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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 ${}_{2}{F}_{1}$ hypergeometric functions and for two of its special cases, the complete elliptic integrals of the first and second kind. A typical one is $-g\left(x\right)>{}_{2}{F}_{1}\left(a,b;a-1b:X\right)>-g\left(x\right)/B\left(a,b\right)$ for $0, where $g\left(x\right)={x}^{-1}log\left(1-x\right)$. The lower estimate is sharp at $x=1$ and the upper estimate is sharp at $x=0$. For $a,b>1$, $0, these inequalities swap and for $a=b=1$ there is equality.
MSC:
 33E05 Elliptic functions and integrals 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$