Log-concavity And The Multiplicative Properties of Restricted Partition Functions

Abstract

The partition function $p(n)$ and many of its related restricted partition functions have recently been shown independently to satisfy log-concavity: $p(n{)}^2\ge p(n-1)p(n+1)$ for $n\ge 26$, and satisfy the inequality: $p(n)p(m)\ge p(n+m)$ for $n\ge m\ge 2$ with only finitely many instances of equality or failure. This paper proves that this is no coincidence, that any log-concave sequence $\{x_n\}$ satisfying a particular initial condition likewise satisfies the inequality $x_n{x}_m\ge x_{n+m}$. This paper further determines that these conditions are sufficient but not necessary and considers various examples to illuminate the situation.

results of this article were first reported on July, 2023. In this version we have updated our main theorem and provided new examples