A new class of generalized close-to-starlike functions defined by the Srivastava-Attiya operator
type: Item
instance of: scholarly article
title: A new class of generalized close-to-starlike functions defined by the Srivastava-Attiya operator
zbMATH Open document ID: 1293.30038
publication date: 2013-05-14T00:00:00Z
review text: Let \(A\) denote the class of functions of the form \[ \displaystyle{f(z)=z+\sum^{\infty}_{n=2}a_{n}z^{n}}, \] which are analytic in the open unit disk \[ \displaystyle{U=\{z:z \in\mathbb C \,\, \text{and} \,\, |z|<1\}}. \tag{1} \] Let \(S^{*}(\alpha)\) and \(K\) denote the subclasses of \(A\) consisting of functions which are starlike of order \(\alpha\) in \(U\) and convex in \(U\), respectively. We denote by \(P\) the class of analytic functions normalized by \(p(0)=1\) and having positive real part in \(U\). A function \(f\) in the class \(A\) is in the class \(CS^{*}\) of close-to-starlike functions in \(U\) if it satisfies the following inequality (see [\textit{M. O. Reade}, Mich. Math. J. 3, 59--62 (1956; Zbl 0070.07302)], see also [\textit{S. Owa} et al., Appl. Math. Lett. 15, No. 1, 63--69 (2002; Zbl 1038.30011)]): \[ \displaystyle{\int^{\theta_{2}}_{\theta_{1}} \Re e \left( \frac{zf^{\prime}(z)}{f(z)}\right)d \theta>-\pi}, \tag{2} \] where \(z=re^{i\theta}\), \(0<r<1\) and \(0\leq \theta_{1}<\theta_{2}\leq \theta_{1}+2\pi\). The general Hurwitz-Lerch Zeta function \(\Phi(z,s,a)\) defined by [\textit{H. M. Srivastava} and \textit{Juensang Choi}, Series associated with the zeta and related functions. Dordrecht: Kluwer Academic Publishers (2001; Zbl 1014.33001), p. 121; \textit{H. M. Srivastava} and \textit{Junesang Choi}, Zeta and q-zeta functions and associated series and integrals. Amsterdam: Elsevier (2012; Zbl 1239.33002), p. 194] \[ \displaystyle{\Phi(z,s,a):=\sum^{\infty}_{n=0}\frac{z^{n}}{(n+a)^{s}}}, \tag{3} \] where \(a \in\mathbb C\setminus Z^{-}_{0}\), \(s\in \mathbb C \) when \(|z|<1\) and \(\Re(s)>1\) when \(|z|=1\), and (as usual) \[ \mathbb Z^{-}_{0}:=\mathbb Z\setminus\mathbb N, \,\,\,\mathbb Z:=\{0, \pm 1, \pm 2,...\} \,\, \text{and} \,\, \mathbb N:=\{1,2,3,...\}. \] \textit{H. M. Srivastava} and \textit{A. A. Attiya} [Integral Transforms Spec. Funct. 18, No. 3, 207--216 (2007; Zbl 1112.30007)] introduced the linear operator \[ J_{s,b}:A\to …
reviewed by: Mugur Acu
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 11M35
zbMATH DE Number: 6162869
zbMATH Keywords: starlike functions
zbMATH Keywords: convex functions
zbMATH Keywords: close-to-starlike functions
author: H. M. Srivastava
author: Dorina Raducanu
author: Grigore Ştefan Sălăgean
MaRDI profile type: Publication
full work available at URL: s10114-013-2462-z
OpenAlex ID: W2051125952
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DOI: 10.1007/S10114-013-2462-Z
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6162869