  We consider transfer operators acting on spaces of holomorphic functions, and
provide explicit bounds for their eigenvalues. More precisely, if D is any open
set in C^d, and L is a suitable transfer operator acting on Bergman space
A^2(D), its eigenvalue sequence lambda_n(L) is bounded by |lambda_n(L)| \leq
A\exp(-a n^{1/d}), where a, A are explicitly given.
