Let X be a Riemann surface, and H∞(X) the ring of bounded holomorphic functions in X. We offer here a question on divisibility in H∞(X), and then give in Section 2 a condition in which the answer is yes (Corollary 2 to Lemma 1). In Section 3 we use part 2 to prove a theorem on the separation of points by H∞(X). In Section 4 we study X/H∞(X).

If f is meromorphic in X and z ∈ X, then by o(f, z) we mean the order of f at z. (We agree that o(f, z) = ∞ if f ≡ 0.) Let h be memomorphic in X; then h might be said to be of bounded type if h = f/g where f,g ∈ H∞(X), g ≠ 0.

1.1. We will denote by B the open unit ball in Cn, and we will denote by H(B) the class of all holomorphic functions on B. Let

Thus N(B) is convex (and compact in the compact open topology). We think that the structure of N(B) is of interest and importance. Thus we proved in [1] that if

(1.1)

if

(1.2)

and if n≧ 2, then g is an extreme point of N(B). We will denote by E(B) the class of all extreme points of N(B). If n = 1 and if (1.2) holds, then as is well known g ∈ E(B) if and only if

(1.3)

Let X and Y be sets, let λ be a bounded positive measure on X, and let μ be a bounded positive measure on Y. Furthermore let M be a subalgebra of L∞(λ), let p ∈ (0, ∞), and let A be a linear transformation of M into Lp(μ) such that

for all f in M.

In § 2 of this paper we will prove the following theorem.

Let R be an open Riemann surface. ƒ belongs to H1(R) if ƒ is holomorphic on R and if the subharmonic function |ƒ| has a harmonie majorant on R. Let p be in R and define ||ƒ|| to be the value at p of the least harmonic majorant of |ƒ|. ||ƒ|| is a norm on the linear space H1(R), and with this norm H1(R) is a Banach space (7). The unit ball of H1(R) is the closed convex set of all ƒ in H1(R) with ||ƒ|| ⩽ 1. Problem: What are the extreme points of the unit ball of H1(R)? de Leeuw and Rudin have given a complete solution to this problem where R is the open unit disk (1).

Let a be the Lebesgue measure on the unit circle |z| = 1 with

and let Lp be the space of complex-valued σ-measurable functions f such that

is finite. Hp is the closure in Lp of the algebra of analytic polynomials