scholarly journals Carathéodory properties of Gaussian hypergeometric function associated with differential inequalities in the complex plane

2021 ◽  
Vol 6 (12) ◽  
pp. 13143-13156
Author(s):  
Georgia Irina Oros ◽  
Keyword(s):  
Hypergeometric Function ◽  
Complex Plane ◽  
Differential Inequalities ◽  
Gaussian Hypergeometric Function ◽  
Inclusion Relations ◽  
Carathéodory Function ◽  
Differential Superordination

<abstract><p>The results presented in this paper highlight the property of the Gaussian hypergeometric function to be a Carathéodory function and refer to certain differential inequalities interpreted in form of inclusion relations for subsets of the complex plane using the means of the theory of differential superordination and the method of subordination chains also known as Löwner chains.</p></abstract>

DEVELOPMENT OF A GAUSSIAN HYPERGEOMETRIC FUNCTION CODE IN COMPLEX DOMAINS

1993 ◽  
Vol 04 (04) ◽  
pp. 805-840 ◽  
Author(s):  
YUPAI P. HSU
Keyword(s):  
Hypergeometric Function ◽  
Complex Plane ◽  
Gaussian Hypergeometric Function ◽  
Trigonometric Identity ◽  
Arbitrary Complex ◽  
Function Code

The analytic properties of the Gaussian hypergeometric function is reviewed and applied to the development of a Fortran function code. The code developed can be used to evaluate the hypergeometric function on the whole complex plane with arbitrary complex parametric values. In the process of numerically verifying the code, a trigonometric identity involving the hypergeometric function listed in most of the mathematical handbooks is found to be incorrectly stated.

scholarly journals Univalence Conditions for Gaussian Hypergeometric Function Involving Differential Inequalities

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 904
Author(s):  
Georgia Irina Oros
Keyword(s):  
Hypergeometric Function ◽  
Analytic Functions ◽  
Special Function ◽  
Differential Inequalities ◽  
Gaussian Hypergeometric Function ◽  
Geometrical Properties

In their paper published in 1990, Miller and Mocanu have investigated the special function Gaussian hypergeometric function in view of its relation to the theory of analytic functions, stating conditions for this function to be univalent using a, b, c ∈ ℝ, c ≠ 0, −1, −2, ... The study done in this paper extends the results on the univalence of the considered function taking a, b, c ∈ ℂ; with c ≠ 0, −1, −2, ... two criteria being stated in the corollaries of the proved theorems. An interpretation of the univalence results from the sets inclusion view is also given, underlining the geometrical properties of the outcomes. Examples showing how the univalence results can be applied are also included.

scholarly journals Study on new integral operators defined using confluent hypergeometric function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Georgia Irina Oros
Keyword(s):  
Hypergeometric Function ◽  
Complex Plane ◽  
Convex Functions ◽  
Univalent Functions ◽  
Integral Operators ◽  
Confluent Hypergeometric Function ◽  
Differential Inequalities ◽  
Starlike And Convex Functions ◽  
Inclusion Properties ◽  
Admissible Functions

AbstractTwo new integral operators are defined in this paper using the classical Bernardi and Libera integral operators and the confluent (or Kummer) hypergeometric function. It is proved that the new operators preserve certain classes of univalent functions, such as classes of starlike and convex functions, and that they extend starlikeness of order $\frac{1}{2}$ 1 2 and convexity of order $\frac{1}{2}$ 1 2 to starlikeness and convexity, respectively. For obtaining the original results, the method of admissible functions is used, and the results are also written as differential inequalities and interpreted using inclusion properties for certain subsets of the complex plane. The example provided shows an application of the original results.

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

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 259
Author(s):  
Georgia Irina Oros
Keyword(s):  
Hypergeometric Function ◽  
Complex Plane ◽  
Complex Analysis ◽  
Type Theorem ◽  
Confluent Hypergeometric Function ◽  
Differential Subordination ◽  
Real Plane ◽  
Sandwich Type ◽  
Differential Superordination ◽  
Dual Notion

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.

scholarly journals MATHIEU-TYPE SERIES BUILT BY (p, q)-EXTENDED GAUSSIAN HYPERGEOMETRIC FUNCTION

2017 ◽  
Vol 54 (3) ◽  
pp. 789-797 ◽  
Author(s):  
Junesang Choi ◽  
Rakesh Kumar Parmar ◽  
Tibor K. Pogany
Keyword(s):  
Hypergeometric Function ◽  
Gaussian Hypergeometric Function ◽  
Type Series

A Class of Generalized Hypergeometric Functions in Several Variables

1992 ◽  
Vol 44 (6) ◽  
pp. 1317-1338 ◽  
Author(s):  
Zhimin Yan
Keyword(s):  
Asymptotic Behavior ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Hypergeometric Function ◽  
Unique Solution ◽  
Hypergeometric Functions ◽  
Integral Representations ◽  
Gaussian Hypergeometric Function ◽  
Generalized Hypergeometric Functions ◽  
Several Variables

AbstractWe study a class of generalized hypergeometric functions in several variables introduced by A. Korânyi. It is shown that the generalized Gaussian hypergeometric function is the unique solution of a system partial differential equations. Analogues of some classical results such as Kummer relations and Euler integral representations are established. Asymptotic behavior of generalized hypergeometric functions is obtained which includes some known estimates.

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