On n-Polynomial convexity and some related inequalities

• Published in: AIMS mathematics Vol. 5; no. 2; pp. 1304 - 1318
• Main Authors: Toplu, Tekin, Kadakal, Mahir, İşcan, İmdat
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2020
• Subjects: convex function; hermite-hadamard inequality; hölder integral inequality; hölder-i̇şcan integral inequality; n-polynomial convexity
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2020089

In this paper, we introduce and study the concept of n-polynomial convexity functions and their some algebric properties. We prove two Hermite-Hadamard type inequalities for the newly introduced class of functions. In addition, we obtain some refinements of the Hermite-Hadamard inequality for functions whose first derivative in absolute value, raised to a certain power which is greater than one, respectively at least one, is n-polynomial convexity. Also, we compare the results obtained with both Hölder, Hölder-İşcan inequalities and power-mean, improved-power-mean integral inequalities and show that the result obtained with Hölder-İşcan and improved power-mean inequalities give better approach than the others. Some applications to special means of real numbers are also given.