An inequality involving alternating binomial sums

TL;DR

This paper proves a new inequality involving alternating binomial sums using the variance of the logarithm of the maximum of i.i.d. exponential variables, building on previous work related to Stirling numbers and collecting processes.

Contribution

It introduces a novel inequality involving alternating binomial sums and logarithmic sums, derived through probabilistic methods, expanding the theoretical understanding of such sums.

Findings

Established a new inequality involving alternating binomial sums

Connected the inequality to the variance of the maximum of exponential variables

Extended previous work on Stirling numbers and collecting processes

Abstract

In this letter, we prove an inequality involving alternating binomial logarithmic sums by exploiting the variance of the logarithm of the maximum of independent and identically distributed exponential random variables. This inequality was introduced in our recent work [On the minimum of independent collecting processes via the Stirling numbers of the second kind, Statist. Probab. Lett., 185 (2022)].