Fekete-Szeg\"{o} inequality for certain Subclasses of analytic functions related with Crescent-Shaped domain and application of Poison distribution series

Abstract

The purpose of this paper is to define a newclass of analytic, normalizedfunctions in the open unit disk $\Delta=\{ z:z\in \mathbb{C}\quad \text{and}\quad \left\vertz\right\vert <1\}$ subordinating with crescent shaped regions, and to derive certain  coefficient estimates $a_2$ , $a_3$ and Fekete-Szeg\"{o} inequality for $f\in\mathscr{M}_q(\alpha,\beta,\lambda)$. A similar result have been  done for the function $ f^{-1}. $ Further application of our results to certain functions defined by convolution products with a normalizedanalytic function is given,  in particular we obtainFekete-Szeg\"{o} inequalities for certainsubclasses of functions defined through Poisson distribution series.