Let p ¯ ( n ) 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., ( - 1 ) r - 1 Δ r log p ¯ ( n ) , by studying the inequality of the following form log ( 1 + C ( r ) n r - 1 / 2 - 1 + C 1 ( r ) n r ) < ( - 1 ) r - 1 Δ r log p ¯ ( n ) < log ( 1 + C ( r ) n r - 1 / 2 ) for n ≥ N ( r ) , where C ( r ) , C 1 ( r ) , and N ( r ) 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 ( - 1 ) r - 1 Δ r log p ¯ ( n ) than 0. By settling the problem, we are able to show that lim n → ∞ ( - 1 ) r - 1 Δ r log p ¯ ( n ) = π 2 ( 1 2 ) r - 1 n 1 2 - r . .
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