Fractional Integral of a Confluent Hypergeometric Function Applied to Defining a New Class of Analytic Functions

Alina Alb Lupaş,Georgia Irina Oros

doi:10.3390/sym14020427

Abstract

The study on fractional integrals of confluent hypergeometric functions provides interesting subordination and superordination results and inspired the idea of using this operator to introduce a new class of analytic functions. Given the class of functions An=f∈HU:fz=z+an+1zn+1+…,z∈U written simply A when n=1, the newly introduced class involves functions f∈A considered in the study due to their special properties. The aim of this paper is to present the outcomes of the study performed on the new class, which include a coefficient inequality, a distortion theorem and extreme points of the class. The starlikeness and convexity properties of this class are also discussed, and partial sums of functions from the class are evaluated in order to obtain class boundary properties.

Highlights

The study presented in this paper is done in the general, well-known space on Geometric Function Theory
Inspired by earlier work done by applying this function on different hypergeometric functions seen in papers, such as [5,6], a new operator was defined combining confluent hypergeometric function with the remarkable fractional integral in [7]
For the study done on this class, we considered recently published results, such as the geometric properties proved by the authors related to classes of p-valent functions [14], q-starlike functions involving Janowski functions [15] or harmonic p-valent functions based on hypergeometric functions [16]

Summary

Introduction

The study presented in this paper is done in the general, well-known space on Geometric Function Theory. Inspired by earlier work done by applying this function on different hypergeometric functions seen in papers, such as [5,6], a new operator was defined combining confluent hypergeometric function with the remarkable fractional integral in [7]. The new operator was introduced with inspiration from the numerous studies conducted on confluent hypergeometric functions from different perspectives, such as using it together with other remarkable functions (for example in [8,9]) or studying its univalence properties as seen in [10,11]. For the study done on this class, we considered recently published results, such as the geometric properties proved by the authors related to classes of p-valent functions [14], q-starlike functions involving Janowski functions [15] or harmonic p-valent functions based on hypergeometric functions [16].

Study Regarding Function Class Coefficients

Starlikeness and Convexity Properties

Conclusions