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Landen inequalities for zero-balanced hypergeometric functions. (English) Zbl 1241.26021
Summary: For zero-balanced Gaussian hypergeometric functions $F(a, b; a + b; x)$, $a, b > 0$, we determine maximal regions of $ab$ plane where well-known Landen identities for the complete elliptic integral of the first kind turn on respective inequalities valid for each $x \in (0, 1)$. Thereby an exhausting answer is given to the open problem from the work by {\it G. D. Anderson, M. K. Vamanamurthy}, and {\it M. Vuorinen} [Conformal invariants, inequalities, and quasiconformal maps. Wiley (1997; Zbl 0885.30012), p. 79].

##### MSC:
 26D15 Inequalities for sums, series and integrals of real functions 33C05 Classical hypergeometric functions, ${}_2F_1$
Full Text:
##### References:
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