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.
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.
Consider $f(x) = x^c (\log x)^b$, where $b>1$. This function is $O(x^{(c+a)})$ but it is not $O(x^c \log x)$.
Consider $f(x) = (\log x)^2$.