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Means and coefficients of functions which omit a sequence of values

Published online by Cambridge University Press:  24 October 2008

Albert Baernstein II  and
Richard Rochberg
Albert Baernstein II
Affiliation:
Washington University, St Louis, MO, 63130 U.S.A.
Richard Rochberg
Affiliation:
Washington University, St Louis, MO, 63130 U.S.A.
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Extract

Suppose that f is analytic in the unit disk D. If its range f(D) is contained in a simply connected proper subdomain of the plane, then the principle of subordination and the distortion theorem for univalent functions show that

where M(r, f) denotes the maximum modulus of f. Cartwright (2) studied functions which, instead of omitting all values on a continuum stretching to infinity, omit only a sequence of values. She assumed that the sequence {wn} satisfies

and

and proved that if f(D) contains none of the points {wn} thenm

for every ε > 0. Cartwright's proof was based on the Ahlfors Distortion Theorem, and is quite complicated. A much simpler proof was given by Pommerenke in (10). The key idea in his proof will also be used in the present paper.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 81 , Issue 1 , January 1977 , pp. 47 - 57
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

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