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 \xₙ\ satisfying a particular initial condition likewise satisfies the inequality xₙxₘ x₍+₌. This paper further determines that these conditions are sufficient but not necessary and considers various examples to illuminate the situation.