Some bounds for the Mellin transform and applications to the Riemann zeta and Macdonald functions

Abstract

Requiring monotonicity of the product of the exponential with a function, we prove an inequality for the Mellin transform of the function. For a certain class of functions, the log-convex property is proved. Applications of the result in proving inequalities and the log-convex property for the extended zeta function, the Riemann zeta function, and the Macdonald function are discussed.