Let $l \in (0,1)$, $w>0$, $K$ and $T$ positive integers.
The function $f(x) = (1-l x)^K(1+w x)^{(T-K)}$, restricted to $x \in [0,1]$ is then logconcave in x. Thus, we can separate the domain into regions where the function is increasing/decreasing.
Let $x^*$ denote the maximum of this function. Then, if $x<x^*$, there is an inverse function $g_{<x^*}(y)$. Analogously, for $x>x^*$, we have the inverse $g_{>x^*}(y)$.
My question: is there a representation of this function in terms of special functions, such as Beta, Gamma, etc?
The context is that I want to do a change of variables inside the integral:
$\int_{0}^1\psi(f(x))dx$
and write it as something like:
$\int_{g_{<x^*}(0)}^{g_{<x^*}(x^*)}\psi(y)g'_{<x^*}(y)dy$ + $\int_{g_{>x^*}(x^*)}^{g_{>x^*}(1)}\psi(y)g'_{>x^*}(y)dy$