Cumulative Riemann sums, distribution functions, and a universal inequality

TL;DR

This paper explores a universal inequality linking discrete Riemann sums and distribution functions, providing a unified perspective that connects classical inequalities, Riemann sums, and majorization theory.

Contribution

It introduces a general discrete theorem for monotone functions derived from a distribution-free continuous identity, unifying various classical inequalities.

Findings

Establishes a universal inequality for decreasing functions and Riemann sums.

Connects discrete inequalities with distribution functions and probability transforms.

Discusses links with majorization theory and Karamata's 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.