On bounded univalent functions that omit two given values

Dimitrios Betsakos

ArticleOriginal scientific text

Title

On bounded univalent functions that omit two given values

Authors ¹

Affiliations

Department of Mathematics, University of Helsinki, FIN-00014, Helsinki, Finland

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,b} with 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

Keywords

ENG

conformal radius, quadratic differential, univalent functions, symmetrization

V. N. Dubinin, Symmetrization in the geometric theory of functions of a complex variable, Russian Math. Surveys 49 (1994), 1-79.
W. K. Hayman, Multivalent Functions, 2nd ed., Cambridge Univ. Press, Cambridge, 1994.
J. A. Jenkins, On the existence of certain general extremal metrics, Ann. of Math. 66 (1957), 440-453.
J. A. Jenkins, Univalent Functions and Conformal Mappings, Springer, Berlin, 1965.
J. A. Jenkins, A criterion associated with the schlicht Bloch constant, Kodai Math. J. 15 (1992), 79-81.
G. V. Kuz'mina, Covering theorems for functions meromorphic and univalent within a disk, Soviet Math. Dokl. 3 (1965), 21-25.
G. V. Kuz'mina, Moduli of Families of Curves and Quadratic Differentials, Proc. Steklov Inst. Math. 139 (1982).
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).

Title

On bounded univalent functions that omit two given values

Affiliations

Abstract

Keywords

Bibliography