Related to my finance research, I am trying to find a non-trivial upper bound of \begin{equation*} \left( \frac{ \left(1 - r^{\frac{1}{K-1}}\right)K}{ \left(1 - r^{\frac{K}{K-1}} \right) } -1\right)\cdot \frac{r}{1-r}, \end{equation*} when we consider all $r \in (0,1)$ and $K$ being any integer weakly greater than $2$. (A numerical exercise suggests that it can be bounded from above by $0.5$, and the bound is achieved as $r$ goes to $1$.)
What I know is that it is non-negative and bounded from above by $1$, because it comes from the share of one trader's profit relative to the sum of profits of two or more traders, all of which are positive.
Thanks in advance for your help!