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

30C45Special classes of univalent and multivalent functions
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