Generalizations of some $ q $-integral inequalities of Hölder, Ostrowski and Grüss type

Da Shi; Ghulam Farid; Abd Elmotaleb A. M. A. Elamin; Wajida Akram; Abdullah A. Alahmari; B. A. Younis; Da Shi; Ghulam Farid; Abd Elmotaleb A. M. A. Elamin; Wajida Akram; Abdullah A. Alahmari; B. A. Younis

doi:10.3934/math.20231192

AIMS Mathematics

2023, Volume 8, Issue 10: 23459-23471. doi: 10.3934/math.20231192

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Research article

Generalizations of some $ q $-integral inequalities of Hölder, Ostrowski and Grüss type

1.
School of Computer Science, Chengdu University, Chengdu, China
2.
Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan
3.
Department of Mathematics, College of Science and Humanity, Prince Sattam bin Abdulaziz University, Sulail, Saudi Arabia
4.
Department of Mathematical Sciences, Faculty of Applied Science, Umm Al-Qura University, Makkah 21955, Saudi Arabia
5.
King Khalid University, College of Arts & Sciences, Saudi Arabia

Received: 25 April 2023 Revised: 06 June 2023 Accepted: 15 June 2023 Published: 26 July 2023
MSC : 26A51, 26A33, 33E12

This paper investigates some well-known inequalities for $ q $-$ h $-integrals. These include Hölder, Ostrowski, Grüss and Opial type inequalities. Refinement of the Hadamard inequality for $ q $-$ h $-integrals is also established by applying the definition of strongly convex functions. From main theorems, $ q $-Hölder, $ q $-Ostrowski and $ q $-Grüss inequalities can be obtained in particular cases.
- $ q $-integral,
- $ q $-$ h $-integral,
- Hölder inequality,
- Ostrowski inequality,
- Güss inequality,
- Hadamard inequality
Citation: Da Shi, Ghulam Farid, Abd Elmotaleb A. M. A. Elamin, Wajida Akram, Abdullah A. Alahmari, B. A. Younis. Generalizations of some $ q $-integral inequalities of Hölder, Ostrowski and Grüss type[J]. AIMS Mathematics, 2023, 8(10): 23459-23471. doi: 10.3934/math.20231192

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Abstract

This paper investigates some well-known inequalities for $ q $-$ h $-integrals. These include Hölder, Ostrowski, Grüss and Opial type inequalities. Refinement of the Hadamard inequality for $ q $-$ h $-integrals is also established by applying the definition of strongly convex functions. From main theorems, $ q $-Hölder, $ q $-Ostrowski and $ q $-Grüss inequalities can be obtained in particular cases.

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