Applications of Inequalities in the Complex Plane Associated with Confluent Hypergeometric Function

Abstract

The idea of inequality has been extended from the real plane to the complex plane through the notion of subordination introduced by Professors Miller and Mocanu in two papers published in 1978 and 1981. With this notion came a whole new theory called the theory of differential subordination or admissible functions theory. Later, in 2003, a particular form of inequality in the complex plane was also defined by them as dual notion for subordination, the notion of differential superordination and with it, the theory of differential superordination appeared. In this paper, the theory of differential superordination is applied to confluent hypergeometric function. Hypergeometric functions are intensely studied nowadays, the interest on the applications of those functions in complex analysis being renewed by their use in the proof of Bieberbach’s conjecture given by de Branges in 1985. Using the theory of differential superodination, best subordinants of certain differential superordinations involving confluent (Kummer) hypergeometric function are stated in the theorems and relation with previously obtained results are highlighted in corollaries using particular functions and in a sandwich-type theorem. An example is also enclosed in order to show how the theoretical findings can be applied.

Study on new integral operators defined using confluent hypergeometric function

“…The original subordination results presented in this paper are also given as differential inequalities in the complex plane which are interpreted in terms of inclusion relations involving the subsets of C. This was already done for the results obtained in [14], and the technique can be seen in another very recent paper [2], which shows that it is a perspective in trend with an interesting outcome.…”

“…Using Theorem 2.6a from [11, p. 57] known as Marx-Strohhäcker result [8,17], knowing that the function is convex gives the certainty that it is starlike of order 1 2 . In [14] the theory of differential superordination was used to obtain that KHF is a Carathéodory function and differential inequalities associated to the results were interpreted as inclusions for certain subsets of the complex plane. A sandwich-type result was stated providing a link between [13] and [14].…”

“…In [14] the theory of differential superordination was used to obtain that KHF is a Carathéodory function and differential inequalities associated to the results were interpreted as inclusions for certain subsets of the complex plane. A sandwich-type result was stated providing a link between [13] and [14]. KHF has already been used in defining a new operator using fractional integral in [7].…”

Some New Extensions on Fractional Differential and Integral Properties for Mittag-Leffler Confluent Hypergeometric Function

“…The Mittag-Leffler function is a precise function that appears often in the study of fractional integrals and derivatives; see, for example, Ghanim and Al-Janaby [13,14], Ghanim et al [15], Oros [16,17], Haubold et al [18], Paneva-Konovska [19], Mainardi and Gorenflo [20], Mathai and Haubold [21], Srivastava [22,23], and Srivastava et al [24].…”