On <em>n</em>-Polynomial convexity and some related inequalities

Tekin Toplu; Mahir Kadakal; İmdat İşcan; Tekin Toplu; Mahir Kadakal; İmdat İşcan

doi:10.3934/math.2020089

AIMS Mathematics

2020, Volume 5, Issue 2: 1304-1318. doi: 10.3934/math.2020089

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Research article

On n-Polynomial convexity and some related inequalities

1.
Institute of Science, Giresun University, 28200, Giresun-Turkey
2.
Department of Mathematics, Faculty of Arts and Sciences, Giresun University, 28200, Giresun-Turkey

Received: 28 October 2019 Accepted: 17 January 2020 Published: 20 January 2020
MSC : 26A51, 26D10, 26D15

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.
- convex function,
- n-polynomial convexity,
- Hölder integral inequality,
- Hölder-İşcan integral inequality,
- Hermite-Hadamard inequality
Citation: Tekin Toplu, Mahir Kadakal, İmdat İşcan. On n-Polynomial convexity and some related inequalities[J]. AIMS Mathematics, 2020, 5(2): 1304-1318. doi: 10.3934/math.2020089

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Abstract

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.

References

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