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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2401.05522 |
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| _version_ | 1866914637303250944 |
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| author | Mukherjee, Gargi |
| author_facet | Mukherjee, Gargi |
| contents | Let $\overline{p}(n)$ denote the overpartition function. In this paper, we study the asymptotic growth of finite difference of logarithm of $\sqrt[n]{\overline{p}(n)/n^α}$ for $α$ being a non-negative real number, namely $(-1)^{r}Δ^r \log \sqrt[n]{\overline{p}(n)/n^α}$ by presenting an inequality of it with a symmetric upper and lower bound. Consequently, we arrive at log-convexity of $\sqrt[n]{\overline{p}(n)}$ and $\sqrt[n]{\overline{p}(n)/n}$, previously studied by the author. The another main objective of this paper is to introduce the notion of the reverse higher order Turán inequalities and we prove this for $\sqrt[n]{\overline{p}(n)/n^α}$, which not only generalize the study of Sun, Chen, and Zheng but also depicts the non real-rootedness of the Jensen polynomial associated with the sequence mentioned before. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05522 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic Growth of $(-1)^{r} Δ^r \log \sqrt[n]{\overline{p}(n)/n^α}$ and the Reverse Higher Order Turán Inequalities for $\sqrt[n]{\overline{p}(n)/n^α}$ Mukherjee, Gargi Number Theory Let $\overline{p}(n)$ denote the overpartition function. In this paper, we study the asymptotic growth of finite difference of logarithm of $\sqrt[n]{\overline{p}(n)/n^α}$ for $α$ being a non-negative real number, namely $(-1)^{r}Δ^r \log \sqrt[n]{\overline{p}(n)/n^α}$ by presenting an inequality of it with a symmetric upper and lower bound. Consequently, we arrive at log-convexity of $\sqrt[n]{\overline{p}(n)}$ and $\sqrt[n]{\overline{p}(n)/n}$, previously studied by the author. The another main objective of this paper is to introduce the notion of the reverse higher order Turán inequalities and we prove this for $\sqrt[n]{\overline{p}(n)/n^α}$, which not only generalize the study of Sun, Chen, and Zheng but also depicts the non real-rootedness of the Jensen polynomial associated with the sequence mentioned before. |
| title | Asymptotic Growth of $(-1)^{r} Δ^r \log \sqrt[n]{\overline{p}(n)/n^α}$ and the Reverse Higher Order Turán Inequalities for $\sqrt[n]{\overline{p}(n)/n^α}$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2401.05522 |