A nonnormal Blaschke-quotient

Joseph A. Cima

Vol. 15 (1965), No. 3, 767–773

DOI: 10.2140/pjm.1965.15.767

Abstract

We shall call the meromorphic functions of the form F(z) = B₁(z)∕B₂(z) Blaschke-quotients, where B₁(z) and B₂(z) are Blaschke products in |z| < 1 with zeros at {a_n} and {b_k} respectively. Although there is a characterization of meromorphic functions which are normal there is no characterization of the Blaschke-quotients which are normal in terms of the non-Euclidean (hyperbolic) distances between the zeros {a_n} and {b_k}. In this paper we show by construction that even if the zeros of a Blaschke-quotient are separated by a positive non-Euclidean distance the Blaschke-quotient need not be normal.

Mathematical Subject Classification

Primary: 30.65

Milestones

Received: 17 April 1964

Revised: 3 August 1964

Published: 1 September 1965