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 \(n\in\mathbb N\setminus\{1\}\), \(\alpha\in[0, 1)\), \[ | a_n| \leq {2(1-\alpha)\over n-1}\left| \left({n+b\over 1+b}\right)^s\right| \prod_{j=2}^{n-1}\left( 1+{2(1-\alpha)\over j-1}\right). \]


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


Zbl 1112.30007
Full Text: DOI


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