Regularity properties and integral inequalities related to (k; h₁; h₂)-convexity of functions

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.