Cumulative Riemann sums, distribution functions, and a universal inequality

Abstract

We study discrete expressions of the form $$ T_n(g)=\sum_{i=1}^n a_i g(S_i), \qquad S_i=\sum_{j=1}^i a_j, $$ where $a_i>0$ and $\sum_{i=1}^n a_i=1$. If $g:[0,1]\to\mathbb{R}$ is a decreasing integrable function, we have $$ \sum_{i=1}^n a_i g(S_i) \le \int_0^1 g(x)\,dx, $$ from which classical inequalities can be obtained, for instance for the choice $g(x)=1-x^k$. Although elementary, this inequality admits a natural interpretation in terms of Riemann sums, Abel summation, and the probability integral transform. The aim of this paper is to present a unified perspective emphasizing that the discrete inequality is a consequence of a distribution-free continuous identity. Beyond the specific example, we establish a general discrete theorem for monotone functions and discuss connections with majorization theory and Karamata's inequality.