The weighted $l_1$ norm minimization can enhance the sparsity performance in comparison with traditional $l_1$ norm minimization, that is $$ \min_{x\in \mathbb{R}^{n\times n}} ||wx||_1 > \min_{x\in \mathbb{R}^{n\times n}} ||x||_1, $$
where $A>B$ denotes that the entry $A$ has much more sparsity than the entry $B$.
How can I prove this in case the weights are from the derivative of the function $\phi(x)=\frac{x}{x+\epsilon}$, $\epsilon>0$?
Thank you.