A reverse Denjoy theorem III

Abstract

Suppose that C 1 and C 2 are two simple curves joining 0 to ∞, non-intersecting in the finite plane except at 0 and enclosing a domain D which is such that, for all large r, the set {θ: re iθ ∈ $$ \bar D $$ } has measure at most 2α, where 0 < α < π. Suppose also that u is a non-constant subharmonic function in the plane such that u(z) = Φ(|z|) for all large z ∈ C 1 ∪ C 2∪ ∼D, where Φ(|z|) is a convex, non-decreasing function of |z| and ∼D is the complement of D. Let A D (r, u) = inf{u(z): z ∈ $$ \bar D $$ and |z| = r}. It is shown that if A D (r, u) = O(1) then lim infr→∞ B(r, u)/r π/(2α) > 0.