Journal of Convex Analysis 33 (2026), No. 1&2, 441--454
Copyright Heldermann Verlag 2026
Inequalities for Generalized Convex Functions with the Use of the Hesselager Theorem
Magdalena Mamcarz
Institute of Mathematics, University of Silesia, Katowice, Poland
magdalena.mamcarz@us.edu.pl
Tomasz Szostok
Institute of Mathematics, University of Silesia, Katowice, Poland
tomasz.szostok@us.edu.pl
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.
Keywords: Chebyshev systems, higher-order convexity, generalized convexity, stochastic orderings.
MSC: 60E15, 26A51,26D10, 39B62.
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