Improvements of the Hermite-Hadamard inequality

Abstract

The article provides refinements and generalizations of the Hermite-Hadamard inequality for convex functions on the bounded closed interval of real numbers. Improvements are related to the discrete and integral part of the inequality. MSC: 26A51; 26D15

Application of Functionals in Creating Inequalities

“…Letting approach zero, we reach the conclusion ( ( )) = ( ). In this case, trivial inequality ( ) ≤ ( ) ≤ ( ) represents formula (10). Formula (10) can be expressed in the form which includes the convex combination of interval endpoints and .…”

“…The first term of formula (34) is equal to ( ( )), the second term is equal to ( ( )), and the third term is equal to sec ( ( )). Thus, formula (34) fits into the frame of formula (10), and it is valid for a continuous convex function .…”

Improvements of the Hermite-Hadamard inequality for the simplex

“…A significant generalization of the Jensen inequality for multivariate convex functions can be found in [1]. Improvements of the Hermite-Hadamard inequality for univariate convex functions were obtained in [11]. As for the Hermite-Hadamard inequality for multivariate convex functions, one may refer to [2, 4, 5, 12–16], and [17].…”

On Some New Weighted Inequalities for Differentiable Exponentially Convex and Exponentially Quasi-Convex Functions with Applications

“…Interestingly, both sides of the above integral inequality characterize convex functions. For some interesting details and applications of HH inequality, we refer readers to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”