Vol. 47, No. 2, 1973

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The behavior of the norm of an automorphism of the unit disk

Dennis Michael Girard

Vol. 47 (1973), No. 2, 443–456
Abstract

For f(z) analytic on the closed unit disk,

      ∑
f(z) =   akzk,

let

∥f ∥ = ∑ |ak|.

In this paper the following result is obtained: Theorem. Let f(z) be an automorphism of the unit disk:

f (z) = eiζ-z −-α-,0 < |α| < 1,ζ real.
1− αz

Then

            √-
√1-∥fn∥ ∼ -82-21-(1− β2)1∕2F (1, 3; 3;1 − β2 )
n       Γ (4)            2 4 2

as n →∞ where F = 2F1 is the hypergeometric function and

   1 − |α|
β = 1-+-|α|.

Mathematical Subject Classification
Primary: 42A28
Secondary: 30A24
Milestones
Received: 8 May 1972
Revised: 2 January 1972
Published: 1 August 1973
Authors
Dennis Michael Girard