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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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But $m$ is a fixed number. Also the final estimate seems to be very "slow" . I'm working in some ideas....tkx anyway! – Dubglass Aug 3 '12 at 13:45
I didn't realize that $m$ is supposed to be fixed. I'll update my answer when I get a chance. – Jonathan Aug 3 '12 at 14:12

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