On Caputo fractional derivative inequalities by using strongly $ (\alpha, h-m) $-convexity

Tao Yan; Ghulam Farid; Sidra Bibi; Kamsing Nonlaopon; Tao Yan; Ghulam Farid; Sidra Bibi; Kamsing Nonlaopon

doi:10.3934/math.2022565

AIMS Mathematics

2022, Volume 7, Issue 6: 10165-10179. doi: 10.3934/math.2022565

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On Caputo fractional derivative inequalities by using strongly $ (\alpha, h-m) $-convexity

1.
School of Computer Science, Chengdu University, Chengdu, China
2.
Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan
3.
Govt. Girls Primary School, Kamra Khurd, Attock 43570, Pakistan
4.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 23 December 2021 Revised: 13 February 2022 Accepted: 22 February 2022 Published: 22 March 2022
MSC : 26D10, 26A33, 31A10

In the literature of mathematical inequalities, one can have different variants of the well-known Hadamard inequality for CFD (Caputo fractional derivatives). These variants include generalizations, extensions and refinements which have been proved by defining new classes of functions. This paper aims to formulate inequalities of the Hadamard type which will simultaneously produce refinements and generalizations of many fractional versions of such inequalities that already exist in the literature. The error bounds of some existing inequalities are also proved by applying well-known identities. The results given in this paper are improvements of several well-known Hadamard type Caputo fractional derivative inequalities.
- Caputo fractional derivative,
- Hadamard inequality,
- strongly $ (\alpha, h-m) $-convex function,
- convex function,
- strongly convex function
Citation: Tao Yan, Ghulam Farid, Sidra Bibi, Kamsing Nonlaopon. On Caputo fractional derivative inequalities by using strongly $ (\alpha, h-m) $-convexity[J]. AIMS Mathematics, 2022, 7(6): 10165-10179. doi: 10.3934/math.2022565

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Abstract

In the literature of mathematical inequalities, one can have different variants of the well-known Hadamard inequality for CFD (Caputo fractional derivatives). These variants include generalizations, extensions and refinements which have been proved by defining new classes of functions. This paper aims to formulate inequalities of the Hadamard type which will simultaneously produce refinements and generalizations of many fractional versions of such inequalities that already exist in the literature. The error bounds of some existing inequalities are also proved by applying well-known identities. The results given in this paper are improvements of several well-known Hadamard type Caputo fractional derivative inequalities.

References

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