Inequalities for Generalized Convex Functions with the Use of the Hesselager Theorem

We observe that the theorem of Hesselager may be used to obtain new proofs of Hermite-Hadamard inequalities for functions that are convex with respect to a Chebyshev system. Such inequalities were proved by Bessenyei and Páles for integrals with density function, in our approach, the first function from the system involved must be constant but the main theorem is valid for the integral with respect to any measure. Additionally, we present examples of inequalities that follow from Hesselager's theorem but are not of the Hermite-Hadamard type.