Univalence criteria in multiply-connected domains

Osgood, Brad G.

doi:10.1090/S0002-9947-1980-0574792-7

Univalence criteria in multiply-connected domains
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by Brad G. Osgood PDF

Trans. Amer. Math. Soc. 260 (1980), 459-473 Request permission

Abstract:

Theorems due to Nehari and Ahlfors give sufficient conditions for the univalence of an analytic function in relation to the growth of its Schwarzian derivative. Nehari’s theorem is for the unit disc and was generalized by Ahlfors to any simply-connected domain bounded by a quasiconformal circle. In both cases the growth is measured in terms of the hyperbolic metric of the domain. In this paper we prove a corresponding theorem for finitely-connected domains bounded by points and quasiconformal circles. Metrics other than the hyperbolic metric are also considered and similar results are obtained.

References

Lars V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963), 291–301. MR 154978, DOI 10.1007/BF02391816
L. Ahlfors and G. Weill, A uniqueness theorem for Beltrami equations, Proc. Amer. Math. Soc. 13 (1962), 975–978. MR 148896, DOI 10.1090/S0002-9939-1962-0148896-1
A. F. Beardon and F. W. Gehring, Schwarzian derivatives, the Poincaré metric and the kernel function, Comment. Math. Helv. 55 (1980), no. 1, 50–64. MR 569245, DOI 10.1007/BF02566674
A. F. Beardon and Ch. Pommerenke, The Poincaré metric of plane domains, J. London Math. Soc. (2) 18 (1978), no. 3, 475–483. MR 518232, DOI 10.1112/jlms/s2-18.3.475
P. R. Garabedian and M. Schiffer, Identities in the theory of conformal mapping, Trans. Amer. Math. Soc. 65 (1949), 187–238. MR 28944, DOI 10.1090/S0002-9947-1949-0028944-8
F. W. Gehring, Univalent functions and the Schwarzian derivative, Comment. Math. Helv. 52 (1977), no. 4, 561–572. MR 457701, DOI 10.1007/BF02567390
F. W. Gehring and J. Väisälä, The coefficients of quasiconformality of domains in space, Acta Math. 114 (1965), 1–70. MR 180674, DOI 10.1007/BF02391817
Einar Hille, Remarks on a paper be Zeev Nehari, Bull. Amer. Math. Soc. 55 (1949), 552–553. MR 30000, DOI 10.1090/S0002-9904-1949-09243-1

Über den Zusammenhang einiger Charakteristiken eines einfach zusammenhängenden Bereiches mit der Kreisabbildung

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Olli Lehto, Domain constants associated with Schwarzian derivative, Comment. Math. Helv. 52 (1977), no. 4, 603–610. MR 457703, DOI 10.1007/BF02567393

Quasiconformal mappings in the plane

O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463
Zeev Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551. MR 29999, DOI 10.1090/S0002-9904-1949-09241-8
George Springer, Fredholm eigenvalues and quasiconformal mapping, Bull. Amer. Math. Soc. 69 (1963), 810–811. MR 161975, DOI 10.1090/S0002-9904-1963-11043-5
Kurt Strebel, On the maximal dilation of quasiconformal mappings, Proc. Amer. Math. Soc. 6 (1955), 903–909. MR 73702, DOI 10.1090/S0002-9939-1955-0073702-X