Lower bound for cyclic sums with one-sided maximal averages in denominators
| Autor: | Sadov, Sergey |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $\mathbf{x}=(x_1,\dots,x_n)$ be an $n$-tuple of positive real numbers and the sequence $(x_i)_{i\in\mathbb{Z}}$ be its $n$-periodic extension. Given an $n$-tuple $\mathbf{r}=(r_1,\dots,r_n)$ of positive integers, let $a_i$ be the arithmetic mean of $x_{i+1},\dots,x_{i+r_i}$. We form the cyclic sums $S_n(\mathbf{x},\mathbf{r})=\sum_{i=1}^n x_i/a_{i}$, following the pattern of the long studied Shapiro sums, which correspond to all $r_i=2$, and more general Diananda sums, where all $r_i$ are equal. We find the asymptotics of the $\mathbf{r}$-independent lower bounds $A_{n,*}=\inf_{\mathbf{r}}\inf_{\mathbf{x}} S_n(\mathbf{x},\mathbf{r})$ as $n\to\infty$: it is $A_{n,*}=e\log n - A+O(1/\log n)$. Comment: 21 pages, 1 table, 1 figure |
| Databáze: | arXiv |
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