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1996 | 69 | 1 | 19-17
Tytuł artykułu

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

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EN
Abstrakty
EN
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 $V(f,e^{iθ})=∫_{0}^{1}|f'(re^{iθ})|dr$ 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 $(1 - r)|f'(re^{iθ})| ≠ o(1)$ 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.
Rocznik
Tom
69
Numer
1
Strony
19-17
Opis fizyczny
Daty
wydano
1996
otrzymano
1994-03-27
Twórcy
  • Análisis Matemático Facultad de Ciencias Universidad de Málaga 29071 Málaga, Spain
Bibliografia
  • [1] A. Beurling, Ensembles exceptionnels, Acta Math. 72 (1940), 1-13.
  • [2] J. Clunie and T. H. MacGregor, Radial growth of the derivative of univalent functions, Comment. Math. Helv. 59 (1984), 362-375.
  • [3] E. F. Collingwood and A. J. Lohwater, The Theory of Cluster Sets, Cambridge University Press, London, 1966.
  • [4] P. L. Duren, Univalent Functions, Springer, New York, 1983.
  • [5] T. M. Flett, On the radial order of a univalent function, J. Math. Soc. Japan 11 (1959), 1-3.
  • [6] F. W. Gehring, On the radial order of subharmonic functions, ibid. 9 (1957), 77-79.
  • [7] D. Girela, On analytic functions with finite Dirichlet integral, Complex Variables Theory Appl. 12 (1989), 9-15.
  • [8] D. J. Hallenbeck and K. Samotij, Radial growth and variation of Dirichlet finite holomorphic functions in the disk, Colloq. Math. 58 (1990), 317-325.
  • [9] A. J. Lohwater and G. Piranian, On the derivative of a univalent function, Proc. Amer. Math. Soc. 4 (1953), 591-594.
  • [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] N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. (3) 51 (1986), 369-384.
  • [12] R. Nevanlinna, Analytic Functions, Springer, New York, 1970.
  • [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.
  • [14] M. Tsuji, Beurling's theorem on exceptional sets, Tôhoku Math. J. 2 (1950), 113-125.
  • [15] M. Tsuji, Potential Theory in Modern Function Theory, Chelsea, New York, 1975.
  • [16] A. Zygmund, On certain integrals, Trans. Amer. Math. Soc. 55 (1944), 170-204.
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bwmeta1.element.bwnjournal-article-cmv69i1p19bwm
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