Let $\mathcal H$ be a complex Hilbert space and let $A$ be a bounded linear transformation on $\mathcal H$. For a complex-valued function $f$, which is analytic in a domain $\mathbb D$ of the complex plane containing the spectrum of $A$, let $f(A)$ denote the operator on $\mathcal H$ defined by means of the \emph{Riesz--Dunford integral}. In the present paper, several (presumably new) versions of Pick's theorems are proved for $f(A)$, where $A$ is a dissipative operator (or a proper contraction) and $f$ is a suitable analytic function in the domain $\mathbb D$.