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

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.