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 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>

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 On Certain Sufficient Condition Involving Gaussian Hypergeometric Functions

2009 ◽  
Vol 2009 ◽  
pp. 1-15
Author(s):  
H. Silverman ◽  
Thomas Rosy ◽  
S. Kavitha
Keyword(s):  
Unit Disk ◽  
Hypergeometric Functions ◽  
Open Unit ◽  
Sufficient Condition ◽  
Open Unit Disk ◽  
New Class ◽  
Inclusion Properties ◽  
Complex Order ◽  
Gaussian Hypergeometric Functions

The authors define a new subclass of of functions involving complex order in the open unit disk . For this new class, we obtain certain inclusion properties involving the Gaussian hypergeometric functions.

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