Suppose that $X$ is a vector space with two norms $\|\cdot\|_a$ and $\|\cdot\|_b$. The mapping 
$$
  f(x) = \frac{\|x\|_{a}}{\|x\|_{b}} x
$$
is a radial and maps the unit ball of $(X,\|\cdot\|_a)$ onto the unit ball of $(X,\|\cdot\|_b)$.

We can assume that the two norms are equivalent. 

What is the smallest Lipschitz constant of $f$? That is, what is the smallest constant $L \geq 0$ in the following inequality:
$$
	\| f(x) - f(y) \|_b \leq L \| x - y \|_a
$$
It is possible to get an estimate in terms of the equivalence constants of the norms (and along rays from the origin, it is just an equivalence constant) but I am explicitly looking for the most tight estimates known.