1
$\begingroup$

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.

  • 0
    I think another example is (x^c)logxlogx2010-12-16

2 Answers 2

6

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

  • 0
    I don't quite see how f is big O of x^(c+a) for all a>0. How does one show it?2010-11-23
0

Consider $f(x) = (\log x)^2$.