Some results on thin sets in a half plane

Howard Lawrence Jackson

Annales de l'institut Fourier (1970)

  • Volume: 20, Issue: 2, page 201-218
  • ISSN: 0373-0956

Abstract

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When one is restricted to a Stolz domain in a half plane we prove that internal thinness of a set at the origin structly implies minimal thinness there. Furthermore this result extends to the half plane itself. We also work out some relations among the concepts of minimal thinness, semi-thinness and finite logarithmic length. Finally we show that a theorem of Ahlfors and Heins can be improved.

How to cite

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Jackson, Howard Lawrence. "Some results on thin sets in a half plane." Annales de l'institut Fourier 20.2 (1970): 201-218. <http://eudml.org/doc/74015>.

@article{Jackson1970,
abstract = {When one is restricted to a Stolz domain in a half plane we prove that internal thinness of a set at the origin structly implies minimal thinness there. Furthermore this result extends to the half plane itself. We also work out some relations among the concepts of minimal thinness, semi-thinness and finite logarithmic length. Finally we show that a theorem of Ahlfors and Heins can be improved.},
author = {Jackson, Howard Lawrence},
journal = {Annales de l'institut Fourier},
keywords = {partial differential equations},
language = {eng},
number = {2},
pages = {201-218},
publisher = {Association des Annales de l'Institut Fourier},
title = {Some results on thin sets in a half plane},
url = {http://eudml.org/doc/74015},
volume = {20},
year = {1970},
}

TY - JOUR
AU - Jackson, Howard Lawrence
TI - Some results on thin sets in a half plane
JO - Annales de l'institut Fourier
PY - 1970
PB - Association des Annales de l'Institut Fourier
VL - 20
IS - 2
SP - 201
EP - 218
AB - When one is restricted to a Stolz domain in a half plane we prove that internal thinness of a set at the origin structly implies minimal thinness there. Furthermore this result extends to the half plane itself. We also work out some relations among the concepts of minimal thinness, semi-thinness and finite logarithmic length. Finally we show that a theorem of Ahlfors and Heins can be improved.
LA - eng
KW - partial differential equations
UR - http://eudml.org/doc/74015
ER -

References

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  1. [1] L. V. AHLFORS and M. HEINS, Questions of regularity connected with the Phragmén-Lindelöf principle, Annals of Math. 50, N°. 2 (1949), 341-346. Zbl0036.04702MR10,522c
  2. [2] M. BRELOT, Points irréguliers et transformations continues en théorie du potentiel, Jour. Math. Pures et Appliquées 19 (1940), 319-337. Zbl0024.40301MR3,47bJFM66.0447.01
  3. [3] M. BRELOT, Étude comparée des deux types d'effilement, Annales de l'Inst. Fourier, Grenoble, 15 (1965), 155-168. Zbl0127.05305MR32 #5912
  4. [4] M. BRELOT, Aspect statistique et comparé des deux types d'effilement, Anais da Academia Brasileira de Ciencias, 37 (1965), No. 1, 1-15. Zbl0137.35902MR33 #4303
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  8. [8] C. CONSTANTINESCU and A. CORNEA, Ideale Ränder Riemannscher Flächen, Springer, Berlin, 1963. Zbl0112.30801
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  10. [10] M. ESSÉN, A Generalization of the Ahlfors-Heins theorem, Bull. Amer. Math. Soc. 75 (1969), 127-131. Zbl0184.33003MR38 #1276
  11. [11] K. GOWRISANKARAN, Extreme harmonic functions and boundary value problems, Annales de l'Inst. Fourier, Grenoble 13 (1963), 307-356. Zbl0134.09503MR29 #1350
  12. [12] W. K. HAYMAN, Questions of regularity connected with the Phragmén-Lindelöf principle, Jour. Math. Pures et Appliquées 35 (1956), 115-126. Zbl0073.29304MR17,1073e
  13. [13] J. LELONG, Propriétés des fonctions surharmoniques positives dans un demi-espace, Comptes Rend. Acad. Sc. 226 (1948), 1161-1163. Zbl0030.35903MR10,39d
  14. [14] J. LELONG, Étude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace, Annales de l'École Normale Sup., 66 (1949), 125-159. Zbl0033.37301
  15. [15] L. NAÏM, Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Annales de l'Inst. Fourier, Grenoble, 7 (1957), 183-285. Zbl0086.30603MR20 #6608
  16. [16] M. TSUJI, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959. Zbl0087.28401MR22 #5712

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