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 ≥ p(n-1)p(n+1), and satisfy the inequality: p(n)p(m) ≥ p(n+m) with only finitely many instances of equality or failure. This paper proves that this is no coincidence, that any log-concave sequence ₙ\ satisfying a particular initial condition likewise satisfies the inequality xₙxₘ ≥ xₙ₊ₘ. This paper further determines that these conditions are sufficient but not necessary and considers various examples to illuminate the situation.