Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity

Hengxiao Qi; Muhammad Yussouf; Sajid Mehmood; Yu-Ming Chu; Ghulam Farid; Hengxiao Qi; Muhammad Yussouf; Sajid Mehmood; Yu-Ming Chu; Ghulam Farid

doi:10.3934/math.2020386

AIMS Mathematics

2020, Volume 5, Issue 6: 6030-6042. doi: 10.3934/math.2020386

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Research article

Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity

1.
Party School of Shandong Provincial Committee of the Communist Party of China (Shandong Administration College), Jinan 250014, China
2.
Department of Mathematics, University of Sargodha, Sargodha, Pakistan
3.
Govt Boys Primary school Sherani, Hazro, Attock, Pakistan
4.
Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China
5.
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, P. R. China
6.
Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan

Received: 20 May 2020 Accepted: 17 July 2020 Published: 23 July 2020
MSC : 26B25, 26A33, 26A51, 33E12

In this paper, we establish generalized fractional versions of Hermite-Hadamard inequalities for exponentially (α, h - m)-convex functions, exponentially (h - m)-convex functions and exponentially (α, m)-convex functions. These inequalities arise when using the generalized fractional integral operators containing Mittag-Leffler function via a monotonically increasing function. The presented results hold at the same time for various kinds of convexities and well-known fractional integral operators. Moreover, the established inequalities reproduce several known results which are part of the existing literature.
- convex function,
- Hermite-Hadamard inequality,
- generalized fractional integral operators,
- Mittag-Leffler function
Citation: Hengxiao Qi, Muhammad Yussouf, Sajid Mehmood, Yu-Ming Chu, Ghulam Farid. Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity[J]. AIMS Mathematics, 2020, 5(6): 6030-6042. doi: 10.3934/math.2020386

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Abstract

In this paper, we establish generalized fractional versions of Hermite-Hadamard inequalities for exponentially (α, h - m)-convex functions, exponentially (h - m)-convex functions and exponentially (α, m)-convex functions. These inequalities arise when using the generalized fractional integral operators containing Mittag-Leffler function via a monotonically increasing function. The presented results hold at the same time for various kinds of convexities and well-known fractional integral operators. Moreover, the established inequalities reproduce several known results which are part of the existing literature.

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