Parameterized Yang-Hilbert-Type Integral Inequalities and Their Operator Expressions

TL;DR

This paper develops parameterized Yang-Hilbert-type integral inequalities with optimal constants, explores their equivalent forms and operator expressions, and applies them to Hardy-type inequalities and operators, with various examples.

Contribution

It introduces new parameterized inequalities with best constants, including their operator forms and applications to Hardy-type inequalities, expanding the theoretical framework.

Findings

Derived two parameterized inequalities with optimal constants

Established operator expressions and equivalent forms

Provided examples with specific kernels

Abstract

Applying methods of Real Analysis and Functional Analysis, we build two weight functions with parameters and provide two kinds of parameterized Yang-Hilbert-type integral inequalities with the best constant factors. Equivalent forms, the reverses, and the operator expressions are also given. In particular, the Hardy-type inequalities and Hardy-type operators are studied. Additionally, a number of examples with two kinds of particular kernels are considered.