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Let $c\in(0,1)$, $m\geq 1$ be positive integer and $\{a_{n}\}$ a decreasing sequence of positive real numbers. Suppose that $a_{n^{m}}\leq K c^{n}n^{-m/2}, \forall n\in\mathbb{N}, $for some $K>0$. How to describe the behavior of the whole sequence $\{a_{n}\}$?

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Since the sequence is decreasing, $ a_i \leq a_j $ for $j \leq i$. Putting in the inequality given, we have $ a_i \leq K \min_{n^m \leq i} c^n n^{-m/2}. $ For fixed $m$, we can take $n = \lfloor i^\frac{1}{m} \rfloor$, and plugging this into the inequality yields $ a_i \leq K \frac{c^{\lfloor i^\frac{1}{m} \rfloor}}{\lfloor i^\frac{1}{m} \rfloor^{m/2}}. $

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    I didn't realize that $m$ is supposed to be fixed. I'll update my answer when I get a chance.2012-08-03