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Functional inequalities for hypergeometric functions and complete elliptic integrals. (English) Zbl 0764.33009
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.
 33E05 Elliptic functions and integrals 33C05 Classical hypergeometric functions, $${}_2F_1$$