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I was wondering if $f(x)=O(x^{c+a})$ for all $a>0$ then is it necessarily true that $f(x)=O(x^c\log x)$? I suspect it's not true but want to know why. (I know the converse is true.)

Any help is much appreciated, Thank you.

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    I think another example is (x^c)logxlogx2010-12-16

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