A Hardy–Hilbert-type integral inequality involving two multiple upper-limit functions

Abstract

Abstract By means of the weight functions, the idea of introducing parameters and the technique of real analysis, a new Hardy–Hilbert-type integral inequality with the homogeneous kernel 1 ( x + y ) λ ( λ > 0 ) $\frac{1}{(x + y)^{\lambda}}\ (\lambda > 0)$ involving two multiple upper-limit functions is obtained. The equivalent statements of the best possible constant factor related to the beta and gamma functions are considered. As applications, the equivalent forms and the case of a nonhomogeneous kernel are deduced. Some particular inequalities and the operator expressions are provided.

Keywords

• Weight function
• Hardy–Hilbert-type integral inequality
• Multiple upper-limit function
• Parameter
• Gamma function
• Beta function