# On bounded univalent functions that omit two given values

Colloquium Mathematicae (1999)

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

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topBetsakos, 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

top- [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] W. K. Hayman, Multivalent Functions, 2nd ed., Cambridge Univ. Press, Cambridge, 1994.
- [3] J. A. Jenkins, On the existence of certain general extremal metrics, Ann. of Math. 66 (1957), 440-453. Zbl0082.06301
- [4] J. A. Jenkins, Univalent Functions and Conformal Mappings, Springer, Berlin, 1965.
- [5] J. A. Jenkins, A criterion associated with the schlicht Bloch constant, Kodai Math. J. 15 (1992), 79-81. Zbl0764.30018
- [6] G. V. Kuz'mina, Covering theorems for functions meromorphic and univalent within a disk, Soviet Math. Dokl. 3 (1965), 21-25.
- [7] G. V. Kuz'mina, Moduli of Families of Curves and Quadratic Differentials, Proc. Steklov Inst. Math. 139 (1982).
- [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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