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Some geometric properties of certain class of Le Roy type functions

Suhas B Mahesh, Karthik V Pai, Abhinav Sharma

TL;DR

The paper addresses when Le Roy type Mittag-Leffler functions and their multivariate generalizations are subordinate to the exponential function, thereby being exponentially starlike and exponentially convex. It develops a framework based on coefficient bounds, digamma inequalities, and subordination lemmas to derive explicit parameter-inequality criteria involving Gamma function terms. The authors first treat univariate Le Roy functions and then extend to a multivariate 3k-parameter family, establishing sufficient conditions for exponential subordination. They then translate these results into concrete criteria for exponential starlikeness and convexity, both in the unit disk and in the half-disk, for both univariate and multivariate cases. Overall, the work advances geometric function theory for Le Roy type functions with practical criteria for their exponential-type geometric properties.

Abstract

The main goal of this paper is to obtain sufficient conditions so that Le Roy type functions and multivariate Le Roy type functions satisfy subordination of exponential function. Moreover conditions on parameters have been derived to claim them being exponential starlike and exponential convex for both of the functions. Starlikeness and convexity have also been studied for multivariate Le Roy type functions.

Some geometric properties of certain class of Le Roy type functions

TL;DR

The paper addresses when Le Roy type Mittag-Leffler functions and their multivariate generalizations are subordinate to the exponential function, thereby being exponentially starlike and exponentially convex. It develops a framework based on coefficient bounds, digamma inequalities, and subordination lemmas to derive explicit parameter-inequality criteria involving Gamma function terms. The authors first treat univariate Le Roy functions and then extend to a multivariate 3k-parameter family, establishing sufficient conditions for exponential subordination. They then translate these results into concrete criteria for exponential starlikeness and convexity, both in the unit disk and in the half-disk, for both univariate and multivariate cases. Overall, the work advances geometric function theory for Le Roy type functions with practical criteria for their exponential-type geometric properties.

Abstract

The main goal of this paper is to obtain sufficient conditions so that Le Roy type functions and multivariate Le Roy type functions satisfy subordination of exponential function. Moreover conditions on parameters have been derived to claim them being exponential starlike and exponential convex for both of the functions. Starlikeness and convexity have also been studied for multivariate Le Roy type functions.

Paper Structure

This paper contains 5 sections, 15 theorems, 124 equations.

Table of Contents

  1. General Introduction and Motivation
  2. Useful definitions and lemmas
  3. exponential subordination
  4. Exponential Convexity and Exponential Starlikeness
  5. Starlikeness and Convexity of Multivariate Le Roy Type Functions

Key Result

Lemma 1

Convexity and Starlikeness Let $f(z) \in \mathcal{A}$ and $|f'(z)-1| < \frac{2}{\sqrt{5}}$$\forall z \in \mathcal{D},$ then f(z) is starlike in $\mathcal{D}.$

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5: Ozaki's Lemma
  • Lemma 6
  • ...and 17 more