November 30, 2025Open Access

Two New Contributions to the Three-dimensional Hardy-Hilbert-type Integral Inequalities

Key Points

Integral inequalities are examined to yield significant theorems in mathematical analysis.
The findings include a remarkable instance of a maximum function in three-dimensional contexts, enriching existing knowledge.
These theorems are validated without using special functions, emphasizing straightforward proof methods.
Implications extend to various mathematical fields, calling for further exploration of integral inequalities.

Abstract

The Hardy-Hilbert integral inequality is one of the most celebrated results in mathematical analysis, inspiring numerous variants and extensions. In this paper, we further advance the study of three-dimensional Hardy-Hilbert-type integral inequalities by proving two new theorems. One of these is notable for its inclusion of a maximum function, a feature rarely encountered in this three-dimensional context. The associated constant factors are determined explicitly and detailed proofs are provided, without recourse to special functions.

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