On the mean integral of the reciprocal and ratio of trigonometrically convex functions

Trigonometrically convex functions have recently emerged as an important class of functions in the field of convex analysis. Despite the growing interest, several aspects of their theory remain unexplored. In this article, we first establish lower and upper bounds for the mean integral of the reciprocal of a function that is either trigonometrically convex or trigonometrically concave. Building on these results, we further investigate the mean integral of the ratio of two functions, each of which is assumed to be either trigonometrically convex or concave. These investigations provide new knowledge about the integral behavior and comparative structure of such functions.