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Neighborhood properties of certain classes of multivalently analytic functions associated with the convolution structure. (English) Zbl 1258.30006

Let \(T_{n}(p)\) denote the class of functions of the form \[ \displaystyle{f(z)=z^{p}-\sum^{\infty}_{k=n+p} a_{k}z^{k}},\tag{1} \] \(a_{k}\geq 0\), \(p,n \in \mathbb{N} := \{1, 2, 3, \dots\}\), which are analytic and \(p\)-valent in the open unit disk \(\mathbb{U}= \{z \in \mathbb{C}:|z|<1\}\). A function \(f\in T_{n}(p)\) is said to be \(p\)-valently starlike of order \(\alpha\), \(0 \leq \alpha<p\) (\(f \in T^{*}_{n,p}(\alpha)\)), if it satisfies the inequality \[ \displaystyle{\mathrm{Re} \left(\frac{zf^{\prime}(z)}{f(z)} \right)>\alpha} \] for \(z \in \mathbb{U}\). Furthermore, a function \(f \in T_{n}(p)\) is said to be \(p\)-valently convex of order \(\alpha\), \(0\leq \alpha<p\) (\(f\in C_{n,p}(\alpha)\)), if it satisfies the inequality \[ \displaystyle{\mathrm{Re} \left(1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)} \right)>\alpha} \] for \(z \in \mathbb{U}\). Denote by \(f*g\) the Hadamard product (or convolution) of the functions \(f\) and \(g\), that is, if \(f\) is given by (1) and \(g\) is given by \[ \displaystyle{g(z)=z^{p}+ \sum^{\infty}_{k=n+p}b_{k}z^{k}}, \tag{2} \] where \(p,n \in \mathbb{N}\), then \[ \displaystyle{(f*g)(z):=z^{p}- \sum^{\infty}_{k=n+p} a_{k}b_{k}z^{k}=:(g*f)(z)}. \] Let \(S_{n,p}(g; \lambda,\mu, \alpha)\) denote the subclass of \(T_{n}(p)\) consisting of functions \(f\) which satisy the inequality \[ \displaystyle{\mathrm{Re} \left(\frac{z(F_{\lambda, \mu}*g)^{\prime}(z)}{(F_{\lambda, \mu}*g)(z)} \right) > \alpha ,} \] where \(0\leq \mu \leq \lambda \leq 1\), \(0\leq \alpha<p\), \(z \in\mathbb{U}\) and \[ F_{\lambda,\mu}(z)=\lambda \mu z^{2}f^{\prime\prime}(z)+(\lambda-\mu)zf^{\prime}(z)+(1-\lambda+\mu)f(z). \] Let \(R_{n,p}(g;\lambda,\mu,\alpha,m,u)\) denote the subclass of \(T_{n}(p)\) consisting of functions \(f\) which satisfy the following non-homogenous Cauchy-Euler differential equation: \[ \displaystyle{z^{m} \frac{d^{m}w}{dz^{m}}+C^{1}_{m}(u+m-1)z^{m-1} \frac{d^{m-1}w}{dz^{m-1}}+ \dots + C^{m}_{m} w \prod^{m-1}_{j=0}(u+j) =h(z) \prod^{m-1}_{j=0}(u+j+p)}, \] where \(w=f(z)\in T_{n}(p)\), \(h\in S_{n,p}(g;\lambda,\mu,\alpha)\), \(m \in \mathbb{N}^{*}\) and \(u\in(-p, \infty)\). The main object of this paper is to investigate the various properties and characteristics of functions belonging to the above-defined classes \(S_{n,p}(g;\lambda,\mu,\alpha)\) and \(R_{n,p}(g;\lambda,\mu,\alpha,m,u)\). Apart from deriving coefficient bounds and distortion inequalities for each of these function classes, the authors establish several inclusion relationships involving the \((n,\delta)\)-neighborhoods of functions belonging to the general function classes which are introduced above.
Reviewer: Mugur Acu (Sibiu)

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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References:

[1] Altintaş, O., Neighborhoods of certain p-valently analytic functions with negative coefficients, Appl. math. comput., 187, 47-53, (2007) · Zbl 1119.30003
[2] Altintaş, O., On a subclass of certain starlike functions with negative coefficients, Math. japon., 36, 489-495, (1991) · Zbl 0739.30011
[3] Altıntaş, O.; Irmak, H.; Srivastava, H.M., Fractional calculus and certain starlike functions with negative coefficients, Comput. math. appl., 30, 2, 9-15, (1995) · Zbl 0838.30011
[4] Altintaş, O.; Owa, S., Neighborhoods of certain analytic functions with negative coefficients, Internat. J. math. math. sci., 19, 797-800, (1996) · Zbl 0915.30008
[5] Altintaş, O.; Özkan, Ö.; Srivastava, H.M., Neighborhoods of a certain family of multivalent functions with negative coefficients, Comput. math. appl., 47, 1667-1672, (2004) · Zbl 1068.30006
[6] Goodman, A.W., Univalent functions, vols. 1, 2, (1983), Polygonal Publishing House Washington, New Jersey
[7] Orhan, H.; Kamalı, M., Fractional calculus and some properties of certain starlike functions with negative coefficients, Appl. math. comput., 136, 269-279, (2003) · Zbl 1033.30012
[8] Owa, S., On certain classes of p-valent functions with negative coefficients, Simon stevin, 59, 385-402, (1985) · Zbl 0593.30018
[9] Ruscheweyh, S., Neighborhoods of univalent functions, Proc. amer. math. soc., 81, 521-527, (1981) · Zbl 0458.30008
[10] Silverman, H., Univalent functions with negative coefficients, Proc. amer. math. soc., 51, 109-116, (1975) · Zbl 0311.30007
[11] Srivastava, H.M.; Altıntaş, O.; Kirci Serenbay, S., Coefficient bounds for certain subclasses of starlike functions of complex order, Appl. math. lett., 24, 1359-1363, (2011) · Zbl 1216.30021
[12] Srivastava, H.M.; Owa, S.; Chatterjea, S.K., A note on certain classes of starlike functions, Rend. sem. mat. univ. Padova, 77, 115-124, (1987) · Zbl 0596.30018
[13] Yamakawa, R., Certain subclasses of p-valently starlike functions with negative coefficients, (), 393-402 · Zbl 0992.30502
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