Generalized Čebyšev and Grüss Type Results in Weighted Lebesgue Spaces

Abstract

The classical Grüss and related inequalities have spurred a range of improvements, refinements, generalizations, and extensions. In the present article, we provide generalizations of Sokolov’s inequality in weighted Lebesgue LωΩ,A,μ spaces by employing the weighted Sonin’s identity and Čebyšev functional. As a result, we provide a generalized Grüss inequality in which the bounding constants are improved with bounding functions in LωpΩ,A,μ spaces. As an application, we provide several new bounds for Jensen–Grüss type differences.