Let $f:\mathbb{R}^n\to\mathbb{R}$ such that $f$ is $L$-smooth, i.e. $\|\nabla f(y) - \nabla f(x)\| \leq L\|y-x\|$ with the usual Euclidean norm.
Define $f_\gamma:\mathbb{R}^n\to\mathbb{R}$ such that $f_\gamma(x) = \frac{1}{\gamma}e^{\gamma f(x)}$, $\gamma > 0$. Then, $\nabla f_\gamma(x) = e^{\gamma f(x)}\nabla f(x)$.
I'm wondering what the Lipschitz constant of $\nabla f_\gamma(x)$ would be, i.e. what is $L_\gamma$ such that $\|\nabla f_\gamma(y) - \nabla f_\gamma(x)\| \leq L_\gamma\|y-x\|$. Tried quite a few different things but the presence of $e^{\gamma f(x)}$ in the gradient ruins my approaches.