  We study entropy numbers and box dimension of (the image of) homogeneous
polynomials and holomorphic functions between Banach spaces. First, we see that
entropy numbers and box dimensions of subsets of Banach spaces are related. We
show that the box dimension of the image of a ball under a homogeneous
polynomial is finite if and only if it spans a finite-dimensional subspace, but
this is not true for holomorphic functions. Furthermore, we relate the entropy
numbers of a holomorphic function to those of the polynomials of its Taylor
series expansion. As a consequence, if the box dimension of the image of a ball
by a holomorphic function $f$ is finite, then the entropy numbers of the
polynomials in the Taylor series expansion of $f$ at any point of the ball
belong to $\ell_p$ for every $p>1$.
