Inequalities for higher order differences of the logarithm of the overpartition function and a problem of Wang-Xie-Zhang (Q2679118)

In this article, the author gives an affirmative solution to a problem proposed by \textit{L. X. W. Wang} et al. [Adv. Appl. Math. 92, 51--72 (2018; Zbl 1375.05026)], which state as ``Does there exist a positive number \(A\) such that \(n^{r-1/2}(-1)^{r-1}\Delta^{r}\log\overline{p}(n)>{A}\), for any \(r\ge{1}\) and all sufficiently large \(n\)?'' The solution is given in form of theorems \(1.6\) and \(1.8\). Also, the author shows that \(\lim_{n\to{\infty}}(-1)^{r-1}\Delta^{r}\log\overline{p}(n)=\frac{\pi}{2}(\frac{1}{2})_{r-1} n^{\frac{1}{2}-r}\). All the results are explained properly with supporting arguments.