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Subordination Involving Gauss Hypergeometric Function

Anish Kumar, Sourav Das

TL;DR

The paper addresses geometric properties of the normalized Gauss hypergeometric function in the unit disk, aiming to establish sufficient parameter conditions for exponential starlikeness, exponential convexity, and Janowski starlikeness/convexity. It uses differential subordination together with key lemmas on exponential image inclusion and admissible subordination to derive concrete parameter regimes and to relate the Gauss hypergeometric function to exponential classes $\mathcal{S}_{e}$ and $\mathcal{K}_{e}$. Main contributions include H1-type conditions ensuring $F(u,v;w;z) \in \mathcal{P}_{e}$, H2-type conditions ensuring the normalized $\mathbb{F}(u,v;w;z) \in \mathcal{K}_{e}$, and duality arguments yielding exponential starlikeness for $zF(u,v;w;z)$, as well as Janowski starlikeness/convexity results via a constructive differential-subordination framework. These results extend geometric function theory for Gauss hypergeometric functions and suggest applications in fractional and quantum calculus contexts, while highlighting open problems like lemniscate starlikeness under subordination.

Abstract

The primary objective of this work is to obtain some sufficient conditions so that normalized Gauss hypergeometric function satisfies exponential starlikeness and convexity in the unit disk. Moreover, conditions on parameter of this function has been derived for being Janowski convexity and starlikeness with the help of differential subordination. Results established in this work are presumably new and their significance is illustrated by several consequences.

Subordination Involving Gauss Hypergeometric Function

TL;DR

The paper addresses geometric properties of the normalized Gauss hypergeometric function in the unit disk, aiming to establish sufficient parameter conditions for exponential starlikeness, exponential convexity, and Janowski starlikeness/convexity. It uses differential subordination together with key lemmas on exponential image inclusion and admissible subordination to derive concrete parameter regimes and to relate the Gauss hypergeometric function to exponential classes and . Main contributions include H1-type conditions ensuring , H2-type conditions ensuring the normalized , and duality arguments yielding exponential starlikeness for , as well as Janowski starlikeness/convexity results via a constructive differential-subordination framework. These results extend geometric function theory for Gauss hypergeometric functions and suggest applications in fractional and quantum calculus contexts, while highlighting open problems like lemniscate starlikeness under subordination.

Abstract

The primary objective of this work is to obtain some sufficient conditions so that normalized Gauss hypergeometric function satisfies exponential starlikeness and convexity in the unit disk. Moreover, conditions on parameter of this function has been derived for being Janowski convexity and starlikeness with the help of differential subordination. Results established in this work are presumably new and their significance is illustrated by several consequences.
Paper Structure (5 sections, 7 theorems, 57 equations)

This paper contains 5 sections, 7 theorems, 57 equations.

Table of Contents

  1. Introduction and Motivation
  2. Useful Lemmas
  3. Exponential Starlikeness and Convexity of Gauss Hypergeometic fucntion
  4. Janowski starlikeness and convexity of Gauss hypergeometric function
  5. conclusion

Key Result

Lemma 1

Medirata Let $S_{1}$ be the image set of the function $f(z)$ and $S_{2}$ be the image set of the function $e^z$ then $S_{1} \subset S_{2}$ if and $f(0)=1$, where e is the Euler's number.

Theorems & Definitions (10)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • Theorem 3
  • proof
  • Corollary 2