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Inequalities involving a logarithmically convex function and their applications to special functions. (English) Zbl 1132.26337
Summary: It has been shown that if \( f\) is a differentiable, logarithmically convex function on nonnegative semi-axis, then the function \( [f(x)]^a/f(ax)\), (\( a \geq 1\)) is decreasing on its domain. Applications to inequalities involving gamma function, Riemann’s zeta function, and the complete elliptic integrals of the first kind are included.

MSC:
26D07 Inequalities involving other types of functions
26D20 Other analytical inequalities
33B15 Gamma, beta and polygamma functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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