This study concerns the class $A_K^D$ of functions $x$ analytic in a domain $D$ of an open Riemann surface and satisfying there the inequality $|x|<1$ with metric defined by the norm of the space $C(K)$ of functions continuous on the compact subset $K\subset D$. The asymptotic formula $$ \lim_{n\to\infty}[d_n(A_K^D)]^{1/n}=e^{-1/\tau}, $$ is established, where $D$ is a finitely connected domain of Carathéodory type, $K\subset D$ is a regular compact subset such thatdsetmnk is connected, and $\tau=\tau(D,K)$ is the flux of harmonic measure of the set $\partial D$ relative to the $D\setminus K$ through any rectifiable contour separating $\partial D$ and $K$.