scholarly journals Monotonicity, convexity properties and inequalities involving Gaussian hypergeometric functions with applications

2021 ◽  
Vol 7 (4) ◽  
pp. 4974-4991
Author(s):  
Ye-Cong Han ◽  
◽  
Chuan-Yu Cai ◽  
Ti-Ren Huang ◽  
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Elliptic Integral ◽  
Gaussian Hypergeometric Function ◽  
Gaussian Hypergeometric Functions ◽  
Convexity Properties

<abstract><p>In this paper, we mainly prove monotonicity and convexity properties of certain functions involving zero-balanced Gaussian hypergeometric function $ F(a, b; a+b; x) $. We generalize conclusions of elliptic integral to Gaussian hypergeometric function, and get some accurate inequalities about Gaussian hypergeometric function.</p></abstract>

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.

scholarly journals Starlike and Convex Properties for Hypergeometric Functions

2008 ◽  
Vol 2008 ◽  
pp. 1-11 ◽  
Author(s):  
Oh Sang Kwon ◽  
Nak Eun Cho
Keyword(s):  
Integral Operator ◽  
Hypergeometric Function ◽  
Convex Functions ◽  
Hypergeometric Functions ◽  
Gaussian Hypergeometric Function ◽  
Starlike And Convex Functions

The purpose of the present paper is to give some characterizations for a (Gaussian) hypergeometric function to be in various subclasses of starlike and convex functions. We also consider an integral operator related to the hypergeometric function.

scholarly journals Landen Inequalities for Zero-Balanced Hypergeometric Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Slavko Simić ◽  
Matti Vuorinen
Keyword(s):  
Open Problem ◽  
Hypergeometric Functions ◽  
Elliptic Integral ◽  
Complete Elliptic Integral ◽  
Turn On ◽  
Gaussian Hypergeometric Functions

For zero-balanced Gaussian hypergeometric functionsF(a,b;a+b;x),a,b>0, we determine maximal regions ofabplane where well-known Landen identities for the complete elliptic integral of the first kind turn on respective inequalities valid for eachx∈(0,1). Thereby an exhausting answer is given to the open problem from the work by Anderson et al., 1990.

scholarly journals On Mathieu-type series for the unified Gaussian hypergeometric functions

2020 ◽  
Vol 14 (1) ◽  
pp. 138-149
Author(s):  
Rakesh Parmar ◽  
Tibor Pogány
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Integral Form ◽  
Type A ◽  
Gauss Hypergeometric Function ◽  
Type Series ◽  
Gaussian Hypergeometric Functions ◽  
Special Cases

The main purpose of this paper is to present closed integral form expressions for the Mathieu-type a-series and for the associated alternating versions whose terms contain a generalized p-extended Gauss' hypergeometric function. Related bounding inequalities for the p-generalized Mathieu-type series are also obtained. Finally, a set of various (known or new) special cases and consequences of the results earned are presented.

scholarly journals Sufficiency for Gaussian hypergeometric functions to be uniformly convex

1999 ◽  
Vol 22 (4) ◽  
pp. 765-773 ◽  
Author(s):  
Yong Chan Kim ◽  
S. Ponnusamy
Keyword(s):  
Hypergeometric Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Hypergeometric Functions ◽  
Uniformly Convex ◽  
Gaussian Hypergeometric Functions

LetF(a,b;c;z)be the classical hypergeometric function andfbe a normalized analytic functions defined on the unit disk𝒰. Let an operatorIa,b;c(f)be defined by[Ia,b;c(f)](z)=zF(a,b;c;z)*f(z). In this paper the authors identify two subfamilies of analytic functionsℱ1andℱ2and obtain conditions on the parametersa,b,csuch thatf∈ℱ1impliesIa,b;c(f)∈ℱ2.

scholarly journals Two integral transform pairs involving hypergeometric functions

1965 ◽  
Vol 7 (1) ◽  
pp. 42-44 ◽  
Author(s):  
Jet Wimp
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Integral Transform ◽  
Confluent Hypergeometric Function ◽  
Gaussian Hypergeometric Function ◽  
Special Cases

In this note, we first establish an integral transform pair where the kernel of each integral involves the Gaussian hypergeometric function. Special cases of Theorem 1 have been studied by several authors [1, 2, 5, 6]. In Theorem 2 a similar integral transform pair involving a confluent hypergeometric function is given.

Functional inequalities for Gaussian hypergeometric function and generalized elliptic integral of the first kind

2021 ◽  
Vol 71 (3) ◽  
pp. 667-682
Author(s):  
Shen-Yang Tan ◽  
Ti-Ren Huang ◽  
Yu-Ming Chu
Keyword(s):  
Hypergeometric Function ◽  
Elliptic Integral ◽  
Functional Inequalities ◽  
Gaussian Hypergeometric Function

Abstract In the article, we present several new functional inequalities for the Gaussian hypergeometric function and generalized elliptic integral of the first kind.

Recursion formulas for multivariable hypergeometric functions

2015 ◽  
Vol 08 (04) ◽  
pp. 1550082 ◽  
Author(s):  
Vivek Sahai ◽  
Ashish Verma
Keyword(s):  
Hypergeometric Function ◽  
Radiation Field ◽  
Hypergeometric Functions ◽  
Integral Transforms ◽  
Recursion Formulas ◽  
Appl Math ◽  
Appell Functions

Recently, Opps, Saad and Srivastava [Recursion formulas for Appell’s hypergeometric function [Formula: see text] with some applications to radiation field problems, Appl. Math. Comput. 207 (2009) 545–558] presented the recursion formulas for Appell’s function [Formula: see text] and also gave its applications to radiation field problems. Then Wang [Recursion formulas for Appell functions, Integral Transforms Spec. Funct. 23(6) (2012) 421–433] obtained the recursion formulas for Appell functions [Formula: see text] and [Formula: see text]. In our investigation here, we derive the recursion formulas for 14 three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions.

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