Annals of Communications in Mathematics

ABSTRACT.In this paper, we introduce the concept of ordered quasi-hyperideals of regular ordered semihypergroups, and study the basic results on ordered quasi-hyperideals of ordered semihypergroups. We also investigate regular ordered semihypergroups in terms of its ordered quasi-hyperideals, ordered right hyperideals and ordered left hyperideals. We prove that:(i) A partially ordered semihypergroup \( S \) is regular if and only if for every ordered bi-hyperideal \( B \), every ordered hyperideal \( I \) and every ordered quasi-hyperideal \( Q \), we have \[B \cap I \cap Q \subseteq (B \circ I \circ Q),\]and(ii) A partially ordered semihypergroup \( S \) is regular if and only if for every ordered quasi-hyperideal \( Q \), every ordered left hyperideal \( L \) and every ordered right-hyperideal \( R \), we have \[R \cap Q \cap L \subseteq (R \circ Q \circ L).\]

AbstractThe Hilbert integral inequality is a well-known and widely studied result in analysis that has inspired many refinements and modifications. In this paper, we present a new logarithmic modification of this inequality. Our approach is based on a trigonometric method that offers a fresh perspective on existing standard techniques. As a consequence, we also derive another integral inequality. All arguments are presented in full detail.