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Interpolation by Linear Sums of Harmonic Measures

Published online by Cambridge University Press:  20 November 2018

Marvin Ortel
Marvin Ortel*
Affiliation:
Stanford University, Stanford, California 94305
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Let α be an open arc on the unit circle

and for z=reiθ, 0 ≤ r < 1, let

(1)

The function ω(z; α) is called the harmonic measure of the arc α with respect to the unit disc, (Nevanlinna 2); it is harmonic and bounded in the unit disc and possesses (Fatou) boundary values 1 and 0 at interior points of α and the complementary arc β respectively.

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 2 , June 1975 , pp. 249 - 253
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Ahlfors, L. and Sario, L., Riemann Surfaces, II, 3E, Princeton University Press, Princeton, 1960.Google Scholar
2. Nevanlinna, R., Analytic Functions, Springer, Berlin, 1970.Google Scholar
3. Walsh, J. L., The critical points of linear combinations of harmonic functions, Bull. Amer. Math. Soc, vol. 54 (1948), 191-195.Google Scholar
4. Walsh, J. L., The location of critical points of analytic and harmonic functions, American Mathematical Society, New York, 1950.Google Scholar