Increasing property and logarithmic convexity concerning Dirichlet beta function, Euler numbers, and their ratios

Abstract

In the paper, by virtue of an integral representation of the Dirichlet beta function, with the aid of a relation between the Dirichlet beta function and the Euler numbers, and by means of a monotonicity rule for the ratio of two definite integrals with a parameter, the author finds increasing property and logarithmic convexity of two functions and two sequences involving the Dirichlet beta function, the Euler numbers, and their ratios.

Several Functions Originating from Fisher–Rao Geometry of Dirichlet Distributions and Involving Polygamma Functions

“…Lemma 12 has been applied in [33,[42][43][44] and generalized in [45][46][47]. There have been a number of papers, plenty of studies in the literature, and many mathematicians contributing to various monotonicity rules, and we just take two examples [48,49] here.…”