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 $f(0)=f'(0)-1=0$\ in the open unit disk $\Delta=\{ z:z\in \mathbb{C}\quad \text{and}\quad \left\vert z\right\vert <1\}$ subordinating generalized telephone numbers, to derive certain coefficient estimates $a_2,a_3$ and Fekete-Szeg\"{o} inequality for $f\in\mathfrak{G}_{\vartheta}^{\kappa}(\mathcal{X},\varkappa)$. A similar results have been done for the function $ f^{-1} $ and $\log\dfrac{f(z)}{z}.$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.