For $n$ a positive integer at least equal to $2$, define the two following functions as follows : $ \begin{align*} f_{n}(x) & = \pi/4 + \arctan(\sqrt[n]{x})-\arctan(1/x) \text{ for all nonzero }x, \\ g_{n}(x) & = x^{n+1} +x^n+x-1 \text{ for all real }x. \end{align*} $ We can find, using differentiation for instance, that these two functions have each one unique real root and, using the intermediate value theorem, that it lies in the interval $(0,1)$. If we define the sequence $u_{n}$ as the real root of $f_{n}$ and $v_{n}$ as the real root of $g_{n}$, my question is to prove that the equality $u_{n}=v_{n}^n$ holds for all $n\geq 2$.
EDITED : I corrected the statement from $g_{n}(x) = x^{n+1} +x^4+x-1$ to $g_{n}(x) = x^{n+1} +x^n+x-1$.