Refined Weighted Schweitzer-Type Inequalities With Applications to Statistical Moments

In this paper, we establish a weighted version of the classical Schweitzer integral inequality for positive bounded functions. We first derive a sharp upper bound for the product of two weighted integrals involving a function and its reciprocal. We then obtain a refined form of the inequality containing an explicit nonnegative remainder term, which provides additional information on the deviation from the extremal case. Applications to probability theory are also presented, demonstrating the usefulness of the obtained inequalities in statistical analysis. In addition, a connection between the obtained estimates and statistical dispersion is discussed through a variance-type interpretation.