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Inequalities for functions regular in the unit circle

Published online by Cambridge University Press:  24 October 2008

C. N. Linden
C. N. Linden
Affiliation:
University CollegeSwansea
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Extract

Let f(z) = u(z) + iv(z) be a function regular for |z| < 1 with real and imaginary parts u(z) and u(z) respectively. The following theorems have been proved by Miss M. L. Cartwright(1) and (2).

Theorem A. if v(0) = 0, 0 < α < ∞ and

for 0 ≤ r < 1, 0 ≤ θ < 2π then

for 0 ≤ r < 1, 0 ≤ θ < 2π

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 58 , Issue 1 , January 1962 , pp. 26 - 37
Copyright
Copyright © Cambridge Philosophical Society 1962

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References

REFERENCES

(1)Cartwright, M. L.Quart. J. Math. Oxford, Ser. (1), 4 (1933), 246–57.Google Scholar
(2)Cartwright, M. L.Quart. J. Math. Oxford, Ser. (1), 6 (1935), 94–105.CrossRefGoogle Scholar
(3)Hayman, W. K.Proc. London Math. Soc. (3), 2 (1952), 469–512.CrossRefGoogle Scholar
(4)Linden, C. N.Quart. J. Math. Oxford, Ser. (2), 7 (1956), 196–216.CrossRefGoogle Scholar
(5)Titchmarsh, E. C.The theory of functions, 2nd ed. (Oxford, 1949).Google Scholar