I'm trying to find out if there is some sort of functional relationship between $\displaystyle \pi$ and $\displaystyle \psi$ such that $\displaystyle \sum\limits_{k=1}^{G}{{{\pi }_{k}}\left[ 1-{{\left( 1-{{\psi }_{k}} \right)}^{h}} \right]}=\frac{h}{b+h}\$. In other words, under what conditions (i.e. relationship between $\displaystyle \pi$ and $\displaystyle \psi$ would this equation be true?

With some manipulating, $\displaystyle \sum\limits_{k=1}^{G}{{{\pi }_{k}}\left( 1-{{e}^{-{{C}_{k}}h}} \right)}\text{ where }{{C}_{k}}=-\log \left( 1-{{\psi }_{k}} \right)\

$, and also $\displaystyle \exp (-\log (a))=\frac{1}{a}\$, but I'm having trouble figuring out how to move forward from there. The left-hand side can be expressed in the form of a dot-product as well. This isn't math homework, it's biological research - so my apologies if the framing isn't particularly clear.

Thanks in advance for your ideas and comments.