×

Two-point distortion theorems for harmonic mappings. (English) Zbl 1207.30035

The two-point distortion of a function \(f(z)\) gives the lower bound of the distance \(|f(z_1) - f(z_2)|\) with respect to the distance between \(z_1\) and \(z_2\). If the distortion is positive, then \(f\) is injective (univalent). Some earlier results gave sufficient conditions for curves in \(\mathbb R^n\) to be injective and for a lift of a harmonic mapping \(f(z) = u(z) + iv(z)\) to a suitable minimal surface to be injective.
Using a real analog of the Schwarzian derivative, the authors improve these results estimating the distortion (which implies injectivity).

MSC:

30C99 Geometric function theory
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
PDF BibTeX XML Cite
Full Text: arXiv Euclid

References:

[1] L. V. Ahlfors, Cross-ratios and Schwarzian derivatives in \(\mathbbR^n\) , Complex Analysis: Articles dedicated to Albert Pfluger on the occasion of his 80th birthday, Birkhäuser Verlag, Basel, 1988, pp. 1–15. · Zbl 0675.30021
[2] G. Birkhoff and G.-C. Rota, Ordinary differential equations , 4th ed., Wiley, New York, 1989. · Zbl 0183.35601
[3] C. Blatter, Ein Verzerrungssatz für schlichte Funktionen , Comment. Math. Helv. 53 (1978), 651–659. · Zbl 0398.30004
[4] M. Chuaqui, P. Duren and B. Osgood, The Schwarzian derivative for harmonic mappings , J. Analyse Math. 91 (2003), 329–351. · Zbl 1054.31003
[5] M. Chuaqui, P. Duren and B. Osgood, Univalence criteria for lifts of harmonic mappings to minimal surfaces , J. Geom. Analysis 17 (2007), 49–74. · Zbl 1211.30011
[6] M. Chuaqui, P. Duren and B. Osgood, Injectivity criteria for holomorphic curves in \(\mathbbC^n\) , Pure Appl. Math. Q. 7 (2011), 223–251. · Zbl 1252.30003
[7] M. Chuaqui and J. Gevirtz, Simple curves in \(\mathbbR^n\) and Ahlfors’ Schwarzian derivative , Proc. Amer. Math. Soc. 132 (2004), 223–230. JSTOR: · Zbl 1045.53003
[8] M. Chuaqui and B. Osgood, Sharp distortion theorems associated with the Schwarzian derivative , J. London Math. Soc. 48 (1993), 289–298. · Zbl 0792.30013
[9] M. Chuaqui and B. Osgood, Finding complete conformal metrics to extend conformal mappings , Indiana Univ. Math. J. 47 (1998), 1273–1292. · Zbl 0937.30026
[10] M. Chuaqui and C. Pommerenke, Characteristic properties of Nehari functions , Pacific J. Math. 188 (1999), 83–94. · Zbl 0931.30016
[11] P. L. Duren, Univalent functions , Springer, New York, 1983.
[12] P. Duren, Harmonic mappings in the plane , Cambridge University Press, Cambridge, UK, 2004. · Zbl 1055.31001
[13] M. Essén and F. R. Keogh, The Schwarzian derivative and estimates of functions analytic in the unit disc , Math. Proc. Cambridge Philos. Soc. 78 (1975), 501–511. · Zbl 0313.30019
[14] Z. Nehari, The Schwarzian derivative and schlicht functions , Bull. Amer. Math. Soc. 55 (1949), 545–551. · Zbl 0035.05104
[15] Z. Nehari, Some criteria of univalence , Proc. Amer. Math. Soc. 5 (1954), 700–704. · Zbl 0057.31102
[16] V. V. Pokornyi, On some sufficient conditions for univalence , Dokl. Akad. Nauk SSSR 79 (1951), 743–746 (in Russian).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.