## A new class of analytic functions defined by means of a convolution operator involving the Hurwitz-Lerch zeta function.(English)Zbl 1130.30003

The authors introduce and investigate a new class of analytic functions defined by the convolution operator $$J_{s, b}(f)$$ introduced by the second author and A. A. Attiya in [Integral Transforms Spec. Funct. 18, No. 3, 207–216 (2007; Zbl 1112.30007)]. Let $$\mathcal A$$ denote the class of functions $f(z)=z+\sum_{k=2}^\infty a_kz^k$ analytic in $$\mathbb U=\{z\in\mathbb C: | z| <1\}$$. For $$0\leq \alpha<1$$, define $$S^*(\alpha)=\left\{f\in{\mathcal A}: \operatorname{Re}\left( {zf'(z)\over f(z)}\right)>\alpha \right\}$$. Let, for $$z\in\mathbb U$$, $G_{s, b}(z)=(1+b)^s\left(\Phi(z, s, b)-b^{-s}\right),$ where $$\Phi(z, s, a)$$ is the Hurwitz-Lerch zeta-function defined, for $$a\in\mathbb C \setminus\mathbb Z_0^{-}$$ and $$s\in\mathbb C$$ if $$| z| <1$$ and $$\operatorname{Re}(s)>$$1 if $$| z| =1$$, by $\Phi(z, s, a)=\sum_{n=0}^\infty {z^n\over (n+a)^s}.$ Then the operator $$J_{s, b}(f)(z): {\mathcal A} \to{\mathcal A}$$ is given by $J_{s, b}(f)(z)=G_{s, b}(z)*f(z).$ More precisely, the authors introduce and study the class $S_{s, b}^*(\alpha)=\left\{f\in{\mathcal A}: J_{s, b}(f)\in S^*(\alpha)\right\},$ and for the functions of this class they obtain coefficient inequalities, prove distortion inequalities, study extreme points and the Fekete-Szegö inequality. For example, for $$n\in\mathbb N\setminus\{1\}$$, $$\alpha\in[0, 1)$$, $| a_n| \leq {2(1-\alpha)\over n-1}\left| \left({n+b\over 1+b}\right)^s\right| \prod_{j=2}^{n-1}\left( 1+{2(1-\alpha)\over j-1}\right).$

### MSC:

 30C10 Polynomials and rational functions of one complex variable 11M35 Hurwitz and Lerch zeta functions

Zbl 1112.30007
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### References:

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