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Minimum growth of harmonic functions and thinness of a set

Published online by Cambridge University Press:  24 October 2008

Jang-Mei G. Wu
Affiliation:
University of Illinois, Urbana, IL 61801, U.S.A.

Extract

In [3], Barth, Brannan and Hayman proved that if u(z) is any non-constant harmonic function in ℝ2, ø(r) is a positive increasing function of r for r ≥ 1 and

then there exists a path going from a finite point to ∞, such that u(z) > ø(|z|) on Γ. Moreover, they showed by example that the integral condition above cannot be relaxed.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1] Ancona, A., Hayman, W. K. and Wu, J.-M.. The growth of harmonic functions along a path in space. J. London Math. Soc. (2), 24 (1981), 313524.Google Scholar
[2] Arsove, M. and Huber, A.. Local behavior of subharmonic functions. Indiana Math. J. 22 (1973), 11911199.CrossRefGoogle Scholar
[3] Barth, K. F., Brannan, D. A. and Hayman, W. K.. The growth of plane harmonic functions along an asymptotic path. Proc. London Math. Soc. (3) 37 (1978), 363384.CrossRefGoogle Scholar
[4] Cámera, G. A.. Minimum growth rate of subharmonic functions. Acta Cient. Venezolana 30 (1979), 349359.Google Scholar
[5] Cámera, G. A.. Subharmonic functions on sets of finite measure. Quart. J. Math. Oxford (2), 33 (1982), 2743.CrossRefGoogle Scholar
[6] Helms, L. L.. Introduction to Potential Theory (Wiley, 1969).Google Scholar
[7] Landkof, N. S.. Foundation of Modern Potential Theory (Springer-Verlag, 1972).CrossRefGoogle Scholar
[8] Tsuji, M.. Potential Theory in Modern Function Theory (Chelsea, 1975).Google Scholar