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Convolution features of univalent meromorphic functions generated by Barnes-Mittag-Leffler function

Tuğba Yavuz, Şahsene Altınkaya

Abstract

The Mittag-Leffler function plays an important role in Geometric Function Theory, particularly in the study of analytic and meromorphic function classes. Among its various generalizations, the Barnes-Mittag-Leffler function stands out due to its intricate structure and applications in diverse mathematical fields. In this paper, our main focus is to investigate the convolution properties of these functions and establish conditions that ensure specific geometric characteristics. Additionally, we explore membership relations for functions in these classes. The results obtained in this work are novel, and their significance is demonstrated through various illustrative consequences and corollaries, emphasizing their potential impact in function theory and its applications.

Convolution features of univalent meromorphic functions generated by Barnes-Mittag-Leffler function

Abstract

The Mittag-Leffler function plays an important role in Geometric Function Theory, particularly in the study of analytic and meromorphic function classes. Among its various generalizations, the Barnes-Mittag-Leffler function stands out due to its intricate structure and applications in diverse mathematical fields. In this paper, our main focus is to investigate the convolution properties of these functions and establish conditions that ensure specific geometric characteristics. Additionally, we explore membership relations for functions in these classes. The results obtained in this work are novel, and their significance is demonstrated through various illustrative consequences and corollaries, emphasizing their potential impact in function theory and its applications.

Paper Structure

This paper contains 6 sections, 7 theorems, 58 equations.

Table of Contents

  1. 1. Introduction
  2. 2. Preliminaries
  3. 3. Convolution Properties
  4. 4. Integral Representation Formula
  5. 5. Conclusion
  6. Acknowledgements

Key Result

Theorem 7

Let $\Theta$ be an analytic function in $\mathbb{B}$ and be defined on $\partial\mathbb{B}=\left\{ z\in \mathbb{C}: \left\vert z\right\vert =1 \right\}$. The function $\mathcal{B}_{K,\vartheta}^a \left( f\right)$ is in $S^{\lambda}_{K, \vartheta} ( \Theta)$ if and only if where

Theorems & Definitions (13)

  • Remark 6
  • Theorem 7
  • proof
  • Theorem 12
  • proof
  • Theorem 14
  • proof
  • Corollary 16
  • Theorem 17
  • proof
  • ...and 3 more