Gamma-Bazilevic functions related with generalized telephone numbers

Abstract

The purpose of this paper is to consider coefficient estimates in a class of functions ${\mathfrak{G}}_{\vartheta }^{\kappa }(\mathcal{X},\varkappa )$ consisting of analytic functions $f$ normalized by $f(0)=f^{\prime }(0)-1=0$\ in the open unit disk $\Delta =\{z:z\in \mathbb{C}\text{and}\left|z\right|<1\}$ subordinating generalized telephone numbers, to derive certain coefficient estimates $a_2,a_3$ and Fekete-Szegö inequality for $f\in {\mathfrak{G}}_{\vartheta }^{\kappa }(\mathcal{X},\varkappa )$. A similar results have been done for the function $f^{-1}$ and $\log \frac{f(z)}{z}.$Similarly application of our results to certain functions defined by using convolution products with a normalized analytic function is given, and in particular we state Fekete-Szeg"{o} inequalities for subclasses described through Poisson Borel and Pascal distribution series.

Analytic functions, starlike functions, convex functions, subordination, Fekete-Szegö inequality, Poisson distribution series, Borel distribution series Hadamard product