Functional inequalities for hypergeometric functions and complete elliptic integrals. (English) Zbl 0764.33009

Anderson, G. D.; Vamanamurthy, M. K.; Vuorinen, M.

SIAM J. Math. Anal. 23, No. 2, 512-524 (1992).

The authors obtain a number of inequalities for the classical \(_ 2F_ 1\) hypergeometric functions and for two of its special cases, the complete elliptic integrals of the first and second kind. A typical one is \(-g(x)>{_ 2F_ 1}(a,b;a-1b:X)>-g(x)/B(a,b)\) for \(0<a,b,x<1\), where \(g(x)=x^{-1}\log(1-x)\). 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.

Reviewer: R.Askey (Madison)

MSC:

33E05	Elliptic functions and integrals
33C05	Classical hypergeometric functions, \({}_2F_1\)

Keywords:

inequalities; hypergeometric functions; complete elliptic integrals