A new class of generalized close-to-starlike functions defined by the Srivastava-Attiya operator
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 \[ J_{s,b}:A\to A \] defined in terms of the Hadamard product (or convolution) by \[ J_{s,b}(f)(z):=G_{s,b}(z)*f(z), \,\,\, z\in U,\, b\in\mathbb C\setminus \mathbb Z^{-}_{0}, s\in \mathbb C, \tag{4} \] where \[ G_{s,b}(z):=(1+b)^{s}[\Phi(z,s,b)-b^{-s}], \,\, z\in U. \tag{5} \] If \(f\) is of the form (1), then it can easily be seen from (3) to (5) that \[ \displaystyle{J_{s,b}(f)(z)=z+\sum^{\infty}_{n=2} \left( \frac{1+b}{n+b} \right)^{s} a_{n}z^{n}}. \] The authors introduce a new class \(CS^{*}_{s,b}\) of functions as follows. Definition. A function \(f\in A\) is said to be in the class \(CS^{*}_{s,b}\) if \(J_{s,b}(f)\) is in the class \(CS^{*},\) that is, if \[ \displaystyle{\int^{\theta_{2}}_{\theta_{1}} \Re \left( \frac{zJ_{s,b}(f)(z)}{J_{s,b}(f)(z)}\right) d\theta>-\pi}, \tag{8} \] where \(z=re^{i\theta}\), \(0<r<1\) and \(0\leq \theta_{1}< \theta_{2} \leq \theta_{1}+2\pi.\) The function class \(CS^{*}_{s,b}\) can be defined in another equivalent form as follows. Definition. A function \(f\) is said to belong to the class \(CS^{*}_{s,b}\) if and only if there exists a function \(g\in S^{*}\) such that \[ \displaystyle{\Re \left( \frac{J_{s,b}(f)(z)}{g(z)} \right)>0, \,\, z \in U}. \tag{9} \] The main object of this paper is to present a basic investigation of the general class \(CS^{*}_{s,b}\) defined above by means of the Srivastava-Attiya operator \(J_{s,b}\). In particular, the authors derive an inclusion result, coefficient inequalities and other interesting properties.
scientific article; zbMATH DE number 6162869
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