论文标题
log-convexity和分区功能
Log-convexity and the overpartition function
论文作者
论文摘要
令$ \ overline {p}（n）$表示分支机构功能。在本文中，我们获得了序列$δ^{2} \ log \ \ \ \ sqrt [n-1] {\ edimelline {p}（n-1）/（n-1）/（n-1）^α} $的不等式的不等式\ bigGl（1+ \ frac {3π} {4n^{5/2}} - \ frac {11+5α} {n^{11/4}} \ biggr）<Δ^{2} {2} \ log \ \ \ \ \ \ \ \ \ \ \ \ \ \ sqrt [n-1] \ bigGl（1+ \ frac {3π} {4n^{5/2}} \ biggr）\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ geq n（α），\ end end {equation {qore {qore*} $α$，其中$α$ and Innegative norke and $ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n（$α）尊重$ n $。因此，这种不平等意味着$ \ log $ -convexity的$ \ bigl \ {\ sqrt [n] {\ overline {p}（n）/n} \ bigr \} _ {n \ geq 19} $ $ \ bigl \ {\ sqrt [n] {\ overline {p}（n）} \ bigr \} _ {n \ geq 4} $。此外，它还建立了$δ^{2} \ log \ \ \ \ sqrt [n-1] {\ overline {p}（n-1）/（n-1）/（n-1）^α} $的渐近生长\ sqrt [n] {\ overline {p}（n）/n^α} = \ dfrac {3π} {4 n n^{5/2}}}。
Let $\overline{p}(n)$ denote the overpartition function. In this paper, we obtain an inequality for the sequence $Δ^{2}\log \ \sqrt[n-1]{\overline{p}(n-1)/(n-1)^α}$ which states that \begin{equation*} \log \biggl(1+\frac{3π}{4n^{5/2}}-\frac{11+5α}{n^{11/4}}\biggr) < Δ^{2} \log \ \sqrt[n-1]{\overline{p}(n-1)/(n-1)^α} < \log \biggl(1+\frac{3π}{4n^{5/2}}\biggr) \ \ \text{for}\ n \geq N(α), \end{equation*} where $α$ is a non-negative real number, $N(α)$ is a positive integer depending on $α$ and $Δ$ is the difference operator with respect to $n$. This inequality consequently implies $\log$-convexity of $\bigl\{\sqrt[n]{\overline{p}(n)/n}\bigr\}_{n \geq 19}$ and $\bigl\{\sqrt[n]{\overline{p}(n)}\bigr\}_{n \geq 4}$. Moreover, it also establishes the asymptotic growth of $Δ^{2} \log \ \sqrt[n-1]{\overline{p}(n-1)/(n-1)^α}$ by showing $\underset{n \rightarrow \infty}{\lim} Δ^{2} \log \ \sqrt[n]{\overline{p}(n)/n^α} = \dfrac{3 π}{4 n^{5/2}}.$
