments, it is very difficult to calculate. We use the continuous relaxation spectrum to express the Mittag-Leffler function in integral form. This problem has been solved. A numerical example illustrates the effectiveness of the result.Lemniscate convexity of generalized Bessel functions.https://zbmath.org/1449.300072021-01-08T12:24:00+00:00"Madaan, Vibha"https://zbmath.org/authors/?q=ai:madaan.vibha"Kumar, Ajay"https://zbmath.org/authors/?q=ai:kumar.ajay.1|kumar.ajay"Ravichandran, V."https://zbmath.org/authors/?q=ai:ravichandran.vIn this paper the authors study lemniscate convexity of generalized Bessel functions. First, they recall the classes of lemniscate convex, lemniscate starlike and lemniscate Carathéodory functions in the unit disc D. Then, in a series of theorems, they state conditions on parameters so that several Bessel functions belong to these classes. In addition to above results, they also show that the Lommel function of the first kind, Alexander and Libera transform of this function belong to class of lemniscate convex functions.
Reviewer: Yusuf Avci (Istanbul)On the solution of Fermat-type differential-difference equations.https://zbmath.org/1449.300692021-01-08T12:24:00+00:00"Liu, Dan"https://zbmath.org/authors/?q=ai:liu.dan"Deng, Bingmao"https://zbmath.org/authors/?q=ai:deng.bingmao"Yang, Degui"https://zbmath.org/authors/?q=ai:yang.deguiSummary: In this paper, we mainly discuss the entire solutions of finite order of the following Fermat type differential-difference equation \[[f^{ (k)} (z)]^2 + [{\Delta_c}f (z)]^2 = 1,\] and the systems of differential-difference equations of the form \[\begin{cases}[{f_1^{ (k)}} (z)]^2 + [{\Delta_c}{f_2} (z)]^2 = 1, \\ [{f_2^{ (k)}} (z)]^2 + [{\Delta_c}{f_1} (z)]^2 = 1.\end{cases}\] Our results can be proved to be the sufficient and necessary solutions to both the equation and the system of equations.Further results on meromorphic functions and their \(n\)th order exact differences with three shared values.https://zbmath.org/1449.300652021-01-08T12:24:00+00:00"Chen, Shengjiang"https://zbmath.org/authors/?q=ai:chen.shengjiang"Xu, Aizhu"https://zbmath.org/authors/?q=ai:xu.aizhu"Lin, Xiuqing"https://zbmath.org/authors/?q=ai:lin.xiuqingSummary: Let \(E (a,f)\) be the set of \(a\)-points of a meromorphic function \(f (z)\) counting multiplicities. We prove that if a transcendental meromorphic function \(f (z)\) of hyper order is strictly less than 1 and its \(n\)th exact difference \(\Delta_c^nf (z)\) satisfies \(E (1,f) = E (1,\Delta_c^nf)\), \(E (0,f) \subset E (0, \Delta_c^nf)\) and \(E (\infty, f) \supset E (\infty, \Delta_c^nf)\), then \(\Delta_c^nf (z) \equiv f (z)\). This result improves a more recent theorem by using a simple method.On uniqueness problem of meromorphic functions sharing values with their \(q\)-shifts.https://zbmath.org/1449.300752021-01-08T12:24:00+00:00"Zhang, Shuiying"https://zbmath.org/authors/?q=ai:zhang.shuiying"Liu, Huifang"https://zbmath.org/authors/?q=ai:liu.huifangSummary: In this paper, the uniqueness problems on meromorphic function \(f (z)\) of zero order sharing values with their \(q\)-shifts \(f (qz + c)\) are studied. It is shown that if \(f (z)\) and \(f (qz + c)\) share values CM and IM respectively, or share four values partially, then they are identical under an appropriate deficiency assumption.Normality criteria of zero-free meromorphic functions.https://zbmath.org/1449.300792021-01-08T12:24:00+00:00"Xie, Jia"https://zbmath.org/authors/?q=ai:xie.jia"Deng, Bingmao"https://zbmath.org/authors/?q=ai:deng.bingmaoSummary: Let \(k\) be a positive integer, let \(h (z)\not\equiv 0\) be a holomorphic function in a domain \(D\), and let \(\mathcal{F}\) be a family of zero-free meromorphic function in \(D\), all of their poles have order at least \(l\). If, for each \(f\in \mathcal{F}\), \(P (f) (z)-h (z)\) has at most \(k+l-1\) distinct zeros (ignoring multiplicity) in \(D\), where \(P (f) (z) = f^{ (k)} (z) + {a_1} (z)f^{ (k-1)} (z)+ \cdots + {a_k} (z)f (z)\) is a differential polynomial of \(f\) and \({a_j} (z) (j = 1,2,\cdots,k)\) are holomorphic functions in \(D\), then \(\mathcal{F}\) is normal in \(D\).Entire function solutions of two types of Fermat type \(q\)-difference differential equations.https://zbmath.org/1449.300532021-01-08T12:24:00+00:00"Fan, Bo"https://zbmath.org/authors/?q=ai:fan.bo"Ding, Jie"https://zbmath.org/authors/?q=ai:ding.jieSummary: In this paper, using Nevanlinna's value distribution theory and the complex differential equations theory, the existence of finite order transcendental entire function solutions for two types of Fermat type \(q\)-difference differential equations of the following form \[{f^2} (qz+c) + ({f^{ (k)}} (z))^2 = 1,\; [f (qz+c) - f (z)]^2 + ({f^{ (k)}} (z))^2 = 1\] is investigated. Moreover, the precise expression of the solutions is obtained under some assumptions.Reverse Markov inequality on the unit interval for polynomials whose zeros lie in the upper unit half-disk.https://zbmath.org/1449.410072021-01-08T12:24:00+00:00"Komarov, M. A."https://zbmath.org/authors/?q=ai:komarov.mikhail-aSummary: We prove that there is an absolute constant \(A>0\) such that \[\max_{-1\leq x\leq 1}\vert P'(x)\vert \geq A\sqrt{n}\cdot \max_{-1\leq x\leq 1}\vert P'(x)\vert \] for an arbitraray algebraic polynomial \(P\) of degree \(n\) whose zeros lie in the half-disk \(\{z:\vert z\vert \leq 1,\)\; Im\(z\geq 0\}\).Uniqueness of differential polynomials of meromorphic functions IM sharing a value.https://zbmath.org/1449.300632021-01-08T12:24:00+00:00"Zhang, Weijie"https://zbmath.org/authors/?q=ai:zhang.weijie"Wang, Xinli"https://zbmath.org/authors/?q=ai:wang.xinli"Wang, Hanjie"https://zbmath.org/authors/?q=ai:wang.hanjieSummary: This paper study uniqueness of differential polynomials of meromorphic functions sharing 1 IM. Let \(f\left(z \right)\) and \(g\left(z \right)\) be two non-constant meromorphic functions, whose zeros and poles are of multiplicities at least \(s\), where \(s\) is a positive integer. Let \(n \ge 2\) be an integer satisfying \(\left({n + 1} \right)s \ge 24\). If \({f^n}f'\) and \({g^n}g'\) share the value 1 IM, then either \(f\left(z \right) = tg\left(z \right)\) for some \(\left({n + 1} \right)\)-th root of unity \(t\) or \(g\left(z \right) = {c_1}{e^{cz}}\), \(f\left(z \right) = {c_2}{e^{ - cz}}\), where \({c_1}\), \({c_2}\), \(c\) are constants satisfying \({\left({{c_1}{c_2}} \right)^{n + 1}}{c^2} = - 1\).Meromorphic solutions of a class of auxiliary differential equation and its applications.https://zbmath.org/1449.343142021-01-08T12:24:00+00:00"Gu, Yongyi"https://zbmath.org/authors/?q=ai:gu.yongyi"Kong, Yinying"https://zbmath.org/authors/?q=ai:kong.yinyingSummary: This paper introduces a method to find exact solutions of nonlinear partial differential equations -- complex method, and derives meromorphic solutions for a class of algebraic differential equation by the mentioned method. The results are used to seek exact solutions of nonlinear differential equations. Exact solutions of the Vakhnenko-Parkes equation and Dodd-Bullough-Mikhailov equation are obtained.Generalized Schwarzian derivatives and analytic Morrey spaces.https://zbmath.org/1449.300422021-01-08T12:24:00+00:00"Jin, Jianjun"https://zbmath.org/authors/?q=ai:jin.jianjun"Li, Huabing"https://zbmath.org/authors/?q=ai:li.huabing"Tang, Shu'an"https://zbmath.org/authors/?q=ai:tang.shuanSummary: In this paper, we study univalent functions \(f\) for which \(\log f'\) belongs to the analytic Morrey spaces. By using the characterization of higher order derivatives of functions in analytic Morrey spaces, we establish some new descriptions for the analytic Morrey domains in terms of two kinds of generalized Schwarzian derivatives.An equivalent characterization of \(CMO (\mathbb{R}^n)\) with \({A_p}\) weights.https://zbmath.org/1449.300882021-01-08T12:24:00+00:00"Zhong, Minghui"https://zbmath.org/authors/?q=ai:zhong.minghui"Hou, Xianming"https://zbmath.org/authors/?q=ai:hou.xianmingSummary: Let \(1< p < \infty\) and \(\omega\in {A_p}\). The space \(CMO (\mathbb{R}^n)\) is the closure in \(BMO (\mathbb{R}^n)\) of the set of \(C_c^\infty (\mathbb{R}^n)\). In this paper, an equivalent characterization of \(CMO (\mathbb{R}^n)\) with \({A_p}\) weights is established.Pseudostarlike, pseudoconvex and close-to-pseudoconvex Dirichlet series satisfying differential equations with exponential coefficients.https://zbmath.org/1449.300032021-01-08T12:24:00+00:00"Holovata, O. M."https://zbmath.org/authors/?q=ai:holovata.o-m"Mulyava, O. M."https://zbmath.org/authors/?q=ai:mulyava.oksana-m"Sheremeta, M. M."https://zbmath.org/authors/?q=ai:sheremeta.myroslav-mThe paper is concerned with the concepts of pseudostarlikeness, pseudoconvexity and close-to-pseudoconvexity introduced for the Dirichlet series with null abscissa of absolute convergence. The obtained results are used to study the properties of solutions of differential equations with exponential coefficients.
Reviewer: V. V. Vlasov (Moscow)Fekete-Szegö functional problems for certain subclasses of bi-univalent functions involving the Hohlov operator.https://zbmath.org/1449.300232021-01-08T12:24:00+00:00"Long, Pinhong"https://zbmath.org/authors/?q=ai:long.pinhong"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huo"Wang, Wenshuai"https://zbmath.org/authors/?q=ai:wang.wenshuaiSummary: In the paper the new subclasses \(\mathcal{N}_\Sigma^{a, b, c} (\mu, \lambda; \phi)\) and \(\mathcal{M}_\Sigma^{a, b, c} (\lambda; \phi)\) of the function class \(\Sigma\) of bi-univalent functions involving the Hohlov operator are introduced and investigated. Then, the corresponding Fekete-Szegö functional inequalities as well as the bound estimates of the coefficients \({a_2}\) and \({a_3}\) are obtained. Furthermore, several consequences and connections to some of the earlier known results also are given.Transformation formula of Cauchy type singular integral operators with a weakly singular integral operators in Clifford analysis.https://zbmath.org/1449.300832021-01-08T12:24:00+00:00"Huang, Yagai"https://zbmath.org/authors/?q=ai:huang.yagai"Shi, Haipan"https://zbmath.org/authors/?q=ai:shi.haipan"Qiao, Yuying"https://zbmath.org/authors/?q=ai:qiao.yuyingSummary: In this paper we study the transformation problem of a weakly singular integral with the Cauchy type singular operators whose singular point is the integral variable of the weakly singular integral on Lyapunov closed surface in Clifford analysis. Firstly, we discuss the nature of relevant singular integrals and then prove that the two iterated integrals are well defined by using that nature. Next, we divide the integral domain into several parts. Naturally, the integral operators are grouped into two parts. One is with the singular integrals and the other is with non-singular operators. We prove that the limits of the part with the singular operators are zero and the limits of the part without singularity are equal. Finally, we obtain the transformation formula of a weakly singular integral with the Cauchy type singular operators whose singular point is the integral variable of the weakly singular integral.Meromorphic functions concerning difference operator.https://zbmath.org/1449.300732021-01-08T12:24:00+00:00"Waghamore, Harina P."https://zbmath.org/authors/?q=ai:waghamore.harina-pandit"Maligi, Ramya"https://zbmath.org/authors/?q=ai:maligi.ramyaSummary: We deal with a uniqueness question of meromorphic functions sharing a
polynomial with their difference operators and obtain some results, which generalize
and improve the recent result of \textit{S. Majumder} [Appl. Math. E-Notes 17, 114--123 (2017; Zbl 1418.30029)].On the uniqueness of exponential polynomials with constant coefficients concerning shared values CM.https://zbmath.org/1449.300712021-01-08T12:24:00+00:00"Su, Min"https://zbmath.org/authors/?q=ai:su.min"Li, Yuhua"https://zbmath.org/authors/?q=ai:li.yuhua"Gang, Pengfei"https://zbmath.org/authors/?q=ai:gang.pengfeiSummary: In this paper, the authors prove that two non-constant exponential polynomials with constant coefficients must be identical, and provide that they share four distinct CM values while each CM value lies in an angular domain of opening strictly larger than \(\pi\).Cauchy integral formulas for two kinds of functions in Clifford analysis and the related problems.https://zbmath.org/1449.300822021-01-08T12:24:00+00:00"Chen, Xue"https://zbmath.org/authors/?q=ai:chen.xue"Zhang, Tingting"https://zbmath.org/authors/?q=ai:zhang.tingting"Xie, Yonghong"https://zbmath.org/authors/?q=ai:xie.yonghongSummary: This paper mainly studies Cauchy integral formulas for two kinds of functions and the related problems. Firstly, the Cauchy integral formula for right hypergenic functions in Clifford analysis is given. Then, the properties of the right hypergenic quasi-Cauchy type integral are studied. Finally, the Cauchy integral formula for bihypergenic functions in Clifford analysis is given.The uniqueness of meromorphic functions that concern multiple values and deficiencies on annuli.https://zbmath.org/1449.300602021-01-08T12:24:00+00:00"Tan, Yang"https://zbmath.org/authors/?q=ai:tan.yangSummary: In this paper, in order to further enrich the uniqueness theory of meromorphic functions on annuli and find better uniqueness conditions, we investigate the influences of multiple values and deficiencies on the uniqueness of meromorphic functions on annuli, and obtain two uniqueness theorems of meromorphic functions and their derivatives which concern multiple values and deficiencies. These results enrich the uniqueness theory of meromorphic functions on annuli.A subclass of harmonic univalent functions.https://zbmath.org/1449.300312021-01-08T12:24:00+00:00"Sun, Zuochen"https://zbmath.org/authors/?q=ai:sun.zuochen"Wang, Qihan"https://zbmath.org/authors/?q=ai:wang.qihan"Long, Boyong"https://zbmath.org/authors/?q=ai:long.boyongSummary: A class of Salagean-type harmonic univalent functions is investigated. Quasiconformality, convexity and convolution of this class of functions are obtained. Some related results are improved.Value distribution of difference analogues of Hayman problem.https://zbmath.org/1449.300612021-01-08T12:24:00+00:00"Wang, Xinxin"https://zbmath.org/authors/?q=ai:wang.xinxin"Ye, Yasheng"https://zbmath.org/authors/?q=ai:ye.yashengSummary: In this paper, we study difference analogues of some classical results about Hayman problem. By the Nevanlinna theory, we obtain the precise estimation of the lower bound of the zero counting function of a class of difference polynomials and improve some previously results.Relationship between \({Q_K} (p,q)\) spaces and \({\mathcal{B}^\alpha}\) spaces on the unit ball.https://zbmath.org/1449.300872021-01-08T12:24:00+00:00"Hu, Rong"https://zbmath.org/authors/?q=ai:hu.rongSummary: Different function spaces have certain inclusion or equivalence relations. This paper introduces a class of Möbius Banach spaces \({Q_K} (p,q)\) of analytic function on the unit ball of \({\mathbb{C}^n}\), discusses the trivialness condition, and studies the inclusion relations between them and a class of \({\mathcal{B}^\alpha}\) spaces, then obtains the equivalent condition of kernel function \(K (r)\) when \({Q_K} (p,q)\) and \({\mathcal{B}^\alpha}\) are equivalent.Properties and applications of meromorphic function with finite logarithmic growth order.https://zbmath.org/1449.300502021-01-08T12:24:00+00:00"Guo, Xiaona"https://zbmath.org/authors/?q=ai:guo.xiaona"Ding, Jie"https://zbmath.org/authors/?q=ai:ding.jieSummary: The properties of logarithmic growth order of some product and sum of meromorphic functions with finite logarithmic growth order are given, and the relations between meromorphic solutions and coefficients of linear \(q\)-difference equation are obtained by using the properties of meromorphic functions with finite logarithmic growth order and the \(q\)-difference form Wiman-Valiron theory.Coefficient bounds for certain subclass of meromorphic bi-univalent functions defined by Hadamard product.https://zbmath.org/1449.300402021-01-08T12:24:00+00:00"Zireh, A."https://zbmath.org/authors/?q=ai:zireh.ahmad"Hajiparvaneh, S."https://zbmath.org/authors/?q=ai:hajiparvaneh.saidehSummary: In this paper, we propose to introduce the coefficient estimates for an interesting subclass \(\mathcal{M}_\Sigma^{h, p} (\lambda, \Theta)\) of meromorphic bi-univalent functions on \(\Delta = \{ z \in \mathbb{C}:1 < | z | < \infty\}\). Also, we find estimates on the coefficients \(|b_0|\), \(|b_1|\) and \(|b_2|\) for functions belonging to this class. Some interesting remarks of the results presented here are also discussed. The results presented in this paper would generalize and improve some recent works.Results on meromorphic function that shares some small functions with its derivative.https://zbmath.org/1449.300572021-01-08T12:24:00+00:00"Sahoo, Pulak"https://zbmath.org/authors/?q=ai:sahoo.pulak"Halder, Samar"https://zbmath.org/authors/?q=ai:halder.samarSummary: This paper deals with the uniqueness problem of meromorphic function that shares a set of small functions with its derivative. The results of this paper generalize the recent results in a literature.Coefficient estimates for a general subclass of analytic function and bi-univalent strongly Bazilevič function.https://zbmath.org/1449.300182021-01-08T12:24:00+00:00"Guo, Dong"https://zbmath.org/authors/?q=ai:guo.dong"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huo"Ao, En"https://zbmath.org/authors/?q=ai:ao.en"Li, Zongtao"https://zbmath.org/authors/?q=ai:li.zongtaoSummary: In this paper, we introduce and investigate an interesting subclass \(B_\Sigma^\alpha (\beta)\) of analytic and bi-univalent strongly Bazilevič function on \(U = \{Z: |z| < 1\}\), obtain estimates for the initial coefficients \(|a_2|\) and \(|a_3|\). In addition, we obtain the Fekete-Szegö inequality for this class. Several known or new consequences of the results are also pointed out.On certain subclasses of analytic functions associated with confluent hypergeometric distribution.https://zbmath.org/1449.300252021-01-08T12:24:00+00:00"Porwal, Saurabh"https://zbmath.org/authors/?q=ai:porwal.saurabh"Magesh, N."https://zbmath.org/authors/?q=ai:magesh.nanjundan"Balaji, V. K."https://zbmath.org/authors/?q=ai:balaji.vittalrao-kupparaoo|balaji.vittalrao-kupparao|balaji.vitalrao-kupparaoSummary: In this paper, we find the conditions under which \(\Omega (a; c; m)f \in S (\lambda, \alpha, k)\), \(\Omega (a; c; m)f \in UCV (\lambda, \alpha, k)\), \(\Omega (a; c; m)f \in UCD (k)\) if \(f \in {\mathcal{R}^\tau} (A, B)\). Also consequences of the results are pointed out.Development of \(N\)-multiple power series into \(N\)-dimensional regular \(C\)-fraction.https://zbmath.org/1449.300052021-01-08T12:24:00+00:00"Kuchminska, Kh. Yo."https://zbmath.org/authors/?q=ai:kuchminska.kh-yo"Vozna, S. M."https://zbmath.org/authors/?q=ai:vozna.s-mThe authors propose a possible algorithm for the expansion of a formal \(N\)-multiple power series into a functional \(N\)-dimensional regular \(C\)-fraction corresponding to this series. In this case, the algorithms used provide the expansion into the corresponding and corresponding two-dimensional fractions for the power and double power series.
Reviewer: L. N. Chernetskaja (Kyïv)On growth of meromorphic solutions of some kind of non-homogeneous linear difference equations.https://zbmath.org/1449.300772021-01-08T12:24:00+00:00"Zheng, Xiu-Min"https://zbmath.org/authors/?q=ai:zheng.xiumin"Zhou, Yan-Ping"https://zbmath.org/authors/?q=ai:zhou.yanpingSummary: In this paper, we investigate the growth of meromorphic solutions of some kind of non-homogeneous linear difference equations with special meromorphic coefficients. When there are more than one coefficient having the same maximal order and the same maximal type, the estimates on the lower bound of the order of meromorphic solutions of the involved equations are obtained. Meanwhile, the above estimates are sharpened by combining the relative results of the corresponding homogeneous linear difference equations.On the equation \(a{f^n} + b (f')^m \equiv 1\).https://zbmath.org/1449.300462021-01-08T12:24:00+00:00"Dang, Guoqiang"https://zbmath.org/authors/?q=ai:dang.guoqiang"Chen, Honghui"https://zbmath.org/authors/?q=ai:chen.honghuiSummary: Let \(n, m\) be two positive integers. Some previous researchers proved the existence of meromorphic solutions for the Fermat-type functional equation \({f^n} + (f')^m \equiv 1\) in 2013. This paper extends their results and obtains all general solutions of \(a{f^n} + b (f')^m \equiv 1\).Some new properties of Morgan-Voyce polynomials.https://zbmath.org/1449.260162021-01-08T12:24:00+00:00"Pei, Yanni"https://zbmath.org/authors/?q=ai:pei.yanni"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.6|wang.yi.1|wang.yi.10|wang.yi.2|wang.yi.3|wang.yi.4|wang.yi.9|wang.yi.7|wang.yi.5|wang.yi.8Summary: We show that zeros of Morgan-Voyce polynomials are dense in the closed interval \([-4, 0]\). We show also that coefficients of Morgan-Voyce polynomials are approximately normally distributed and that the coefficient arrays are totally positive matrices.The Milin coefficients of Bazilevic functions.https://zbmath.org/1449.300382021-01-08T12:24:00+00:00"Niu, Xiaomeng"https://zbmath.org/authors/?q=ai:niu.xiaomengSummary: The central problem of univalent functions theory is coefficient problem. Milin coefficient estimation is an important research topic. Estimating the order of Milin coefficients is a difficult problem which is still unresolved. In this paper, we introduce a subclass of Bazilevic functions. By using some elementary methods in the complex analysis, Milin coefficients of \({B_{\alpha, \beta}} (C, D)\) are discussed. The accurate results are obtained. The results obtained generalize the related results.Weak subordination relationship of analytic functions.https://zbmath.org/1449.300442021-01-08T12:24:00+00:00"Li, Shuhai"https://zbmath.org/authors/?q=ai:li.shuhai"Ao, En"https://zbmath.org/authors/?q=ai:ao.en"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huo"Ma, Li'na"https://zbmath.org/authors/?q=ai:ma.linaSummary: This paper presents the weak subordinate concept of analytic function, proves some basic properties of weak subordination, establishes weak subordinate principle, and popularizes the basic theory of classical subordinate relation. Finally, concrete examples are used to give a method to construct and study the class of generalized analytic function by means of ``weak subordination. The results generalize the basic properties of Janowski function and uniformly convex function, and some interesting new results are obtained.Meromorphic functions on \(\mathbb{C}\) and the Gauss map of complete minimal surfaces.https://zbmath.org/1449.300582021-01-08T12:24:00+00:00"Su, Min"https://zbmath.org/authors/?q=ai:su.min"Li, Yuhua"https://zbmath.org/authors/?q=ai:li.yuhuaSummary: Let \(\vec r: D \to {\mathbb{R}^3}\) be a minimal surface \(M\) with isothermal parameter, where \(D\) is a domain in \({\mathbb{R}^2}\), then the Gauss map of minimal surface is a meromorphic function on \(D\). Suppose that \(g (z)\) is an arbitrary meromorphic function on \(D \subset \mathbb{C}\), whether there exists a complete minimal surface \(M\), such that \(g (z)\) is the Gauss map of \(M\), which is still unsolved. In this paper, we prove that for the meromorphic function \(g (z)\) on \(\mathbb{C}\) with the property that either the exponent of convergence of zeros or the exponent of convergence of poles is less than \(\frac{1}{2}\), \(g (z)\) must be the Gauss map of some complete minimal surface.The approximation of Laplace-Stieltjes transforms with slow growth.https://zbmath.org/1449.300812021-01-08T12:24:00+00:00"Xu, Hongyan"https://zbmath.org/authors/?q=ai:xu.hongyan"Liu, Sanyang"https://zbmath.org/authors/?q=ai:liu.sanyangSummary: The main purpose of this paper is to study the growth and approximation on entire functions represented by Laplace-Stieltjes transforms of finite logarithmic order convergent on the whole complex plane, and obtain some results about the logarithmic order, the logarithmic type, the error, and the coefficients of Laplace-Stieltjes transforms which are generalization and improvement of the previous results.Pre-Schwarzian norm for linear operators of uniformly convex functions of order \(\alpha\) and type \(\beta\).https://zbmath.org/1449.300142021-01-08T12:24:00+00:00"Dziok, Jacek"https://zbmath.org/authors/?q=ai:dziok.jacek"Zayed, Hanaa M."https://zbmath.org/authors/?q=ai:zayed.hanaa-mousaSummary: By making use of the pre-Schwarzian norm given by \[||f||=\sup_{z\in U}(1-|z|^2)\left|\frac{f''(z)}{f'(z)}\right|\] we obtain such norm estimates for Hohlov operator of functions belonging to the class of uniformly convex functions of order \(\alpha\) and type \(\beta\). We also employ an entirely new method to generalize and extend the results of Theorems 1, 2 and 3 in \textit{S. D. Bernardi} [Trans. Am. Math. Soc. 135, 429--446 (1969; Zbl 0172.09703)]. Finally, some inequalities concerning the norm of the pre-Schwarzian derivative for Dziok-Srivastava operator are also considered.Commuting Toeplitz operators and Toeplitz operators with unbounded symbols on generalized Segal-Bargmann space.https://zbmath.org/1449.470642021-01-08T12:24:00+00:00"Wang, Xiaofeng"https://zbmath.org/authors/?q=ai:wang.xiaofeng.1"Xia, Jin"https://zbmath.org/authors/?q=ai:xia.jin"Chen, Jianjun"https://zbmath.org/authors/?q=ai:chen.jianjunSummary: We consider two Toeplitz operators \({T_u}\) and \({T_v}\) on the generalized Fock space over the complex plane \(\mathbb{C}\). Let's assume that \(u\) is a radial function and the two operators commute. Under a certain growth condition at infinity of \(u\) and \(v\), we prove that \(v\) must be a radial function as well. Finally, we also construct a \({S_p}\) class of Toeplitz operators on the generalized Fock space with symbols which are essentially unbounded on any point of the complex plane \(\mathbb{C}\).Perturbation method for solving the nonlinear eigenvalue problem arising from fatigue crack growth problem in a damaged medium.https://zbmath.org/1449.741012021-01-08T12:24:00+00:00"Stepanova, L. V."https://zbmath.org/authors/?q=ai:stepanova.larisa-valentinovna"Igonin, S. A."https://zbmath.org/authors/?q=ai:igonin.sergey-aSummary: An analytical solution of the nonlinear eigenvalue problem arising from the fatigue crack growth problem in a damaged medium in coupled formulation is obtained. The perturbation technique for solving the nonlinear eigenvalue problem is used. The method allows to find the analytical formula expressing the eigenvalue as the function of parameters of the damage evolution law. It is shown that the eigenvalues of the nonlinear eigenvalue problem are fully determined by the exponents of the damage evolution law. In the paper the third-order (four-term) asymptotic expansions of the angular functions determining the stress and continuity fields in the neighborhood of the crack tip are given. The asymptotic expansions of the angular functions permit to find the closed-form solution for the problem considered.Multipliers between range spaces of co-analytic Toeplitz operators.https://zbmath.org/1449.300902021-01-08T12:24:00+00:00"Fricain, Emmanuel"https://zbmath.org/authors/?q=ai:fricain.emmanuel"Hartmann, Andreas"https://zbmath.org/authors/?q=ai:hartmann.andreas"Ross, William T."https://zbmath.org/authors/?q=ai:ross.william-t-junSummary: In this paper we discuss the multipliers between range spaces of co-analytic Toeplitz operators.The growth of Hadamard product of random Dirichlet series.https://zbmath.org/1449.300042021-01-08T12:24:00+00:00"Ying, Rui"https://zbmath.org/authors/?q=ai:ying.rui"Xu, Hongyan"https://zbmath.org/authors/?q=ai:xu.hongyanSummary: By using the theory of random Dirichlet series, and combining the properties of Hadamard product, the growth of the Hadamard product of random Dirichlet series is studied. Some results about \(q\)-order, lower \(q\)-order, \(q\)-type, lower \(q\)-type and double lower \(q\)-type between random Dirichlet series and random Dirichlet-Hadamard product series are obtained.Some results on difference Riccati equations and delay differential equations.https://zbmath.org/1449.300492021-01-08T12:24:00+00:00"Wang, Qiong"https://zbmath.org/authors/?q=ai:wang.qiong"Long, Fang"https://zbmath.org/authors/?q=ai:long.fang"Wang, Jun"https://zbmath.org/authors/?q=ai:wang.jun.2Summary: We investigate difference Riccati equations with rational coefficients and delay differential equations with constant coefficients. For difference Riccati equations with some relation among coefficients, we prove that every transcendental meromorphic solution is of order no less than one. We also consider the rational solutions for delay differential equations.On a property of Toeplitz operators on Bergman space with a logarithmic weight.https://zbmath.org/1449.470622021-01-08T12:24:00+00:00"Sadraoui, Houcine"https://zbmath.org/authors/?q=ai:sadraoui.houcine"Halouani, Borhen"https://zbmath.org/authors/?q=ai:halouani.borhenSummary: An operator \(T\) on a Hilbert space is hyponormal if \(T^*T-TT^*\) is positive. In this work, we consider hyponormality of Toeplitz operators on the Bergman space with a logarithmic weight. Under a smoothness assumption, we give a necessary condition when the symbol is of the form \(f+\overline{g}\) with \(f,g\) analytic on the unit disk. We also find a sufficient condition when \(f\) is a monomial and \(g\) a polynomial.Second Hankel determinants for some Ma-Minda subclasses of \(I\)-univalent functions.https://zbmath.org/1449.300222021-01-08T12:24:00+00:00"Laxmi, K. Rajya"https://zbmath.org/authors/?q=ai:laxmi.kalikota-rajya"Sharma, R. Bharavi"https://zbmath.org/authors/?q=ai:sharma.rayaprolu-bharavi"Ganesh, K."https://zbmath.org/authors/?q=ai:ganesh.kSummary: In this paper, we have investigated second Hankel determinants for some subclasses of Ma-Minda bi-univalent functions in the open unit disc \(\Delta\) and these results are generalization of results in \textit{M. Çağlar} et al. [Turk. J. Math. 41, No. 3, 694--706 (2017; Zbl 1424.30034)] and \textit{E. Deniz} et al. [Appl. Math. Comput. 271, 301--307 (2015; Zbl 1410.30007)].Initial bounds for a subclass of analytic and bi-univalent functions defined by Chebyshev polynomials and \(q\)-differential operator.https://zbmath.org/1449.300162021-01-08T12:24:00+00:00"Guo, Dong"https://zbmath.org/authors/?q=ai:guo.dong"Ao, En"https://zbmath.org/authors/?q=ai:ao.en"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huo"Xiong, Liangpeng"https://zbmath.org/authors/?q=ai:xiong.liangpengSummary: In this paper, we investigate the coefficient estimate and Fekete-Szegö inequality of a subclass of analytic and bi-univalent functions defined by Chebyshev polynomials and \(q\)-differential operator. The results presented in this paper improve or generalize the recent works of other authors.Fekete-Szegö properties for quasi-subordination class of complex order defined by Sălăgean operator.https://zbmath.org/1449.300122021-01-08T12:24:00+00:00"Aouf, M. K."https://zbmath.org/authors/?q=ai:aouf.mohamed-kamal"Mostafa, A. O."https://zbmath.org/authors/?q=ai:mostafa.adela-osman"Madian, S. M."https://zbmath.org/authors/?q=ai:madian.samar-mohamedSummary: In this paper, we introduce the class \(\mathbf{M}_{b,\lambda,n}^q (\varphi,\phi)\) of quasi subordinations defined by Sălăgean operator, which generalizes known classes and contains new classes. Also we will obtain the coefficient bounds for functions belonging to the class \(\mathbf{M}_{b,\lambda,n}^q (\varphi,\phi)\). A number of known or new results are shown to follow upon specializing the parameters involved in our main results.Third Hankel determinant for Ma-Minda bi-univalent functions.https://zbmath.org/1449.300342021-01-08T12:24:00+00:00"Zhang, Haiyan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huoSummary: In this paper, we investigate the third Hankel determinant \({H_3} (1)\) for the class \(H_\sigma^\mu (\lambda, \varphi)(\lambda \ge 1, \mu \ge 1)\) of Ma-Minda bi-univalent functions in the open unit disk \(\mathbb{D} = \{ z:| z | < 1\}\) and obtain the upper bound of the above determinant \({H_3} (1)\).Coefficient bounds for a class of bi-univalent functions defined by Chebyshev polynomials.https://zbmath.org/1449.300332021-01-08T12:24:00+00:00"Wu, Ren Qiqige"https://zbmath.org/authors/?q=ai:wu.ren-qiqige"Ma, Li'na"https://zbmath.org/authors/?q=ai:ma.lina"Li, Shuhai"https://zbmath.org/authors/?q=ai:li.shuhaiSummary: In the paper, we define a class of bi-univalent analytic functions by Chebyshev polynomials and obtain the coefficient bounds and Fekete-Szegö inequality of the class.Fekete-Szegö inequality for a subclass of bi-univalent functions associated with Hohlov operator and quasi-subordination.https://zbmath.org/1449.300192021-01-08T12:24:00+00:00"Guo, Dong"https://zbmath.org/authors/?q=ai:guo.dong"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huo"Ao, En"https://zbmath.org/authors/?q=ai:ao.en"Xiong, Liangpeng"https://zbmath.org/authors/?q=ai:xiong.liangpengSummary: In this paper, we introduce a new subclass of bi-univalent functions defined by quasi-subordination and Hohlov operator and obtain the coefficient estimates and Fekete-Szegö inequality for function in this new subclass. The results presented in this paper improve or generalize the recent works of other authors.Several companions of quasi-Grüss type inequalities for complex functions defined on unit circle.https://zbmath.org/1449.300012021-01-08T12:24:00+00:00"Zhu, Jian"https://zbmath.org/authors/?q=ai:zhu.jian"Chen, Lijuan"https://zbmath.org/authors/?q=ai:chen.lijuan"Xue, Qiaoling"https://zbmath.org/authors/?q=ai:xue.qiaolingSummary: Several companions of quasi-Grüss type inequalities for the Riemann-Stieltjes integral of continuous complex valued integrands defined on the complex unit circle \(C\left[ {0, 2\pi} \right]\) are given, while the integrator \(u\) is the case of bounded variation Lipschitzian. Some inequalities are generalized for second derivative.Zeros of one class of complex differential-difference polynomial on transcendental meromorphic functions.https://zbmath.org/1449.300092021-01-08T12:24:00+00:00"Hao, Xiaoling"https://zbmath.org/authors/?q=ai:hao.xiaoling"Lei, Zongwen"https://zbmath.org/authors/?q=ai:lei.zongwen"Ding, Jie"https://zbmath.org/authors/?q=ai:ding.jieSummary: In this paper, using Nevanlinna value distribution theory, we study the zeros of complex differential-difference polynomial on transcendental meromorphic functions, and extend some results on differential-difference polynomials. By analyzing the zeros and poles, we prove that when \(n\) takes a certain value, the complex difference-differential polynomials take infinitely many zeros, which can be regarded as the differential-difference analogues of Hayman conjecture.