Inequalities for higher order differences of the logarithm of the overpartition function and a problem of Wang–Xie–Zhang

Let [Formula: see text] denote the overpartition function. In this paper, our primary goal is to study the asymptotic behavior of the finite differences of the logarithm of the overpartition function, i.e., [Formula: see text] , by studying the inequality of the following form [Formula: see text] where [Formula: see text] are computable constants depending on the positive integer r, determined explicitly. This solves a problem posed by Wang, Xie and Zhang in the context of searching for a better lower bound of [Formula: see text] than 0. By settling the problem, we are able to show that [Formula: see text]