Sufficient conditions for $f\cdot f^\prime$ to be lipschitz-continuous!

Question

Let $a>0$ (even sufficiently small) and $f:[-a,a]\to\mathbb{R}$ be a Lipschitz-continuous function such that $f^\prime$ is $\alpha$-Hölder continuous with $\alpha<1$ (so not lipschitzian). Without considering $f^{\prime\prime}$, there is any assumption on $f$ implying that the function $f\cdot f^\prime$ is lipschitz continuous? I'm making this question obviously because I doubt that $f\cdot f^\prime$ is automatically lipschitz continuous.