I am sorry for not being elaborate enough about my earlier question (Jacob Manaker, thank you for letting me know about the clarity issue). Let me try to be more specific and state the complete version of my problem. I am striving to figure out under what type of distributional assumptions the following expression (integrations are taken from the lower bound to upper bounds of a random variable) will be negative or positive.
\begin{equation*} \frac{2kn-1}{kn\left( kn-1\right) }+\frac{\int F^{kn-2}(z)\ln (F(z))\left( f(z)\right) ^{2}dz}{\int F^{kn-2}(z)\left( f(z)\right) ^{2}dz}\text{,} \end{equation*} where $k\geq 1$ is the scale factor, $n\geq 2$ is the number of players that are assumed to be identical, and $f(.)$ and $F(.)$ denote the PDF and CDF of some continuous distribution (possibly supported on a semi-infinite or infinite interval). I suspect that the answer has something to do with the hazard rate. In fact, I have established that if the distribution is exponential (i.e., constant hazard rate), the condition simply takes the value of zero. Of course, it can be related to other obvious distributional characteristics, which I may be overlooking.
Note: Please let me know if anything is still unclear. I appreciate any help you can provide.