Imaginary values of meromorphic functions in the disk

Douglas Wayne Townsend

Vol. 96 (1981), No. 1, 225–242

DOI: 10.2140/pjm.1981.96.225

Abstract

Let f be a meromorphic function in the unit disk, and let ϕ(r,f) be the number of solutions of the equation Re f(re^i𝜃) = 0 for 0 ≦ 𝜃 ≦ 2π. In this paper we bound ϕ(r,f) off an exceptional set of r values, and Φ(r,f) = ∫ ₀^rϕ(t,f)(1 −t)⁻¹ dt for all r, in terms of the Nevanlinna characteristic function of f. We then give examples to show that the bounds obtained are the best possible.

Mathematical Subject Classification 2000

Primary: 30D35

Secondary: 30D30

Milestones

Received: 31 May 1978

Revised: 30 May 1980

Published: 1 September 1981