Condition for the Construction of a Hilbert-Type Integral Inequality Involving Upper Limit Functions

Hilbert-type integral inequalities, which feature a symmetric structure, are a significant class of inequalities with broad applications in the study of operator theory. Hilbert-type integral inequalities involving variable upper limit integral functions are a generalized form of the classical Hilbert-type integral inequalities. In this paper, we employ the construction theorem of homogeneous kernel Hilbert-type integral inequalities and the properties of the Gamma function to discuss the conditions for constructing a Hilbert-type integral inequality involving an upper limit integral function and the best constant factor. We derive the necessary and sufficient conditions for constructing this inequality and a formula for calculating the best constant factor, thereby improving the existing results. Finally, as an application, we consider some special cases of parameters.