Log-concavity And The Multiplicative Properties of Restricted Partition Functions

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.