Inequalities involving a logarithmically convex function and their applications to special functions. (English) Zbl 1132.26337

Neuman, Edward

JIPAM, J. Inequal. Pure Appl. Math. 7, No. 1, Paper No. 16, 4 p. (2006).

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 \(L(s, \chi)\)

Keywords:

logarithmically convex functions; inequalities; gamma function; Riemann’s zeta function; complete elliptic integrals of the first kind