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Symmetry
Volume 15
Issue 3
10.3390/sym15030666
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Article

Inequalities for q-h-Integrals via ℏ-Convex and m-Convex Functions

by
Dong Chen
1,
Matloob Anwar
2,*,
Ghulam Farid
3 and
Waseela Bibi
3
1
College of Electronic Information and Electrical Engineering, Chengdu University, Chengdu 610106, China
2
School of Natural Sciences, NUST, Islamabad 44000, Pakistan
3
Department of Mathematics, COMSATS University Islamabad, Attock Campus, Attock 43600, Pakistan
*
Author to whom correspondence should be addressed.
Symmetry 2023, 15(3), 666; https://doi.org/10.3390/sym15030666
Submission received: 1 February 2023 / Revised: 18 February 2023 / Accepted: 3 March 2023 / Published: 7 March 2023
(This article belongs to the Special Issue Symmetry and Integrable System)
Versions Notes

Abstract

This paper investigates several integral inequalities held simultaneously for q and h-integrals in implicit form. These inequalities are established for symmetric functions using certain types of convex functions. Under certain conditions, Hadamard-type inequalities are deducible for q-integrals. All the results are applicable for ℏ-convex, m-convex and convex functions defined on the non-negative part of the real line.
Keywords: q-derivative; q-integral; q-h-integral; Jensen inequality; Hadamard inequality q-derivative; q-integral; q-h-integral; Jensen inequality; Hadamard inequality

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MDPI and ACS Style

Chen, D.; Anwar, M.; Farid, G.; Bibi, W. Inequalities for q-h-Integrals via ℏ-Convex and m-Convex Functions. Symmetry 2023, 15, 666. https://doi.org/10.3390/sym15030666

AMA Style

Chen D, Anwar M, Farid G, Bibi W. Inequalities for q-h-Integrals via ℏ-Convex and m-Convex Functions. Symmetry. 2023; 15(3):666. https://doi.org/10.3390/sym15030666

Chicago/Turabian Style

Chen, Dong, Matloob Anwar, Ghulam Farid, and Waseela Bibi. 2023. "Inequalities for q-h-Integrals via ℏ-Convex and m-Convex Functions" Symmetry 15, no. 3: 666. https://doi.org/10.3390/sym15030666

APA Style

Chen, D., Anwar, M., Farid, G., & Bibi, W. (2023). Inequalities for q-h-Integrals via ℏ-Convex and m-Convex Functions. Symmetry, 15(3), 666. https://doi.org/10.3390/sym15030666

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