Log-convexity and the overpartition function

Abstract

AbstractLet $$\overline{p}(n)$$ 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 .