## 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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