Study of a hyperbolic sine modification of the Hilbert integral inequality

Abstract

In this article, we study a new modification of the classical Hilbert integral inequality based on the hyperbolic sine function. The resulting upper bound involves weighted integral norms of the main functions, with hyperbolic tangent weight functions.  Notably, the constant factor in this inequality remains the classical value π. From our main result, we derive two further propositions. The first proposition is the series analogue of the main integral inequality. The second proposition yields a new one-function integral inequality.