Weak maps and the Tutte Polynomial

Let $M$ and $N$ be matroids such that $N$ is the image of $M$ under a rank-preserving weak map. Generalizing results of Lucas, we prove that, for $x$ and $y$ positive, $T(M;x,y)\ge T(N;x,y)$ if and only if $x+y\ge xy$ or $M\cong N$. We give a number of consequences of this result.