HOLOMORPHIC MOTION AND QUASICONFORMAL EXTENSION

TOSHIYUKI SUGAWA

In this talk, we present applications of holomorphic motions to the quasiconformal extension of univalent functions. Compare the following two results.

Theorem A (Krzyż [2]). Let ω be an analytic function on the unit disk Δ with |ω′(z)| ≤ k, where 0 ≤ k < 1 is a constant. Then the function f(z) = z + ω(1/z) on the outside of Δ can be extended to a k-quasiconformal automorphism of the Riemann sphere.

Theorem B (Fait, Krzyż and Zygmunt [1, Theorem 2’]). Let ω be an analytic function on the unit disk Δ with |ω′(z)| ≤ k, where 0 ≤ k < 1 is a constant. Then the function f(z) = z+ω(z) on Δ can be extended to a k-quasiconformal automorphism of the Riemann sphere.

As the function f(z) = z + k/z shows, Theorem A is best possible, however Theorem B can be improved as follows: Under the same hypothesis as above, the function f(z) = z + ω(z) can be extended to a k′-quasiconformal automorphism of the Riemann sphere, where k′ = k/(2 − k). In fact, this can be easily obtained as a corollary of the following result, which is one of our results presented here.

Theorem. Let k be a constant in [0, 1). For a normalized analytic function f(z) = z + a_{2}z^{2} + … on the unit disk Δ, let p(z) represent one of the quantities zf′(z)/f(z), 1 + zf′′(z)/f′(z) or f′(z). If p(Δ) ⊂ {w ∈ ℂ; |w − (1 + k^{2})/(1 − k^{2})| ≤ 2k/(1 − k^{2})}, the function f can be extended to a k-quasiconformal automorphism of the Riemann sphere.

In the case that p(z) = zf′(z)/f(z) the above result is best possible as the function p(z) = (1 + kz^{2})/(1 − kz^{2}) indicates.

The above theorem can be shown by means of the optimal λ-lemma first proved by Słodkowski [3]. We will present a general principle which enables us to deduce quasiconformal extension criteria from only univalence criteria.

References

[1] M. Fait, J. G. Krzyż, and J. Zygmunt, Explicit quasiconformal extensions for some classes of univalent functions, Comment. Math. Helv. 51 (1976), 279–285.

[2] J. G. Krzyż, Convolution and quasiconformal extension, Comment. Math. Helvetici 51 (1976), 99–104.

[3] Z. Słodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), 347– 355.

Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan

  E-mail address: sugawa@kusm.kyoto-u.ac.jp