H(X) vs. H(f(X))

It is well known that the entropy $H(X)$ of a finite random variable is always greater or equal to the entropy $H(f(X))$ of a function $f$ of $X$, with equality if and only if $f$ is one-to-one. In this paper, we give tights bounds on $H(f(X))$ when the function $f$ is not one-to-one, and we illustrate a few scenarios where this matters. As an intermediate step towards our main result, we prove a lower bound on the entropy of a probability distribution, when only a bound on the ratio between the maximum and the minimum probability is known. Our lower bound improves previous results in the literature, and it could find applications outside the present scenario.