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 subsetK ⊂ D. The asymptotic formula $$\mathop {\lim }\limits_{n \to \infty } [d_n (A_K^D )]^{{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n}} = e^{ - 1/\tau } $$ is established, where D is a finitely connected domain of Caratheodory type,K ⊂ D is a regular compact subset such thatd∖k is connected, and τ = τ (D, K) is the flux of harmonic measure of the set ∂D relative to the domaind∖k through any rectifiable contour separating ∂D and K.