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The class of (k; h1; h2)-convex functions is introduced, together with some particular classes of corresponding generalized convex dominated functions. Few regularity properties of (k; h1; h2)-convex functions are proved by means of Bernstein-Doetsch type results. Also we find conditions in which every local minimizer of a (k; h1; h2)-convex function is global. Classes of (k; h1; h2)-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; h1; h2)- convex dominated functions.