On $\mathscr{M}$-convex functions

Muhammad Uzair Awan; Muhammad Aslam Noor; Tingsong Du; Khalida Inayat Noor; Muhammad Uzair Awan; Muhammad Aslam Noor; Tingsong Du; Khalida Inayat Noor

doi:10.3934/math.2020157

AIMS Mathematics

2020, Volume 5, Issue 3: 2376-2387. doi: 10.3934/math.2020157

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Research article

On $\mathscr{M}$-convex functions

1.
Department of Mathematics, GC University, Faisalabad, Pakistan
2.
Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
3.
Department of Mathematics, College of Science, China Three Gorges University, China

Received: 30 December 2019 Accepted: 25 February 2020 Published: 04 March 2020
MSC : 05A30, 26A33, 26A51, 26D15

In this article, we introduce the notion of $\mathscr{M}$-convex functions, $\log$-$\mathscr{M}$-convex functions and the notion of quasi $\mathscr{M}$-convex functions. We derive some new analogues of Hermite-Hadamard like inequalities associated with $\mathscr{M}$-convex functions by using the concepts of ordinary, fractional and quantum calculus. The main results of this paper may be useful where bounds for natural phenomena described by integrals such as mechanical work are frequently required. These results are also helpful in the field of numerical analysis where error analysis is required.
- convex,
- $\mathscr{M}$-convex,
- $\log$-$\mathscr{M}$-convex,
- quasi $\mathscr{M}$-convex,
- Hermite-Hadamard inequality,
- fractional,
- quantum
Citation: Muhammad Uzair Awan, Muhammad Aslam Noor, Tingsong Du, Khalida Inayat Noor. On $\mathscr{M}$-convex functions[J]. AIMS Mathematics, 2020, 5(3): 2376-2387. doi: 10.3934/math.2020157

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Abstract

In this article, we introduce the notion of $\mathscr{M}$-convex functions, $\log$-$\mathscr{M}$-convex functions and the notion of quasi $\mathscr{M}$-convex functions. We derive some new analogues of Hermite-Hadamard like inequalities associated with $\mathscr{M}$-convex functions by using the concepts of ordinary, fractional and quantum calculus. The main results of this paper may be useful where bounds for natural phenomena described by integrals such as mechanical work are frequently required. These results are also helpful in the field of numerical analysis where error analysis is required.

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