  We show that the metric entropy of a $C^1$ diffeomorphism with a dominated
splitting and the dominating bundle uniformly expanding is bounded from above
by the integrated volume growth of the dominating (expanding) bundle plus the
maximal Lyapunov exponent from the dominated bundle multiplied by its
dimension. We also discuss different types of volume growth that can be
associated with an expanding foliation and relationships between them,
specially when there exists a closed form non-degenerate on the foliation. Some
consequences for partially hyperbolic diffeomorphisms are presented.
