How to prove the $\alpha$ th derivative of $f(y) = y^{\alpha + \gamma} (1-y)^{\alpha + \gamma}$ is Holder's continuous w.r.t. $\gamma$

Question

Given $\alpha$ is an integer and $\gamma \in (0, 1]$, $y \in [0,1]$, I'd like to prove $|f^{(\alpha)}(y_1) - f^{(\alpha)}(y_2)| \leq L|y_1 - y_2|^\gamma$, where $f^{(\alpha)}(y)$ is the $\alpha$ th derivative of $f(y) = y^{\alpha + \gamma} (1-y)^{\alpha + \gamma}$.