Log-convexity and the overpartition function

Let [Formula: see text] denote the overpartition function. In this paper, we obtain an inequality for the sequence [Formula: see text] which states that [Formula: see text] where [Formula: see text] is a non-negative real number, [Formula: see text] is a positive integer depending on [Formula: see text] , and [Formula: see text] is the difference operator with respect to n. This inequality consequently implies [Formula: see text] -convexity of [Formula: see text] and [Formula: see text] . Moreover, it also establishes the asymptotic growth of [Formula: see text] by showing [Formula: see text]