On bounded univalent functions that omit two given values

Dimitrios Betsakos

Colloquium Mathematicae (1999)

  • Volume: 80, Issue: 2, page 253-258
  • ISSN: 0010-1354

Abstract

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Let a,b ∈ z: 0<|z|<1 and let S(a,b) be the class of all univalent functions f that map the unit disk into {a,bwith f(0)=0. We study the problem of maximizing |f’(0)| among all f ∈ S(a,b). Using the method of extremal metric we show that there exists a unique extremal function which maps onto a simply connnected domain D 0 bounded by the union of the closures of the critical trajectories of a certain quadratic differential. If a<0

How to cite

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Betsakos, Dimitrios. "On bounded univalent functions that omit two given values." Colloquium Mathematicae 80.2 (1999): 253-258. <http://eudml.org/doc/210716>.

@article{Betsakos1999,
abstract = {Let a,b ∈ z: 0<|z|<1 and let S(a,b) be the class of all univalent functions f that map the unit disk into \{a,bwith f(0)=0. We study the problem of maximizing |f’(0)| among all f ∈ S(a,b). Using the method of extremal metric we show that there exists a unique extremal function which maps onto a simply connnected domain $D_0$ bounded by the union of the closures of the critical trajectories of a certain quadratic differential. If a<0 },
author = {Betsakos, Dimitrios},
journal = {Colloquium Mathematicae},
keywords = {conformal radius; quadratic differential; univalent functions; symmetrization; harmonic measure},
language = {eng},
number = {2},
pages = {253-258},
title = {On bounded univalent functions that omit two given values},
url = {http://eudml.org/doc/210716},
volume = {80},
year = {1999},
}

TY - JOUR
AU - Betsakos, Dimitrios
TI - On bounded univalent functions that omit two given values
JO - Colloquium Mathematicae
PY - 1999
VL - 80
IS - 2
SP - 253
EP - 258
AB - Let a,b ∈ z: 0<|z|<1 and let S(a,b) be the class of all univalent functions f that map the unit disk into {a,bwith f(0)=0. We study the problem of maximizing |f’(0)| among all f ∈ S(a,b). Using the method of extremal metric we show that there exists a unique extremal function which maps onto a simply connnected domain $D_0$ bounded by the union of the closures of the critical trajectories of a certain quadratic differential. If a<0
LA - eng
KW - conformal radius; quadratic differential; univalent functions; symmetrization; harmonic measure
UR - http://eudml.org/doc/210716
ER -

References

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  1. [1] V. N. Dubinin, Symmetrization in the geometric theory of functions of a complex variable, Russian Math. Surveys 49 (1994), 1-79. Zbl0830.30014
  2. [2] W. K. Hayman, Multivalent Functions, 2nd ed., Cambridge Univ. Press, Cambridge, 1994. 
  3. [3] J. A. Jenkins, On the existence of certain general extremal metrics, Ann. of Math. 66 (1957), 440-453. Zbl0082.06301
  4. [4] J. A. Jenkins, Univalent Functions and Conformal Mappings, Springer, Berlin, 1965. 
  5. [5] J. A. Jenkins, A criterion associated with the schlicht Bloch constant, Kodai Math. J. 15 (1992), 79-81. Zbl0764.30018
  6. [6] G. V. Kuz'mina, Covering theorems for functions meromorphic and univalent within a disk, Soviet Math. Dokl. 3 (1965), 21-25. 
  7. [7] G. V. Kuz'mina, Moduli of Families of Curves and Quadratic Differentials, Proc. Steklov Inst. Math. 139 (1982). 
  8. [8] M. A. Lavrent'ev, On the theory of conformal mappings, Amer. Math. Soc. Transl. (2) 122 (1984), 1-63 (translation of Trudy Fiz.-Mat. Inst. Steklov. 5 (1934), 159-245). 

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