Generalization of some fractional versions of Hadamard inequalities via exponentially $ (\alpha, h-m) $-convex functions

Yu-Pei Lv; Ghulam Farid; Hafsa Yasmeen; Waqas Nazeer; Chahn Yong Jung; Yu-Pei Lv; Ghulam Farid; Hafsa Yasmeen; Waqas Nazeer; Chahn Yong Jung

doi:10.3934/math.2021521

AIMS Mathematics

2021, Volume 6, Issue 8: 8978-8999. doi: 10.3934/math.2021521

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Research article

Generalization of some fractional versions of Hadamard inequalities via exponentially $ (\alpha, h-m) $-convex functions

1.
Department of Mathematics, Huzhou University, Huzhou 313000, China
2.
Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan
3.
Department of Mathematics, Govt. College University Lahore, Lahore, Pakistan
4.
Department of Business Administration, Gyeongsang National University, Jinju 52828, Korea

Received: 10 March 2021 Accepted: 11 June 2021 Published: 16 June 2021
MSC : 26A51, 26A33, 33E12

In this paper we give Hadamard inequalities for exponentially $ (\alpha, h-m) $-convex functions using Riemann-Liouville fractional integrals for strictly increasing function. Results for Riemann-Liouville fractional integrals of convex, $ m $-convex, $ s $-convex, $ (\alpha, m) $-convex, $ (s, m) $-convex, $ (h-m) $-convex, $ (\alpha, h-m) $-convex, exponentially convex, exponentially $ m $-convex, exponentially $ s $-convex, exponentially $ (s, m) $-convex, exponentially $ (h-m) $-convex, exponentially $ (\alpha, h-m) $-convex functions are particular cases of the results of this paper. The error estimations of these inequalities by using two fractional integral identities are also given.
Citation: Yu-Pei Lv, Ghulam Farid, Hafsa Yasmeen, Waqas Nazeer, Chahn Yong Jung. Generalization of some fractional versions of Hadamard inequalities via exponentially $ (\alpha, h-m) $-convex functions[J]. AIMS Mathematics, 2021, 6(8): 8978-8999. doi: 10.3934/math.2021521

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Abstract

In this paper we give Hadamard inequalities for exponentially $ (\alpha, h-m) $-convex functions using Riemann-Liouville fractional integrals for strictly increasing function. Results for Riemann-Liouville fractional integrals of convex, $ m $-convex, $ s $-convex, $ (\alpha, m) $-convex, $ (s, m) $-convex, $ (h-m) $-convex, $ (\alpha, h-m) $-convex, exponentially convex, exponentially $ m $-convex, exponentially $ s $-convex, exponentially $ (s, m) $-convex, exponentially $ (h-m) $-convex, exponentially $ (\alpha, h-m) $-convex functions are particular cases of the results of this paper. The error estimations of these inequalities by using two fractional integral identities are also given.

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