Log-convexity and the overpartition function.

Let p ¯ ( n ) denote the overpartition function. In this paper, we obtain an inequality for the sequence Δ 2 log p ¯ ( n - 1 ) / ( n - 1 ) α n - 1 which states that log ( 1 + 3 π 4 n 5 / 2 - 11 + 5 α n 11 / 4 ) < Δ 2 log p ¯ ( n - 1 ) / ( n - 1 ) α n - 1 < log ( 1 + 3 π 4 n 5 / 2 ) for n ≥ N ( α ) , where α is a non-negative real number, N ( α ) is a positive integer depending on α , and Δ is the difference operator with respect to n . This inequality consequently implies log -convexity of { p ¯ ( n ) / n n } n ≥ 19 and { p ¯ ( n ) n } n ≥ 4 . Moreover, it also establishes the asymptotic growth of Δ 2 log p ¯ ( n - 1 ) / ( n - 1 ) α n - 1 by showing lim n → ∞ Δ 2 log p ¯ ( n ) / n α n = 3 π 4 n 5 / 2 .