Log-convexity and the overpartition function

Abstract

Let p(n) denote the overpartition function. In this paper, we obtain an inequality for the sequence ∆ 2 log n−1 p(n − 1)/(n − 1) α which states thatwhere α is a non-negative real number, N (α) is a positive integer depending on α and ∆ is the difference operator with respect to n. This inequality consequently implies log-convexity of n p(n)/n n≥19 and n p(n) n≥4 . Moreover, it also establishes the asymptotic growth of ∆ 2 log n−1 p(n − 1)/(n − 1) α by showing lim n→∞ ∆ 2 log n p(n)/n α = 3π 4n 5/2 .