A generalization of convexity via an implicit inequality

We unified several kinds of convexity by introducing the class A(zeta,w)([0, 1] x I-2) of (zeta, w)-admissible functions F : [0, 1] x I x I -> R. Namely, we proved that most types of convexity from the literature generate functions F is an element of A(zeta,w)([0, 1] x I-2) for some zeta is an element of C([0, 1]) and w is an element of C-1(I) with w(I) subset of I and w' > 0. We also studied some properties of (zeta, w)-admissible functions and established some integral inequalities that unify various Hermite-Hadamard-type inequalities from the literature.