## 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.

### MSC:

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

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