Weighted Hermite-Hadamard dual inequality in integral form is an important result as its left hand fact Jensen and right the Lah-Ribaric inequality. In this paper new linear inequalities are introduced via extension of Montgomery identity weighted with without Green functions discrete cases.

Abstract In this article we give two possible generalizations of the Hermite–Hadamard integral inequality for class twice differentiable functions, where convexity property target function is not assumed in advance. They represent a refinement case convex/concave functions with numerous applications.

equality holds in either side only for the affine functions (i.e., for the functions of the form mx+ n). The middle point (a + b)/2 represents the barycenter of the probability measure 1 b−adx (viewed as a mass distribution over the interval [a, b]), while a and b represent the extreme points of [a, b]. Thus the Hermite-Hadamard inequality could be seen as a precursor of Choquet’s theory. See [...