Subordination results for spiral-like functions associated with the Srivastava-Attiya operator. (English) Zbl 1244.30015

Murugusundaramoorthy, G.

Integral Transforms Spec. Funct. 23, No. 2, 97-103 (2012).

Let \(U\) be the complex unit disc and \(\mathcal A\) the class of functions \(f\) with \(f(z)=z+a_2z^2+\cdots\). For such functions, the authors define the integral operator \[ \mathcal J_{\mu,b}^{m,k}f(z)=z+\sum_{n=2}^{\infty}C_{n}^{m}(b,\mu) a_{n}z^{n}, \] where \(C_{n}^{m}(b,\mu)=\!\left|\left(\frac{1+b}{n+b}\right)^{\mu}\frac{m!(n+k-2)!}{(k-2)!(n+m-1)!}\right|\), \(b\in\mathbb C\setminus\{\mathbb Z_{0}^{-}\}\), \(\mu\in\mathbb C\), \(k\geq 2\) and \(m>-1\). This is a generalization of other integral operators studied by various authors. Further, for \(0\leq \lambda\), \(\gamma<1\) and \(-\pi/2<\eta<\pi/2\), they define the subclass of \(\mathcal A\), denoted by \(\mathcal G_{\mu,b}^{m,k}(\eta,\gamma,\lambda)\), which consists of functions \(f\) satisfying \[ \mathrm{Re} \left(e^{\mathrm{i}\eta}\frac{z(\mathcal J_{\mu,b}^{m,k}f(z))'}{(1-\lambda)\mathcal J_{\mu,b}^{m,k}f(z)+\lambda(\mathcal J_{\mu,b}^{m,k}f(z))'}\right)>\gamma\cos \eta \] for \(z\in U\). Some subclasses are also given and the main result provides a sufficient condition for a function \(f\in\mathcal A\) to be also in \(\mathcal G_{\mu,b}^{m,k}(\eta,\gamma,\lambda)\). Other properties of the class \(\mathcal G_{\mu,b}^{m,k}(\eta,\gamma,\lambda)\) are given in the second theorem and a corollary following from it.

Reviewer: Eugen Drăghici (Sibiu)

MSC:

30C45

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

Keywords:

univalent functions; starlike functions; spirallike functions; subordinating factor sequence; Hadamard product; generalized Srivastava-Attiya operator; Hurwitz-Lerch zeta function

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