Geometric properties of some generalized Mathieu power series inside the unit disk

TL;DR

This paper investigates the geometric properties of generalized Mathieu power series within the unit disk, providing conditions for their univalence, starlikeness, and close-to-convexity based on classical criteria.

Contribution

It introduces two parametric families of Mathieu-type power series and establishes sufficient conditions for their geometric properties inside the unit disk.

Findings

Functions are close-to-convex or starlike under certain parameter conditions

Provides criteria for univalence of generalized Mathieu power series

Extends classical results to new families of special functions

Abstract

We consider two parametric families of special functions: One is defined by a power series generalizing the classical Mathieu series, and the other one is a generalized Mathieu type power series involving factorials in its coefficients. Using criteria due to Fejer and Ozaki, we provide sufficient conditions for these functions to be close-to-convex or starlike inside the unit disk, and thus univalent.