Some Opial type inequalities in (<em>p, q</em>)-calculus

Chunhong Li; Dandan Yang; Chuanzhi Bai; Chunhong Li; Dandan Yang; Chuanzhi Bai

doi:10.3934/math.2020377

AIMS Mathematics

2020, Volume 5, Issue 6: 5893-5902. doi: 10.3934/math.2020377

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Research article

Some Opial type inequalities in (p, q)-calculus

1.
Editorial Department of Journal of Huaiyin Normal University, Huai an, Jiangsu Province, 223300, China
2.
School of Mathematical Science, Huaiyin Normal University, Huai an, Jiangsu Province, 223300, China

Received: 31 March 2020 Accepted: 28 June 2020 Published: 15 July 2020
MSC : 26D15, 81P68

In this paper, we establish 5 kinds of integral Opial-type inequalities in (p, q)-calculus by means of H?lder's inequality, Cauchy inequality, an elementary inequality and some analysis technique. First, we investigated the Opial inequalities in (p, q)-calculus involving one function and its (p, q) derivative. Furthermore, Opial inequalities in (p, q)-calculus involving two functions and two functions with their (p, q) derivatives are given. Our results are (p, q)-generalizations of some known inequalities, such as Opial-type integral inequalities and (p, q)-Wirtinger inequality.
- (p, q)-derivative,
- (p, q)-integral,
- (p, q)-calculus,
- Opial inequality,
- Opial-type integral inequality
Citation: Chunhong Li, Dandan Yang, Chuanzhi Bai. Some Opial type inequalities in (p, q)-calculus[J]. AIMS Mathematics, 2020, 5(6): 5893-5902. doi: 10.3934/math.2020377

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Abstract

In this paper, we establish 5 kinds of integral Opial-type inequalities in (p, q)-calculus by means of H?lder's inequality, Cauchy inequality, an elementary inequality and some analysis technique. First, we investigated the Opial inequalities in (p, q)-calculus involving one function and its (p, q) derivative. Furthermore, Opial inequalities in (p, q)-calculus involving two functions and two functions with their (p, q) derivatives are given. Our results are (p, q)-generalizations of some known inequalities, such as Opial-type integral inequalities and (p, q)-Wirtinger inequality.

References

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