Contributions to the Integral of the Product of Two Functions in a Trigonometrically Convex Framework

The concept of a trigonometrically convex function consists in replacing the linear weights inherent in the classical concept of convexity with trigonometric ones. It emerged in 2018, combined with a framework of integral inequalities. In this article, we innovate by establishing valuable bounds for the integral of the product of two functions, including at least one such trigonometrically convex function. To derive these bounds, we use a number of analytical tools, including sharp trigonometric inequalities and the Hermite–Hadamard, Chebyshev and Hölder integral inequalities. Our results are flexible; they can accommodate different convexity conditions. Detailed proofs are given for completeness.