On certain subclasses of analytic functions associated with Poisson distribution series

TL;DR

This paper investigates the conditions under which Poisson distribution series belong to specific subclasses of analytic functions with negative coefficients, and explores the properties of an associated integral operator within these classes.

Contribution

It provides necessary and sufficient conditions and inclusion relations for Poisson distribution series to belong to certain analytic function subclasses, and analyzes an integral operator related to these classes.

Findings

Derived conditions for series inclusion in subclasses

Established inclusion relations between classes

Analyzed properties of the integral operator

Abstract

In this paper, we find the necessary and sufficient conditions, inclusion relations for Poisson distribution series $\mathcal{K}(m,z)=z+\sum_{n=2}^\infty \frac{m^{n-1}}{(n-1)!}e^{-m}z^{n}$ belonging to the subclasses $\mathcal{S}(k,\lambda )$ and $\mathcal{C}(k,\lambda )$ of analytic functions with negative coefficients. Further, we consider the integral operator $\mathcal{G}(m,z) = \int_0^z \frac{\mathcal{F}(m,\zeta)}{\zeta} d\zeta$ belonging to the above classes.