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Functions regular in the unit circle

Published online by Cambridge University Press:  24 October 2008

C. N. Linden
Affiliation:
Trinity CollegeCambridge

Extract

Let

be a function regular for | z | < 1. With the hypotheses f(0) = 0 and

for some positive constant α, Cartwright(1) has deduced upper bounds for |f(z) | in the unit circle. Three cases have arisen and according as (1) holds with α < 1, α = 1 or α > 1, the bounds on each circle | z | = r are given respectively by

K(α) being a constant which depends only on the corresponding value of α which occurs in (1). We shall always use the symbols K and A to represent constants dependent on certain parameters such as α, not necessarily having the same value at each occurrence.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1956

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References

REFERENCES

(1)Cartwright, M. L.Quart. J. Math. 4 (1933), 246–57.Google Scholar
(2)Cartwright, M. L.Quart. J. Math. 6 (1935), 95–6.Google Scholar
(3)Valiron, G.The general theory of integral functions (Toulouse, 1923).Google Scholar