Recent zbMATH articles in MSC 30Chttps://zbmath.org/atom/cc/30C2021-03-30T15:24:00+00:00WerkzeugOn the conjecture of Lehmer, limit Mahler measure of trinomials and asymptotic expansions.https://zbmath.org/1455.111432021-03-30T15:24:00+00:00"Verger-Gaugry, Jean-Louis"https://zbmath.org/authors/?q=ai:verger-gaugry.jean-louisSummary: Let \(n\ge 2\) be an integer and denote by \(\theta_n\) the real root in \((0,1)\) of the trinomial \(G_n(X) = -1 + X + X^n\). The sequence of Perron numbers \((\theta_n^{-1})_{n\ge 2}\) tends to \(1\). We prove that the Conjecture of Lehmer is true for \(\{\theta_n^{-1}\mid n\ge 2\}\) by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the roots \(\theta_n\), \(z_{j,n}\), of \(G_n(X)\) lying in \(\vert z\vert < 1\), as a function of \(n\), \(j\) only. This method, not yet applied to Lehmer's problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measures \(M(G_n) = M(\theta_n) = M(\theta_n^{-1})\) of the trinomials \(G_n\) as a function of \(n\) only, without invoking Smyth's Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth's, Boyd's and Flammang's previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for \(\{\theta_n^{-1}\mid n\ge 2\}\), with a minoration of the house \(\overline{\vert\theta_n^{-1}\vert}\), and a minoration of the Mahler measure \(M(G_n)\) better than Dobrowolski's one. The angular regularity of the roots of \(G_n\), near the unit circle, and limit equidistribution of the conjugates, for \(n\) tending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.On Hilbert boundary value problem for Beltrami equation.https://zbmath.org/1455.300112021-03-30T15:24:00+00:00"Gutlyanskii, Vladimir"https://zbmath.org/authors/?q=ai:gutlyanskii.vladimir-ya"Ryazanov, Vladimir"https://zbmath.org/authors/?q=ai:ryazanov.vladimir-i"Yakubov, Eduard"https://zbmath.org/authors/?q=ai:yakubov.eduard-h"Yefimushkin, Artyem"https://zbmath.org/authors/?q=ai:yefimushkin.artyemThe Hilbert boundary value problem looks for an analytic function \(f\) in a domain \(D\) such that \(\lim_{z \rightarrow \zeta}\operatorname{Re}[\overline{\lambda}(\zeta) f(z)] = \varphi(\zeta)\) for all \(\zeta \in \partial D\), for a historic survey see [\textit{N.I. Muskhelisvili}, Singular integral equations. Translated from the second Russian edition.
Groningen: Wolters-Noordhoff Publishing (1967; Zbl 0174.16201)]. A more generalized problem is studied: the analytic function is replaced by a quasiregular function \(f\), i.e., \(f = h \circ g\) where \(h\) is analytic and \(g\) is quasiconformal. Moreover, the boundary regularity conditions are relaxed and it is shown that, if \(D\) is a Jordan domain with the quasihyperbolic boundary condition, \(\partial D\) has a tangent a.e. with respect to the logarithmic capacity, \(\mu: D \rightarrow \mathbb C\) is a Beltrami coefficient, \(\lambda: \partial D \rightarrow \mathbb C, \, |\lambda| = 1\) has countable bounded variation on \(\partial D\) and \(\varphi : \partial D \rightarrow \mathbb C\) is measurable with respect to the logarithmic capacity, then the Hilbert problem has a quasiregular solution. In this case the boundary condition is understood as the limit along nontangential curves except possibly at a set of logarithmic capacity zero. For the quasihyperbolic boundary condition see [\textit{F. W. Gehring} and \textit{O. Martio}, J. Anal. Math. 45, 181--206 (1985; Zbl 0596.30031)]. A quasidisk satisfies this condition. The countable bounded variation of \(\lambda\) on \(\partial D\) means that that there is a countable number of Jordan arcs on \(\partial D\) such that \(\lambda\) has bounded variation on each of them and the rest of points in \(\partial D\) has logarithmic capacity zero. Setting \(\mu = 0\) generalizes the original Hilbert problem to this set-up for analytic functions. The result is also used to study the boundary behavior of the conjugate harmonic function. The methods are also applied to the generalized Neumann and Poincaré boundary value problems. In all these cases the quasihyperbolic boundary condition together with the countable bounded variation of the boundary value function plays an essential role.
Reviewer: Olli Martio (Helsinki)On Newman polynomials without roots on the unit circle.https://zbmath.org/1455.110452021-03-30T15:24:00+00:00"Dubickas, Artūras"https://zbmath.org/authors/?q=ai:dubickas.arturasLet \(d\) be a positive integer. A polynomial \(f(x)= \sum_{j=0}^d a_j x^j\) is called a Newman polynomial if \(a_j \in \{0,1\}\) for each \(j=0,1,\ldots,d.\) In the paper under review the main result is the following.
Let \(a,b,c\) be integers such that \(0 \leq a \leq b \leq c\). Then the Newman polynomial \(1+ \ldots + x^a +x^b +\ldots + x^c\) has a root on the unit circle if and only if \(c= a+b\) or at least one of the inequalities
(1) \(\gcd(a+1, c-b+1)>1,\)
(2) \(\gcd(c+1, c-b+a+2)>1,\)
(3) \(\nu_2 (c-a-b)>\nu_2 (b)\) holds.
Where \(\nu_p (m)\) is the power of \(p\) in the prime factorization of \(m\) for a positive integer \(m\) and a prime number \(p\) and \(\nu_p (0)=0.\)
Reviewer: Salah Najib (Khouribga)Level curves of rational functions and unimodular points on rational curves.https://zbmath.org/1455.110552021-03-30T15:24:00+00:00"Pakovich, Fedor"https://zbmath.org/authors/?q=ai:pakovich.fedor"Shparlinski, Igor E."https://zbmath.org/authors/?q=ai:shparlinski.igor-eIn this paper, the authors improve and generalize the result on common zeros of shifted powers of polynomials presented in the paper: [\textit{N. Ailon} and \textit{Z. Rudnick}, Acta Arith. 113, No. 1, 31--38 (2004; Zbl 1057.11018)]. For this purpose, thr authors reduce this question to a more general question based on the idea of counting intersections of level curves of complex functions. This problem is solved using technical tools of complex analysis and algebraic geometry.
Reviewer: Sonia Pérez Díaz (Madrid)On the coefficient estimates for new subclasses of bi-univalent functions associated with subordination and Fibonacci numbers.https://zbmath.org/1455.300102021-03-30T15:24:00+00:00"Altınkaya, Şahsene"https://zbmath.org/authors/?q=ai:altinkaya.sahseneSummary: The study of operators plays an important role in geometric function theory in complex analysis and its related fields. This chapter considers a few of the classic integral operators defined on analytic functions. It briefly discusses some of the well-known subclasses of the analytic univalent function class \(\mathcal{S}\), including starlike and convex functions. The chapter offers to get the estimates on the Taylor-Maclaurin coefficients and derive the Fekete-Szegő inequalities for functions in the classes \(\mathrm{St}\), \(\Sigma\mu(p\tilde{})\) and \(\mathrm{Kt}\), \(\Sigma\mu(p\tilde{})\). The geometric properties of the function classes \(\mathrm{St}\), \(\Sigma\mu(p\tilde{})\), \(\mathrm{Kt}\), \(\Sigma\mu(p\tilde{})\) vary according the values assigned to the parameters involved. Nevertheless, some results for the special cases of the parameters involved could be presented as illustrative examples.
For the entire collection see [Zbl 1453.00001].Quasiregular curves.https://zbmath.org/1455.300142021-03-30T15:24:00+00:00"Pankka, Pekka"https://zbmath.org/authors/?q=ai:pankka.pekkaThe author extends the notion of a pseudoholomorphic vector of Iwaniec, Verchota, and Vogel to mappings between Riemannian manifolds. Since this class of mappings contains both quasiregular mappings and pseudoholomorphic curves, these mappings are called quasiregular curves. The author proves, for instance, a version of the Liouville theorem to the effect that bounded quasiregular curves \(\mathbb{R}^n \to \mathbb{R}^m\), \(n\le m\), are constant.
Reviewer: Matti Vuorinen (Turku)Bloch functions, asymptotic variance, and geometric zero packing.https://zbmath.org/1455.300122021-03-30T15:24:00+00:00"Hedenmalm, Haakan"https://zbmath.org/authors/?q=ai:hedenmalm.hakanThe main result is a lower estimate of the discrepancy density \(\rho_{\mathbb{H}}\) for ``hyperbolic zero packing'':
\[
3,21\cdot 10^{-5}< \rho_{\mathbb{H}}< 0,121.
\]
Here
\[
\rho_{\mathbb{H}}= \varliminf_{n\to 1-} \left\{\inf_f\frac{1}{\log\frac{1}{1-n^2}} \iint_{|z|<n} [1-(1-|z|^2)|f(z)|]^2 \frac{dxdy}{\pi(1-|z|^2)}\right\}
\]
and the infimum is taken over all polynomials \(f(z)\). This quantity is related to the universal asymptotic variance \(\Sigma^2= 1-\rho_{\mathbb{H}}\). It appears in the context of entire \(k\)-quasiconformal mappings \(\psi:\overline{\mathbb{C}}\to \overline{\mathbb{C}}\) normalized so that
\[
\psi(z)= z+ b_0+ \sum^\infty_{j=1}b_{j} z^{-j},\quad |z|> 1.
\]
(Thus \(\psi\) is analytic outside the unit disk.)
By a theorem of \textit{O. Ivrii} [``Quasicircles of dimension \(1+k^2\) do not exist'', Preprint, \url{arXiv:1511.07240}], the maximal Minkowski dimension \(D(k)\) of a \(k\)-quasicircle has the asymptotic expansion
\[
D(k)= 1+\Sigma^2 k^2+ \mathcal{O}(k^{\frac{8}{3}-o})\quad\text{as }k\to 0+.
\]
A consequence of the present work is that some conjectures are false. The results obtained are likely to have an impact on future research.
Reviewer: Peter Lindqvist (Trondheim)Teichmüller spaces and coefficient problems for univalent holomorphic functions.https://zbmath.org/1455.300092021-03-30T15:24:00+00:00"Krushkal, Samuel L."https://zbmath.org/authors/?q=ai:krushkal.samuel-lThe author proposes the features of Teichmüller spaces to create a powerful tool for solving coefficient problems in geometric function theory. The canonical class \(S\) is formed by univalent functions \(f\), \(f(z)=z+\sum_{n=2}^{\infty}a_nz^n\), on the unit disk \(\mathbb D=\{z\in\mathbb C:|z|<1\}\). The goal of the author is to estimate on \(S\) and other related classes the general nonconstant homogeneous polynomial coefficient functionals \(J(f)=J(a_{m_1},\dots,a_{m_s})\), \(J({\mathbf 0})=0\), where \(2<m_1<\dots<m_s\) and \(\{{\mathbf 0}\}=(0,\dots,0)\). Many distortion functionals on \(S\) are maximized by the so-called Koebe function \(\kappa_0(z)=z(1-z)^{-2}=\sum_{n=1}^{\infty}nz^n\) and rotations \(\kappa_{\theta}(z)=e^{-i\theta}\kappa_0(e^{i\theta}z)\). Denote by \(\hat S(1)\) the completion in the topology of locally uniform convergence on \(\mathbb D\) of the set of univalent functions \(f(z)=\sum_{n=1}^{\infty}a_nz^n\), \(|a_1|=1\), having quasiconformal extensions across \(\mathbb S^1=\partial\mathbb D\) to the sphere \(\hat{\mathbb C}=\mathbb C\cup\{\infty\}\) which satisfy \(f(1)=1\). The following main theorem shows the important role of two sets: the zero set \(\mathcal Z_J=\{f\in\hat S(1):J(f)=0\}\) and the set \(\mathcal K=\{\kappa_{\tau,\theta}(z)=e^{-i\theta}\kappa_0(e^{i\tau}z)\}\).
Theorem 1. Any homogeneous polynomial functional \(J(f)\), whose zero set \(\mathcal Z_J\) is separated from \(\mathcal K\), is maximized on \(\hat S(1)\) only by the functions \(f_0\in\mathcal K\).
Corollary 1. If a functional \(J(f)\) on \(S\) satisfies
\[
\max_{S}|J(f)|=\max_{\hat S(1)}|J(f)|\ \text{ and }\ |J(\kappa_0)|>0,
\]
then every extremal of \(J(f)\) is a rotated Koebe function \(\kappa_{\theta}\) with some \(\theta\in[-\pi,\pi)\). For a homogeneous \(J\), any function \(\kappa_{\theta}\) is extremal for \(J\).
Reviewer: Dmitri V. Prokhorov (Saratov)Numerical methods for biomembranes: conforming subdivision methods versus non-conforming PL methods.https://zbmath.org/1455.490232021-03-30T15:24:00+00:00"Chen, Jingmin"https://zbmath.org/authors/?q=ai:chen.jingmin"Yu, Thomas"https://zbmath.org/authors/?q=ai:yu.thomas-pok-yin"Brogan, Patrick"https://zbmath.org/authors/?q=ai:brogan.patrick"Kusner, Robert"https://zbmath.org/authors/?q=ai:kusner.robert-b"Yang, Yilin"https://zbmath.org/authors/?q=ai:yang.yilin"Zigerelli, Andrew"https://zbmath.org/authors/?q=ai:zigerelli.andrewSummary: The Canham-Helfrich-Evans models of biomembranes consist of a family of geometric constrained variational problems. In this article, we compare two classes of numerical methods for these variational problems based on piecewise linear (PL) and subdivision surfaces (SS). Since SS methods are based on spline approximation and can be viewed as higher order versions of PL methods, one may expect that the only difference between the two methods is in the accuracy order. In this paper, we prove that a numerical method based on minimizing any one of the ``PL Willmore energies'' proposed in the literature would fail to converge to a solution of the continuous problem, whereas a method based on minimization of the bona fide Willmore energy, well-defined for SS but not PL surfaces, succeeds. Motivated by this analysis, we propose also a regularization method for the PL method based on techniques from conformal geometry. We address a number of implementation issues crucial for the efficiency of our solver. A software package called WMINCON accompanies this article, and provides parallel implementations of all the relevant geometric functionals. When combined with a standard constrained optimization solver, the geometric variational problems can then be solved numerically. To this end, we realize that some of the available optimization algorithms/solvers are capable of preserving symmetry, while others manage to break symmetry; we explore the consequences of this observation.Harmonic measure: algorithms and applications.https://zbmath.org/1455.300152021-03-30T15:24:00+00:00"Bishop, Christopher J."https://zbmath.org/authors/?q=ai:bishop.christopher-jThis survey article discusses several results in the field of planar harmonic measure, starting from \textit{N. G. Makarov}'s results [Proc. Lond. Math. Soc. (3) 51, 369--384 (1985; Zbl 0573.30029)] to recent applications involving 4-manifolds, dessins d'enfants and transcendental dynamics. Various areas from analysis, topology and algebra that are influenced by harmonic measure are illustrated.
For the entire collection see [Zbl 1437.00045].
Reviewer: Marius Ghergu (Dublin)Foundations of function theory. An introduction to complex analysis and its applications. 2nd revised and expanded edition.https://zbmath.org/1455.300012021-03-30T15:24:00+00:00"Fritzsche, Klaus"https://zbmath.org/authors/?q=ai:fritzsche.klausThis is the second edition of the book [Zbl 1221.30001], a textbook on complex analysis or theory of functions of one complex variable.
There are many textbooks devoted to this subject in many languages, e.g., [\textit{A. I. Markushevich}, Theory of functions of a complex variable. Three vols. 2nd rev. ed. New York, N.Y.: Chelsea Publishing Company (1977; Zbl 0357.30002); \textit{J. B. Conway}, Functions of one complex variable. New York-Heidelberg-Berlin; Springer- Verlag (1973; Zbl 0277.30001); \textit{S. Stoilow},
Theorie der Funktionen einer komplexen Veränderlichen. Bd. 1: Grundbegriffe und Hauptsätze. Bd. 2: Harmonische Funktionen, Riemannsche Flächen. In Zusammenarbeit mit Cabiria Andreian Cazacu. (Romanian) Bucuresti: Editura Academiei Republicii Populare Romine, 306 S. (1954); 378 S. (1958; Zbl 0102.29102)].
Although the expositions of the classical content of the course are similar to some other books, this book is complemented by new topics.
The handbook consists of six chapters, the last one contains solutions to exercises to the previous five chapters. The standard course of complex analysis is covered by the first three chapters (Holomorphic Functions, Complex Integration, and Isolated Singularities), and by apart from the fifth chapter (Geometric Function Theory), containing material on automorphisms of domains and analytic continuation.
The fourth chapter, Meromorphic Functions, can be used for advanced courses. It contains a theory of infinite products, foundations of elliptic functions. In particular, basic information on the Riemann \(\zeta\)-function and the Riemann Hypothesis is provided.
Special attention should be paid to Chapter 5, where the theory of normal functions, actively used in modern research, is presented. In particular, the Marty criterion of normality in terms of the spherical derivative is proved. Section 5.3 contains a proof of the Riemann mapping theorem, Section 5.5 includes Cathéodory's theorem.
The exposition of the material is modern. The author actively uses topological concepts such as topological space and homotopy. The distinctive feature of the book are applications (Anwendungen) finishing Chapters 1--5. One can find there various applications of function theory to harmonic functions, the Dirichlet problem, the Fourier and Laplace transforms, non-Euclidean geometry, elliptic integrals and functions, Mandelbrot and Julia sets, electrotechnics, etc.
Every subsection finishes with examples and exercises with solutions at the end of the book. There are many figures which make the reading easier.
Reviewer: Igor Chyzhykov (Lviv)Asymptotics for the capacity of a condenser with variable potential levels.https://zbmath.org/1455.300162021-03-30T15:24:00+00:00"Dubinin, V. N."https://zbmath.org/authors/?q=ai:dubinin.vladimir-nAuthor's abstract: The asymptotic formula is obtained for the capacity of a generalized condenser when parts of its plates contract to prescribed points. In contrast to the previous studies of capacity, we consider condensers with variable potential levels and establish the third term of the asymptotics.
Reviewer: Alexander Ulanovskii (Stavanger)Harmonic mappings with analytic part convex in one direction.https://zbmath.org/1455.300072021-03-30T15:24:00+00:00"Prajapat, Jugal K."https://zbmath.org/authors/?q=ai:prajapat.jugal-kishore"Manivannan, M."https://zbmath.org/authors/?q=ai:manivannan.m"Maharana, Sudhananda"https://zbmath.org/authors/?q=ai:maharana.sudhanandaSummary: In this paper, we study a family of sense-preserving harmonic mappings whose analytic part is convex in one direction. We first establish the bounds on the pre-Schwarzian norm. Next, we obtain radius of fully starlike and radius of fully convex for this family of harmonic mappings.Division algebras of slice functions.https://zbmath.org/1455.300402021-03-30T15:24:00+00:00"Ghiloni, Riccardo"https://zbmath.org/authors/?q=ai:ghiloni.riccardo"Perotti, Alessandro"https://zbmath.org/authors/?q=ai:perotti.alessandro"Stoppato, Caterina"https://zbmath.org/authors/?q=ai:stoppato.caterinaThanks to a result by Zorn, the only finite-dimensional alternative division algebras are the reals \(\mathbb{R}\), complex numbers \(\mathbb{C}\), quaternions \(\mathbb{H}\) and octonions \(\mathbb{O}\).
The present paper aims to give a unified approach to the theory of slice regular functions over
\(\mathbb{C},\mathbb{H}\) and \(\mathbb{O}\). As stated by the authors, some of the results were already
obtained in other works and with other methods, while another set of results is completely new.
In particular, after the introduction, the paper has 5 sections:
-- Section 2 contains some background material on division algebras and on slice functions;
-- Section 3 deals with an analysis of the zeroes of slice functions;
-- Section 4 is about reciprocals: this section contains some genuine new result such as a representation formula for the reciprocal and
some topological phenomena (several examples are given);
-- In Section 5 the authors start to deal with global properties of slice regular functions and prove a maximum module principle together with
a minimum one and an open mapping theorem;
-- The last section studies singularities of slice regular functions and the authors prove a Casorati-Weierstrass type theorem and that the
set of slice semi-regular functions (the analogue of meromorphic functions) form a division algebra.
All proofs are given in the case of octonions but they stay valid for \(\mathbb{C}\) or \(\mathbb{H}\).
Reviewer: Amedeo Altavilla (Ancona)Inequalities for the derivative of polynomials with restricted zeros.https://zbmath.org/1455.260152021-03-30T15:24:00+00:00"Rather, N. A."https://zbmath.org/authors/?q=ai:rather.nisar-ahemad|rather.nisar-ahmed|rather.nisar-ahmad"Dar, Ishfaq"https://zbmath.org/authors/?q=ai:dar.ishfaq"Iqbal, A."https://zbmath.org/authors/?q=ai:iqbal.atif|iqbal.adam-s|iqbal.adnan|iqbal.amer|iqbal.ayesha|iqbal.anam|iqbal.azhar.1|iqbal.ather|iqbal.asif|iqbal.akhlad|iqbal.azhar|iqbal.arshad|iqbal.afshanSummary: For a polynomial \(\mathit{P(z)=\sum_{\nu =0}^na_{\nu}z^{\nu}}\) of degree \(\mathit{n}\) having all its zeros in \(\mathit{|z|\leq k,k \geq 1} \), it was shown by \textit{N. A. Rather} and \textit{I. Dar} [``Some applications of the boundary Schwarz lemma for polynomials with restricted zeros'', Appl. Math. E-Notes 20, 422--431 (2020)] that
\[
\max_{|z|=1} |P^{\prime}(z)|\geq \frac{1}{1+k^n}\bigg(n+\frac{k^n|a_n|-|a_0|}{k^n|a_n|+|a_0|}\bigg)\max_{|z|=1}|P(z)|.
\]
In this paper, we shall obtain some sharp estimates, which not only refine the above inequality but also generalize some well known Turán-type inequalities.Taylor coefficients of the conformal map for the exterior of the reciprocal of the multibrot set.https://zbmath.org/1455.370432021-03-30T15:24:00+00:00"Shimauchi, Hirokazu"https://zbmath.org/authors/?q=ai:shimauchi.hirokazuSummary: In this paper we investigate normalized conformal mappings of the exterior of the reciprocal of the Multibrot set and analyze the growth of the denominator of the coefficients. Our inequality improves Ewing and Schober's result which was presented in [\textit{J. H. Ewing} and \textit{G. Schober}, J. Math. Anal. Appl. 170, No. 1, 104--114 (1992; Zbl 0766.30012)]. We use the coefficient formula of the author [in: Topics in finite or infinite dimensional complex analysis. Proceedings of the 19th international conference on finite or infinite dimensional complex analysis and applications (ICFIDCAA), Hiroshima, Japan, December 11--15, 2011. Sendai: Tohoku University Press. 237--248 (2013; Zbl 1333.37055)]. The straightforward adaptation of the proof in this paper slightly improves the main theorem of the author [Osaka J. Math. 52, No. 3, 737--746 (2015; Zbl 1352.37142)].On the functional properties of weak \((p, q)\)-quasiconformal homeomorphisms.https://zbmath.org/1455.300132021-03-30T15:24:00+00:00"Gol'dshtein, Vladimir"https://zbmath.org/authors/?q=ai:goldshtein.vladimir"Ukhlov, Alexander"https://zbmath.org/authors/?q=ai:ukhlov.alexanderThe class of weak \((p,q)\)-quasiconformal homeomorphisms generalizes the notions of conformal dilations and quasiconformal maps. A homeomorphism \(\varphi:\Omega \to \tilde{\Omega}\) is a weak \((p,q)\)-quasiconformal homeomorphism if
\[
\left\Vert
\inf \left\{ k(x) \, : \, |D\varphi(x)| \leq k(x) |J(x,\varphi)|^{1/p} \right\}
\right\Vert_{L^{\frac{pq}{p-q}}(\Omega)} < \infty.
\]
The paper under review provides some function theoretic results for this class of homeomorphisms. In particular, the authors prove a self-improvement result (Theorem 2.5) for the boundedness of the composition operator \(\varphi^*\) defined as \(\varphi^*(f) = f \circ \varphi\) along with several Liouville-type theorems discussing the finiteness of the measure of the domain or target for varying values of \(p\) and \(q\).
The paper concludes with a discussion of the integrability of the derivative of a weak \((p,q)\)-quasiconformal homeomorphism given the boundedness of its composition operator.
Reviewer: Scott Zimmerman (Marion)New results for some generalizations of starlike and convex functions.https://zbmath.org/1455.300052021-03-30T15:24:00+00:00"Ebadian, Ali"https://zbmath.org/authors/?q=ai:ebadian.ali"Hameed Mohammed, Nafya"https://zbmath.org/authors/?q=ai:hameed-mohammed.nafya"Analouei Adegani, Ebrahim"https://zbmath.org/authors/?q=ai:adegani.ebrahim-analouei"Bulboacă, Teodor"https://zbmath.org/authors/?q=ai:bulboaca.teodorSome new coefficients results for functions \(f\), which are analytic in \(|z|<1\) such that \(zf'(z)/f(z)\) or \(1+zf''(z)/f'(z)\) is subordinated to a function \(\phi(z)\), are established. Here \(\phi(z)=(1+sz)^2\) or \(\phi(z)=1+z+z^3/3\) is considered (for some \(s\in(0,\sqrt{2}/2)\)).
Several geometric properties of the functions \(\phi(z)\) are proved.
Reviewer: Janusz Sokol (Rzeszow)On some problems of strongly Ozaki close-to-convex functions.https://zbmath.org/1455.300062021-03-30T15:24:00+00:00"Maleki, Zahra"https://zbmath.org/authors/?q=ai:maleki.zahra"Shams, Saeid"https://zbmath.org/authors/?q=ai:shams.saeid"Ebadian, Ali"https://zbmath.org/authors/?q=ai:ebadian.ali"Analouei Adegani, Ebrahim"https://zbmath.org/authors/?q=ai:adegani.ebrahim-analoueiThe authors estimate coefficient functionals within the class of strongly Ozaki close-to-convex functions. Let \(\mathcal A\) be the class of functions \(f(z)=z+\sum_{n=2}^{\infty}a_nz^n\) analytic in the unit disk \(\mathbb U=\{z\in\mathbb C:|z|<1\}\). Given \(\gamma\in(0,1]\) and \(\nu\in[1/2,1]\), a function \(f\in\mathcal A\) is called strongly Ozaki close-to-convex if \[\left|\arg\left(\frac{2\nu-1}{2\nu+1}+\frac{2}{2\nu+1}\left(1+\frac{zf''(z)}{f'(z)}\right)\right)\right|\leq\frac{\gamma\pi}{2},\;\;\; z\in\mathbb U.\] The class of such functions is denoted by \(\mathcal F_0(\nu,\gamma)\). For \(f\in\mathcal F_0(\nu,\gamma)\) and \(\log f(z)/z=2\sum_{n=1}^{\infty}\gamma_nz^n\), the authors give estimates for \(|\gamma_1|\), \(|\gamma_2|\), \(|\gamma_3|\), \(|a_n|\), \(n\geq2\), and \(|a_3-\mu a_2^2|\), \(\mu\in\mathbb C\).
Reviewer: Dmitri V. Prokhorov (Saratov)A new subclass of meromorphic functions with positive coefficients defined by Bessel function.https://zbmath.org/1455.300082021-03-30T15:24:00+00:00"Venkateswarlu, B."https://zbmath.org/authors/?q=ai:venkateswarlu.bollineni"Reddy, P. Thirupathi"https://zbmath.org/authors/?q=ai:reddy.pinninti-thirupathi"Meng, Chao"https://zbmath.org/authors/?q=ai:meng.chao.1|meng.chao"Shilpa, R. Madhuri"https://zbmath.org/authors/?q=ai:shilpa.r-madhuriFor a normalized meromorphic function \(f(z)=z^{-1}+\sum_{n=1}^\infty a_n z^n \) defined on the punctured unit disk, the Bessel operator \(S_{\tau}^{c}\) is defined by
\( S_{\tau}^{c} f(z)= z^{-1} +\sum_{n=0}^{\infty} \left( (-c/4)^{n+1} /((n+1) !(\tau)_{n+1})\right) a_{n}z^{n}.
\) For \(0 \leq \eta<1\), \(k \geq 0\), \(0 \leq \lambda<\frac{1}{2}\), the authors study the subclass consisting of the functions \(f\) satisfying
\[
-\operatorname{Re}\left(\frac{z\left(S_{\tau}^{c} f(z)\right)^{\prime}}{S_{\tau}^{c} f(z)}+\lambda z^{2} \frac{\left(S_{\tau}^{c} f(z)\right)^{\prime \prime}}{S_{\tau}^{c} f(z)}+\eta\right)
>k\left|\frac{z\left(S_{\tau}^{c} f(z)\right)^{\prime}}{S_{\tau}^{c} f(z)}+\lambda z^{2} \frac{\left(S_{\tau}^{c} f(z)\right)^{\prime \prime}}{S_{\tau}^{c} f(z)}+1\right|.
\] They obtain a coefficient condition for a function to belong to this class and prove several standard results. The assumption \(a_n>0\) is needed in the whole paper but it is not mentioned (except in the title of the paper).
Reviewer: V. Ravichandran (Tiruchirappalli)