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Regularity properties and integral inequalities related to \((k,h_1,h_2)\)-convexity of functions. (English) Zbl 1374.26048

Summary: The class of \((k,h_1,h_2)\)-convex functions is introduced, together with some particular classes of corresponding generalized convex dominated functions. Few regularity properties of \((k,h_1,h_2)\)-convex functions are proved by means of Bernstein-Doetsch type results. Also we find conditions in which every local minimizer of a \((k,h_1,h_2)\)-convex function is global. Classes of \((k,h_1,h_2)\)-convex functions, which allow integral upper bounds of Hermite-Hadamard type, are identified. Hermite-Hadamard type inequalities are also obtained in a particular class of the \((k,h_1,h_2)\)-convex dominated functions.

MSC:

26D15 Inequalities for sums, series and integrals
26A51 Convexity of real functions in one variable, generalizations
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