Constructive Conditions for a High-Dimensional Hilbert-Type Integral Inequality Involving Multivariate Variable Upper Limit Integral Functions and Optimal Constant Factors

Abstract

Hilbert integral inequalities are beautiful inequalities with a symmetric structure, and have attracted much attention because of their important applications in the study of integral operators, and the Hilbert-type integral inequality involving variable upper limit integral functions is a generalized form of the Hilbert integral inequality. In this paper, we use a construction theorem of the Hilbert-type n-ple integral inequality with homogeneous kernels to discuss a high-dimensional Hilbert-type integral inequality involving multivariate variable upper limit integral functions, and obtain the sufficiently necessary conditions for constructing inequalities and the formulas for optimal constant factors, which improve and generalize the existing results.

Keywords

• high-dimensional Hilbert-type integral inequality
• multivariate variable upper limit integral function
• homogeneous kernel
• Gamma function
• optimal constant factor