Let $F(\theta)$ be an arbitrary, strictly increasing and twice differentiable CDF that is defined on the interval $[0, \overline{\theta}]$, where $\overline{\theta}$ may be infinite. Moreover, let $\mathbb{E}(\theta) = 1$.
Let $N \geq 2$ be a natural number, and $\delta \in (0,1)$ real.
I am looking for criteria on $F$ that guarantee a solution $k$ for
$(N-1)[1-\delta+\delta F(N k)] - k \delta F'(N k) =0$, where $k$ can be any real positive number not larger than $\overline{\theta}$.
So far, I just have the following result. It is clear that the left term, $(N-1)[1-\delta+\delta F(N k)]$, is strictly increasing in $k$ and bounded above by $N-1$. Hence, if the probability mass is sufficiencly "concentrated" in some interval (implying that $f(.)$ is large in that interval), a solution $k$ must exist by continuity of $f$ and $F$.
However, it would be nice to have some sharper conditions on $F$ that guarantee a solution. Ideally, I'd wish to have a result that gives an upper bound on $F$'s variance, or similar.
I would greatly appreciate any input or ideas that might lead to a better understanding. Many thanks in advance!
