Given an integer $k$ and $0\leq \alpha \leq 1$, let $f_1(\alpha)=1/k$ and $f_{i+1}(\alpha)=\frac{(k-1)f_i(\alpha) + (f_i(\alpha)^{1/\alpha} + 1)^\alpha}{k}$.
Consider the function $g(\alpha) = \lim_{k\rightarrow +\infty} f_k(\alpha)$. It can be easily verified that $g(1) = 1$, which follows since $f_k(1) = 1$. Similarly, $g(0) = 1-1/e$, which is shown by the fact that $f_k(0) = 1-(1-1/k)^k$.
My question is: can $g(\alpha)$ be expressed in a closed form in terms of $\alpha$?
Following is a plot of $g(\alpha)$ using MATLAB simulation.
