# Weighted Gehring and Muckenhoupt classes and some inclusion properties with norm inequalities

Volume 24, Issue 3, pp 201--215
Publication Date: February 23, 2021 Submission Date: December 13, 2020 Revision Date: January 01, 2021 Accteptance Date: January 06, 2021
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### Authors

S. H. Saker - Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. M. H. Hassan - Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.

### Abstract

In this paper, we will prove some fundamental properties of the power mean operator $\mathcal{H}_{\lambda }w^{p}$ of order $p$, which is defined by% $\mathcal{H}_{\lambda }w^{p}\left( x\right) =\frac{1}{\Lambda \left( x\right) }\int_{0}^{x}\lambda \left( s\right) w^{p}\left( s\right) ds,\text{ for }% p\in \mathbb{R}^{+}\text{,}$ where $\lambda$ and $w$ are nonnegative functions and $\Lambda \left( x\right) =\int_{0}^{x}\lambda \left( s\right) ds$. Then by using these properties we will establish some norm inequalities of the generalized Muckenhoupt and Gehring weights and prove some fundamental relations between them.

### Share and Cite

##### ISRP Style

S. H. Saker, M. H. Hassan, Weighted Gehring and Muckenhoupt classes and some inclusion properties with norm inequalities, Journal of Mathematics and Computer Science, 24 (2022), no. 3, 201--215

##### AMA Style

Saker S. H., Hassan M. H., Weighted Gehring and Muckenhoupt classes and some inclusion properties with norm inequalities. J Math Comput SCI-JM. (2022); 24(3):201--215

##### Chicago/Turabian Style

Saker, S. H., Hassan, M. H.. "Weighted Gehring and Muckenhoupt classes and some inclusion properties with norm inequalities." Journal of Mathematics and Computer Science, 24, no. 3 (2022): 201--215

### Keywords

• Hardy type inequality
• generalized Muckenhoupt class
• generalized Gehring class
• inclusion properties

•  26D10
•  26D15

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