Some new Caputo fractional derivative inequalities for exponentially $ (\theta, h-m) $–convex functions

Imran Abbas Baloch; Thabet Abdeljawad; Sidra Bibi; Aiman Mukheimer; Ghulam Farid; Absar Ul Haq; Imran Abbas Baloch; Thabet Abdeljawad; Sidra Bibi; Aiman Mukheimer; Ghulam Farid; Absar Ul Haq

doi:10.3934/math.2022166

AIMS Mathematics

2022, Volume 7, Issue 2: 3006-3026. doi: 10.3934/math.2022166

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Some new Caputo fractional derivative inequalities for exponentially $ (\theta, h-m) $–convex functions

1.
Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan
2.
Higher Education Department, Govt. Graduate College for Boys Gulberg Lahore, Punjab, Pakistan
3.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
4.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
5.
Govt. Girls Primary School, Kamra Khurd, Attock 43570, Pakistan
6.
COMSATS University Islamabad, Attock Camopus, Kamra Road, Attock 43600, Pakistan
7.
Department of Natural Sciences and Humanities, University of Engineering and Technology (Narowal Campus), Lahore 54000, Pakistan

Received: 31 May 2021 Accepted: 09 November 2021 Published: 23 November 2021
MSC : 26A33, 26A51, 26D07, 26D10, 26D15

Firstly, we obtain some inequalities of Hadamard type for exponentially $ (\theta, h-m) $–convex functions via Caputo $ k $–fractional derivatives. Secondly, using integral identity including the $ (n+1) $–order derivative of a given function via Caputo $ k $-fractional derivatives we prove some of its related results. Some new results are given and known results are recaptured as special cases from our results.
Citation: Imran Abbas Baloch, Thabet Abdeljawad, Sidra Bibi, Aiman Mukheimer, Ghulam Farid, Absar Ul Haq. Some new Caputo fractional derivative inequalities for exponentially $ (\theta, h-m) $–convex functions[J]. AIMS Mathematics, 2022, 7(2): 3006-3026. doi: 10.3934/math.2022166

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Abstract

Firstly, we obtain some inequalities of Hadamard type for exponentially $ (\theta, h-m) $–convex functions via Caputo $ k $–fractional derivatives. Secondly, using integral identity including the $ (n+1) $–order derivative of a given function via Caputo $ k $-fractional derivatives we prove some of its related results. Some new results are given and known results are recaptured as special cases from our results.

References

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