Functional inequalities involving special functions
Let \(_2F_1 (a,b;c;x)\) be a hypergeometric function and let \[ m(r) = {}_2F_1 (a,b;a+b;(1-r^2))/_2F_1 (a,b;a+b;r^2), \quad r \in (0,1). \] The author shows that \[ m(r) + m(s) \geq 2m[ \sqrt(1- \sqrt(( 1-r^2)(1-s^2)))]. \] If \(u_p\) is the generalized and normalized Bessel function and \(\sigma(r) = u_p(1 - r^2)/ u_p (r^2)\), the author proves that \[ \sqrt (\sigma(r_1)\sigma(r_2)) \leq \sigma( \sqrt(r_1 r_2)), \quad c < 0, \quad 2p+b+1 > 0. \]
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