Let $S$ be a higher genus surface, let $f\colon S\to S$ be a diffeomorphism and let $f_*\colon H_1(M)\to H_1(M)$ be the induced homology automorphism. 
Define dilatation of $f_*$ as the largest absolute value of eigenvalues of $f_*$. Fix a negatively curved metric $g$ on $S$. (Say, $g$ is a hyperbolic metric.)

Question: Is it true that for any $\varepsilon>0$ there exists a diffeomorphism $h$ homotopic to $f$ such that 
$$
dil(h,g)<dil(f_*)+\varepsilon?
$$
If not, what can one say about the $inf_h(dil(h,g)-dil(f_*))$?

Here the dilatation of $h$ with respect to $g$ is defined as
$$
dil(h,g)=max_{v\in TS}\frac{\|Dhv\|_g}{\|v\|_g}.
$$