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

Daniel Girela

ArticleOriginal scientific text

Title

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

Authors ¹

Affiliations

Análisis Matemático Facultad de Ciencias Universidad de Málaga 29071 Málaga, Spain

Abstract

A well known result of Beurling asserts that if f is a function which is analytic in the unit disc

Δ = {z \in 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}) = \int_{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}) | \neq 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.

Keywords

ENG

radial variation, Dirichlet integral, capacity, univalent functions

A. Beurling, Ensembles exceptionnels, Acta Math. 72 (1940), 1-13.
J. Clunie and T. H. MacGregor, Radial growth of the derivative of univalent functions, Comment. Math. Helv. 59 (1984), 362-375.
E. F. Collingwood and A. J. Lohwater, The Theory of Cluster Sets, Cambridge University Press, London, 1966.
P. L. Duren, Univalent Functions, Springer, New York, 1983.
T. M. Flett, On the radial order of a univalent function, J. Math. Soc. Japan 11 (1959), 1-3.
F. W. Gehring, On the radial order of subharmonic functions, ibid. 9 (1957), 77-79.
D. Girela, On analytic functions with finite Dirichlet integral, Complex Variables Theory Appl. 12 (1989), 9-15.
D. J. Hallenbeck and K. Samotij, Radial growth and variation of Dirichlet finite holomorphic functions in the disk, Colloq. Math. 58 (1990), 317-325.
A. J. Lohwater and G. Piranian, On the derivative of a univalent function, Proc. Amer. Math. Soc. 4 (1953), 591-594.
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.
N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. (3) 51 (1986), 369-384.
R. Nevanlinna, Analytic Functions, Springer, New York, 1970.
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.
M. Tsuji, Beurling's theorem on exceptional sets, Tôhoku Math. J. 2 (1950), 113-125.
M. Tsuji, Potential Theory in Modern Function Theory, Chelsea, New York, 1975.
A. Zygmund, On certain integrals, Trans. Amer. Math. Soc. 55 (1944), 170-204.

Title

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

Affiliations

Abstract

Keywords

Bibliography