A Modification of Schwarz's Inequality with Applications to Distributions

Abstract

Theorem 1 provides us with a result from which we can derive the modified Schwarz inequality (1.2) or, more generally, (3.8). The formulas hold when $x(t)$ is any nondecreasing function belonging to a certain wide class, and $\bar{\varphi}(t)$ is the right-hand derivative of the "greatest convex minorant" of $\Phi(t)$. The necessary and sufficient conditions for equality to hold are also given. Applications to distribution problems in statistics are discussed in Section 4.