Vol. 87, No. 1, 1980

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Inner functions: noninvariant connected components

Vassili Nestoridis

Vol. 87 (1980), No. 1, 199–209
Abstract

H denotes the family of all inner functions B, such that for every 𝜃 ]0,1[ the pseudo-hyperbolic diameters of the connected components of the set B,𝜃 = {z : |B(z)| < 𝜃} are less than δB,𝜃 < 1.

Family H is open-closed in the space of the inner functions under the uniform topology. The main result states that for every B H the connected component of B contains neither proper multiples of B nor proper divisors of B. A characterization of the elements of H is given, which in particular implies that if the zeros αn, n = 1,2, of an infinite Blaschke product B satisfy condition

lim  ∏  | αn-−-αm-| = 1 then B ∈ H.
n m ⁄=n1 − αnαm

Mathematical Subject Classification 2000
Primary: 30D50
Milestones
Received: 21 November 1978
Revised: 21 June 1979
Published: 1 March 1980
Authors
Vassili Nestoridis