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

Colloquium Mathematicae (1996)

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

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topGirela, 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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- [2] J. Clunie and T. H. MacGregor, Radial growth of the derivative of univalent functions, Comment. Math. Helv. 59 (1984), 362-375. Zbl0549.30012
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- [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. Zbl0712.30036
- [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] 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. Zbl0199.12501
- [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] M. Tsuji, Beurling's theorem on exceptional sets, Tôhoku Math. J. 2 (1950), 113-125. Zbl0041.40601
- [15] M. Tsuji, Potential Theory in Modern Function Theory, Chelsea, New York, 1975. Zbl0322.30001
- [16] A. Zygmund, On certain integrals, Trans. Amer. Math. Soc. 55 (1944), 170-204. Zbl0061.13902

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