A property of $C^{k,\alpha }$ functions

Let $f$ be a nonnegative function of class $C^k$ ($k\ge 2$) such that $f^{(k)}$ is H''older continuous with exponent $\alpha$ in $(0,1]$. If $f^{\mathrm{\prime }}(x)=\cdots =f^{(k)}(x)=0$ when $f(x)=0$, we show that $f^{\mu }$ is differentiable for $\mu \in (1\mathrm{/}(k+\alpha ),1)$ and under an additional condition we show that $(f^{\mu }{)}^{\mathrm{\prime }}$ is H''older continuous with exponent $\beta =\mu (1+\alpha )-1$ (if $\beta \le 1$) at $x\in [0,T]$ when $f(x)=0$. $(f^{\mu }{)}^{\mathrm{\prime }}$ is Lipschitz continuous at $x$ if $f(x)>0$.