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Functional inequalities for the quotients of hypergeometric functions. (English) Zbl 0890.33004
It was proven in [SIAM J. Math. Anal. 21, No. 2, 536-549 (1990; Zbl 0692.33001)] that for the complete elliptic integral of the first kind, \[ \int_0^{\pi/2} {d\theta \over \sqrt{1-r^2\sin^2\theta}} = {\pi \over 2} F\left( {1\over 2}, {1 \over 2}; 1; r^2 \right), \] if we define \[ \mu(r) = { F(1/2,1/2;1;1-r^2) \over F(1/2,1/2;1;r^2) }, \] then \(\mu(r) + \mu(s) \leq 2 \mu(\sqrt{rs})\) for all \(r,s \in (0,1)\). In this paper, the authors prove that for the function \[ m(r) = { F(a,1-a,1;1-r^2) \over F(a,1-a,1;r^2) },\quad 0<a<1, \] the same inequality, \(m(r) + m(s) \leq 2 m(\sqrt{rs})\), holds for all \(r,s \in (0,1)\). They also prove that for \(a \in (0,2)\) and \(b \in (0,2-a)\), \[ {F(a,b;a+b;r^2) \over F(a,b;a+b;1-r^2)} + {F(a,b;a+b;s^2) \over F(a,b;a+b;1-s^2)} \geq 2{F(a,b;a+b;rs) \over F(a,b;a+b;1-rs)}. . \]

MSC:
33C20 Generalized hypergeometric series, \({}_pF_q\)
33E05 Elliptic functions and integrals
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