Concavity of a function

Question

While I am reading a book I couldn't follow the following step.

" By concavity of the function $x \sqrt{\log\frac{1}{x}}$ for $x \in (0,1)$ we have that "

$O(x \sqrt{\log\frac{k}{x}})$ = $O(\sqrt{\log k})$

Can some one help me out.

Answer 1

The book just means that the function has a maximum because it is concave, and therefore there is a suitable constant $c$ such that $\left|x\sqrt{\log(\frac{k}{x})}\right| < c \sqrt{\log(k)}$ for any $x \in (0,1)$. I don't understand why it has to make things so complicated though... All you need for the function to be bounded on $(0,1)$ is for it to be continuous on $(0,1)$ and have limits at $0^+$ and $1^-$.