arXiv 1711.034932017-11math.CA

关于weighted means的Kedlaya型不等式

On Kedlaya type inequalities for weighted means

Zsolt Páles, Paweł Pasteczka

备注：: J. Inequal. Appl. (2018)
发表信息：: J. Inequal. Appl. (2018), no. 99
DOI：: 10.1186/s13660-018-1685-z

摘要：

在2016年，我们证明了对于每个对称的，重复不变的和Jensen凹平均$\mathscr{M}$，Kedlaya型不等式$$ \mathscr{A}\big(x_1,\mathscr{M}(x_1,x_2),\ldots,\mathscr{M}(x_1,\ldots,x_n)\big)\le \mathscr{M} \big(x_1, \mathscr{A}(x_1,x_2),\ldots,\mathscr{A}(x_1,\ldots,x_n)\big) $$对任意$(x_n)$成立（$\mathscr{A}$代表算术平均）。我们将证明这个不等式的加权对应。更准确地说，如果$(x_n)$是具有对应（非归一化）权重$(λ_n)$的向量，并且$\mathscr{M}_{i=1}^n(x_i,λ_i)$表示加权平均值，则在$\mathscr{M}$上的类似条件下，不等式$$ \mathscr{A}_{i=1}^n \big(\mathscr{M}_{j=1}^i (x_j,λ_j),\:λ_i\big) \le \mathscr{M}_{i=1}^n \big(\mathscr{A}_{j=1}^i (x_j,λ_j),\:λ_i\big) $$对于每个$(x_n)$和$(λ_n)$成立，使得序列$(\frac{λ_k}{λ_1+\cdots+λ_k})$递减。

英文摘要：

In 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean $\mathscr{M}$ the Kedlaya-type inequality $$ \mathscr{A}\big(x_1,\mathscr{M}(x_1,x_2),\ldots,\mathscr{M}(x_1,\ldots,x_n)\big)\le \mathscr{M} \big(x_1, \mathscr{A}(x_1,x_2),\ldots,\mathscr{A}(x_1,\ldots,x_n)\big) $$ holds for an arbitrary $(x_n)$ ($\mathscr{A}$ stands for the arithmetic mean). We are going to prove the weighted counterpart of this inequality. More precisely, if $(x_n)$ is a vector with corresponding (non-normalized) weights $(λ_n)$ and $\mathscr{M}_{i=1}^n(x_i,λ_i)$ denotes the weighted mean then, under analogous conditions on $\mathscr{M}$, the inequality $$ \mathscr{A}_{i=1}^n \big(\mathscr{M}_{j=1}^i (x_j,λ_j),\:λ_i\big) \le \mathscr{M}_{i=1}^n \big(\mathscr{A}_{j=1}^i (x_j,λ_j),\:λ_i\big) $$ holds for every $(x_n)$ and $(λ_n)$ such that the sequence $(\frac{λ_k}{λ_1+\cdots+λ_k})$ is decreasing.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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