Let ${\overline{p}}(n)$ denote the overpartition function. In this paper, we study the asymptotic higher-order log-concavity property of the overpartition function in a similar framework done by Hou and Zhang for the partition function. This will enable us to move on further in order to prove log-concavity of overpartitions, explicitly by studying the asymptotic expansion of the quotient ${\overline{p}}(n-1){\overline{p}}(n+1)/{\overline{p}}(n)^2$ up to a certain order. This enables us to additionally prove 2-log-concavity and higher Turán inequalities with a unified approach.

Let $${\overline{p}}(n)$$ 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., $$(-1)^{r-1}\Delta ^r \log {\overline{p}}(n)$$ ( - 1 ) r - 1 Δ r log p ¯ ( n ) , by studying the inequality of the following form $$\begin{aligned} \log \Bigl (1+\dfrac{C(r)}{n^{r-1/2}}-\dfrac{1+C_1(r)}{n^{r}}\Bigr ){} & {} <(-1)^{r-1}\Delta ^r \log {\overline{p}}(n) \\ {}{} & {} <\log \Bigl (1+\dfrac{C(r)}{n^{r-1/2}}\Bigr )\ \text {for}\ n \ge N(r), \end{aligned}$$ 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 $$C(r), C_1(r), \text {and}\ N(r)$$ 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 $$(-1)^{r-1}\Delta ^r \log {\overline{p}}(n)$$ ( - 1 ) r - 1 Δ r log p ¯ ( n ) than 0. By settling the problem, we are able to show that $$\begin{aligned} \lim _{n\rightarrow \infty }(-1)^{r-1}\Delta ^r \log {\overline{p}}(n) =\dfrac{\pi }{2}\Bigl (\dfrac{1}{2}\Bigr )_{r-1}n^{\frac{1}{2}-r}. \end{aligned}$$ lim n → ∞ ( - 1 ) r - 1 Δ r log p ¯ ( n ) = π 2 ( 1 2 ) r - 1 n 1 2 - r .

Let $$\overline{p}(n)$$ p ¯ ( n ) denote the overpartition function. In this paper, we obtain an inequality for the sequence $$\Delta ^{2}\log \ \root n-1 \of {\overline{p}(n-1)/(n-1)^{\alpha }}$$ Δ 2 log p ¯ ( n - 1 ) / ( n - 1 ) α n - 1 which states that $$\begin{aligned}&\log \biggl (1+\frac{3\pi }{4n^{5/2}}-\frac{11+5\alpha }{n^{11/4}}\biggr )< \Delta ^{2} \log \ \root n-1 \of {\overline{p}(n-1)/(n-1)^{\alpha }}\\&< \log \biggl (1+\frac{3\pi }{4n^{5/2}}\biggr ) \ \ \text {for}\ n \ge N(\alpha ), \end{aligned}$$ 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 $$\alpha $$ α is a non-negative real number, $$N(\alpha )$$ N ( α ) is a positive integer depending on $$\alpha $$ α , and $$\Delta $$ Δ is the difference operator with respect to n. This inequality consequently implies $$\log $$ log -convexity of $$\bigl \{\root n \of {\overline{p}(n)/n}\bigr \}_{n \ge 19}$$ { p ¯ ( n ) / n n } n ≥ 19 and $$\bigl \{\root n \of {\overline{p}(n)}\bigr \}_{n \ge 4}$$ { p ¯ ( n ) n } n ≥ 4 . Moreover, it also establishes the asymptotic growth of $$\Delta ^{2} \log \ \root n-1 \of {\overline{p}(n-1)/(n-1)^{\alpha }}$$ Δ 2 log p ¯ ( n - 1 ) / ( n - 1 ) α n - 1 by showing $$\underset{n \rightarrow \infty }{\lim } \Delta ^{2} \log \ \root n \of {\overline{p}(n)/n^{\alpha }} = \dfrac{3 \pi }{4 n^{5/2}}.$$ lim n → ∞ Δ 2 log p ¯ ( n ) / n α n = 3 π 4 n 5 / 2 .