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Zbl 0764.33009
Anderson, G.D.; Vamanamurthy, M.K.; Vuorinen, M.
Functional inequalities for hypergeometric functions and complete elliptic integrals. (English)
[J] SIAM J. Math. Anal. 23, No.2, 512-524 (1992). ISSN 0036-1410; ISSN 1095-7154/e

The authors obtain a number of inequalities for the classical $\sb 2F\sb 1$ hypergeometric functions and for two of its special cases, the complete elliptic integrals of the first and second kind. A typical one is $-g(x)>{\sb 2F\sb 1}(a,b;a-1b:X)>-g(x)/B(a,b)$ for $0<a,b,x<1$, where $g(x)=x\sp{-1}\log(1-x)$. The lower estimate is sharp at $x=1$ and the upper estimate is sharp at $x=0$. For $a,b>1$, $0<x<1$, these inequalities swap and for $a=b=1$ there is equality.
[R.Askey (Madison)]

MSC 2000:: *33E05 Elliptic functions and integrals
33C05 Classical hypergeometric functions

Keywords: inequalities; hypergeometric functions; complete elliptic integrals

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