New Hermite-Hadamard type inequalities for exponentially convex functions and applications

Shuang-Shuang Zhou; Saima Rashid; Muhammad Aslam Noor; Khalida Inayat Noor; Farhat Safdar; Yu-Ming Chu; Shuang-Shuang Zhou; Saima Rashid; Muhammad Aslam Noor; Khalida Inayat Noor; Farhat Safdar; Yu-Ming Chu

doi:10.3934/math.2020441

AIMS Mathematics

2020, Volume 5, Issue 6: 6874-6901. doi: 10.3934/math.2020441

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Research article

New Hermite-Hadamard type inequalities for exponentially convex functions and applications

1.
School of Science, Hunan City University, Yiyang 413000, China
2.
Department of Mathematics, Government College University Faisalabad, Pakistan
3.
Department of Mathematics, COMSATS University, Islamabad, Pakistan
4.
Department of mathematics, SBK Women University, Quetta, Pakistan
5.
Department of Mathematics, Huzhou University, Huzhou 313000, China
6.
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, P. R. China

Received: 11 April 2020 Accepted: 24 August 2020 Published: 04 September 2020
MSC : 26A33, 26A51, 26D10

The investigation of the proposed techniques is effective and convenient for solving the integrodifferential and difference equations. The present investigation depends on two highlights; the novel Hermite-Hadamard type inequalities for $\mathcal{K}$-conformable fractional integral operator in terms of a new parameter $\mathcal{K}>0$ and weighted version of Hermite-Hadamard type inequalities for exponentially convex functions in the classical sense. By using an integral identity together with the Hölder-İşcan and improved power-mean inequality we establish several new inequalities for differentiable exponentially convex functions. This generalizes the Hadamard fractional integrals and Riemann-Liouville into a single form. Our contribution expands some innovative studies in this line. Moreover, two suitable examples are presented to demonstrate the novelty of the results established, the first one about the contributions of the modified Bessel functions and the other is about $\sigma$-digamma function. Finally, various applications for some special means as arithmetic, geometric and logarithmic are given.
- convex function,
- exponentially convex functions,
- Hermite-Hadamard inequality,
- $\mathcal{K}$-conformable fractional integral,
- weighted inequality
Citation: Shuang-Shuang Zhou, Saima Rashid, Muhammad Aslam Noor, Khalida Inayat Noor, Farhat Safdar, Yu-Ming Chu. New Hermite-Hadamard type inequalities for exponentially convex functions and applications[J]. AIMS Mathematics, 2020, 5(6): 6874-6901. doi: 10.3934/math.2020441

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Abstract

The investigation of the proposed techniques is effective and convenient for solving the integrodifferential and difference equations. The present investigation depends on two highlights; the novel Hermite-Hadamard type inequalities for $\mathcal{K}$-conformable fractional integral operator in terms of a new parameter $\mathcal{K}>0$ and weighted version of Hermite-Hadamard type inequalities for exponentially convex functions in the classical sense. By using an integral identity together with the Hölder-İşcan and improved power-mean inequality we establish several new inequalities for differentiable exponentially convex functions. This generalizes the Hadamard fractional integrals and Riemann-Liouville into a single form. Our contribution expands some innovative studies in this line. Moreover, two suitable examples are presented to demonstrate the novelty of the results established, the first one about the contributions of the modified Bessel functions and the other is about $\sigma$-digamma function. Finally, various applications for some special means as arithmetic, geometric and logarithmic are given.

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