I'm interested in computing the proximal operator for the ratio of $\ell_1, \ell_2$ norms $$ \text{prox}_{\ell_1/\ell_2}(x) = \arg\min_{z} \frac{1}{2}\|z-x\|^2 + \frac{\|z\|_1}{\|z\|_2}. $$ Does this have a closed form expression? If not, are there any fast computational techniques that can be used to solve this optimization problem?
P.S: This norm comes up in sparse signal recovery and seems to have some nice properties.