论文标题
循环总和的下限，在分母中具有单方面最大平均值
Lower bound for cyclic sums with one-sided maximal averages in denominators
论文作者
论文摘要
令$ \ mathbf {x} =（x_1，\ dots，x_n）$为正实数的$ n $ -tuple，序列$（x_i）_ {i \ in \ in \ mathbb {z}} $是$ n $ n $ - periodic extension。给定一个$ n $ -tuple $ \ mathbf {r} =（r_1，\ dots，r_n）$的正整数，让$ a_i $是$ x_ {i+1}的算术平均值，\ dots \ dots，x_ {i+r_i} $。我们形成了$ s_n（\ mathbf {x}，\ mathbf {r}）= \ sum_ {i = 1}^n x_i/a_ {i} $，遵循已长期研究的shapiro sums的模式，与所有$ r_i = 2 $ diananda us_ sums corpor come y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y ye y y y ye，我们找到$ \ mathbf {r} $ - 独立下限$ a_ {n，*} = \ inf _ {\ MathBf {r}} \ inf _ {\ mathbf {x}} s_n（s_n（x}，\ natif {x} $ n Is $ n y IT）， $ a_ {n，*} = e \ log n -a+o（1/\ log n）$。
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 $A_{n,*}=e\log n - A+O(1/\log n)$.
