Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions

Daniel Girela

Colloquium Mathematicae (1996)

  • Volume: 69, Issue: 1, page 19-17
  • ISSN: 0010-1354

Abstract

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A well known result of Beurling asserts that if f is a function which is analytic in the unit disc Δ = z ∈ ℂ : | z | < 1 and if either f is univalent or f has a finite Dirichlet integral then the set of points e i θ for which the radial variation V ( f , e i θ ) = ∫ 0 1 | f ' ( r e i θ ) | d r is infinite is a set of logarithmic capacity zero. In this paper we prove that this result is sharp in a very strong sense. Also, we prove that if f is as above then the set of points e i θ such that ( 1 - r ) | f ' ( r e i θ ) | ≠ o ( 1 ) as r → 1 is a set of logarithmic capacity zero. In particular, our results give an answer to a question raised by T. H. MacGregor in 1983.

How to cite

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Girela, Daniel. "Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions." Colloquium Mathematicae 69.1 (1996): 19-17. <http://eudml.org/doc/210320>.

@article{Girela1996,
abstract = {A well known result of Beurling asserts that if f is a function which is analytic in the unit disc $Δ =\{z ∈ ℂ : |z|<1\} $ and if either f is univalent or f has a finite Dirichlet integral then the set of points $e^\{iθ\}$ for which the radial variation $V(f,e^\{iθ\})=∫_\{0\}^\{1\}|f^\{\prime \}(re^\{iθ\})|dr$ is infinite is a set of logarithmic capacity zero. In this paper we prove that this result is sharp in a very strong sense. Also, we prove that if f is as above then the set of points $e^\{iθ\}$ such that $(1 - r)|f^\{\prime \}(re^\{iθ\})| ≠ o(1)$ as r → 1 is a set of logarithmic capacity zero. In particular, our results give an answer to a question raised by T. H. MacGregor in 1983.},
author = {Girela, Daniel},
journal = {Colloquium Mathematicae},
keywords = {radial variation; Dirichlet integral; capacity; univalent functions; radial growth},
language = {eng},
number = {1},
pages = {19-17},
title = {Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions},
url = {http://eudml.org/doc/210320},
volume = {69},
year = {1996},
}

TY - JOUR
AU - Girela, Daniel
TI - Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions
JO - Colloquium Mathematicae
PY - 1996
VL - 69
IS - 1
SP - 19
EP - 17
AB - A well known result of Beurling asserts that if f is a function which is analytic in the unit disc $Δ ={z ∈ ℂ : |z|<1} $ and if either f is univalent or f has a finite Dirichlet integral then the set of points $e^{iθ}$ for which the radial variation $V(f,e^{iθ})=∫_{0}^{1}|f^{\prime }(re^{iθ})|dr$ is infinite is a set of logarithmic capacity zero. In this paper we prove that this result is sharp in a very strong sense. Also, we prove that if f is as above then the set of points $e^{iθ}$ such that $(1 - r)|f^{\prime }(re^{iθ})| ≠ o(1)$ as r → 1 is a set of logarithmic capacity zero. In particular, our results give an answer to a question raised by T. H. MacGregor in 1983.
LA - eng
KW - radial variation; Dirichlet integral; capacity; univalent functions; radial growth
UR - http://eudml.org/doc/210320
ER -

References

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  1. [1] A. Beurling, Ensembles exceptionnels, Acta Math. 72 (1940), 1-13. Zbl66.0449.01
  2. [2] J. Clunie and T. H. MacGregor, Radial growth of the derivative of univalent functions, Comment. Math. Helv. 59 (1984), 362-375. Zbl0549.30012
  3. [3] E. F. Collingwood and A. J. Lohwater, The Theory of Cluster Sets, Cambridge University Press, London, 1966. Zbl0149.03003
  4. [4] P. L. Duren, Univalent Functions, Springer, New York, 1983. 
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  7. [7] D. Girela, On analytic functions with finite Dirichlet integral, Complex Variables Theory Appl. 12 (1989), 9-15. Zbl0696.30031
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  9. [9] A. J. Lohwater and G. Piranian, On the derivative of a univalent function, Proc. Amer. Math. Soc. 4 (1953), 591-594. Zbl0050.30202
  10. [10] T. H. MacGregor, Radial growth of a univalent function and its derivatives off sets of measure zero, in: Contemp. Math. 38, Amer. Math. Soc., 1985, 69-76. 
  11. [11] N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. (3) 51 (1986), 369-384. 
  12. [12] R. Nevanlinna, Analytic Functions, Springer, New York, 1970. Zbl0199.12501
  13. [13] W. Seidel and J. L. Walsh, On the derivatives of functions analytic in the unit disc and their radii of univalence and of p-valence, Trans. Amer. Math. Soc. 52 (1942), 128-216. Zbl0060.22002
  14. [14] M. Tsuji, Beurling's theorem on exceptional sets, Tôhoku Math. J. 2 (1950), 113-125. Zbl0041.40601
  15. [15] M. Tsuji, Potential Theory in Modern Function Theory, Chelsea, New York, 1975. Zbl0322.30001
  16. [16] A. Zygmund, On certain integrals, Trans. Amer. Math. Soc. 55 (1944), 170-204. Zbl0061.13902

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