Geodesic
The Schwarzian derivatives of harmonic functions and univalence conditions
Problemy analiza, Tome 6 (2017) no. 2, pp. 42-56.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the paper we obtain some analogues of Nehari's univalence conditions for sense-preserving functions that are harmonic in the unit disc $\Bbb D=\{z\in\Bbb C:\,|z|1\}$.
Keywords: harmonic mappings, univalence criteria, Schwarzian derivative. 
@article{PA_2017_6_2_a2,
     author = {S. Yu. Graf},
     title = {The {Schwarzian} derivatives of harmonic functions and univalence conditions},
     journal = {Problemy analiza},
     pages = {42--56},
     publisher = {mathdoc},
     volume = {6},
     number = {2},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PA_2017_6_2_a2/}
}
TY  - JOUR
AU  - S. Yu. Graf
TI  - The Schwarzian derivatives of harmonic functions and univalence conditions
JO  - Problemy analiza
PY  - 2017
SP  - 42
EP  - 56
VL  - 6
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PA_2017_6_2_a2/
LA  - en
ID  - PA_2017_6_2_a2
ER  - 
%0 Journal Article
%A S. Yu. Graf
%T The Schwarzian derivatives of harmonic functions and univalence conditions
%J Problemy analiza
%D 2017
%P 42-56
%V 6
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PA_2017_6_2_a2/
%G en
%F PA_2017_6_2_a2
S. Yu. Graf. The Schwarzian derivatives of harmonic functions and univalence conditions. Problemy analiza, Tome 6 (2017) no. 2, pp. 42-56. http://geodesic.mathdoc.fr/item/PA_2017_6_2_a2/

[1] Ahlfors L., “Cross-ratios and Schwarzian derivatives in $\mathbb{R}^n$”, Complex Analysis, Articles Dedicated to Albert Pfluger on the Occasion of his 80th Birthday, eds. Hersch J., Huber A., Birkhäuser Verlag, Basel, 1989, 1–15 | DOI | MR

[2] Ahlfors L., Weill G., “A uniqueness theorem for Beltrami equations”, Proc. Amer. Math. Soc., 13 (1962), 1975–1978 | DOI | MR

[3] Campbell D. M., “Locally univalent functions with locally univalent derivatives”, Trans. Amer. Math. Soc., 162 (1971), 395–409 | DOI | MR

[4] Chuaqui M., Duren P., Osgood B., “The Schwarzian derivative for harmonic mappings”, J. Anal. Math., 91 (2003), 329–351 | DOI | MR | Zbl

[5] Chuaqui M., Duren P., Osgood B., “Univalence criteria for lifts of harmonic mappings to minimal surfaces”, J. Geom. Anal., 17:1 (2007), 49–74 | DOI | MR | Zbl

[6] Clunie J., Sheil-Small T., “Harmonic univalent functions”, Ann. Acad. Sci. Fenn. Math., 9 (1984), 3–25 | DOI | MR | Zbl

[7] Duren P., Univalent functions, Springer-Verlag, N.Y., 1983, 395 pp. | MR | Zbl

[8] Duren P., Harmonic mappings in the plane, Cambridge, 2004, 214 pp. | MR

[9] Gehring F. W., Pommerenke Ch., “On the Nehari univalence criterion and quasicircles”, Comment. Math. Helv., 59 (1984), 226–242 | DOI | MR | Zbl

[10] Godula J., Starkov V. V., “On the sharpness of some inequalities of D. M. Campbell, Ch. Pommerenke”, Math. Notes, 68:5 (1998), 665–672 (in Russian)

[11] Golusin G. M., Geometrical function theory, M., 1966, 628 pp. (in Russian) | MR

[12] Graf S. Yu., Eyelangoli O. R., “Differential inequalities in linear- and affineinvariant families of harmonic mappings”, Russian Math. (Iz. VUZ), 55:10, pp (2010), 60–62 | DOI | MR

[13] Graf S. Yu., Starkov V. V., Ponnusamy S., “Radii of covering disks for locally univalent harmonic mappings”, Monatsh. Math., 180:3 (2016), 527–548 | DOI | MR | Zbl

[14] Graf S. Yu., “On the Schwarzian norm of harmonic mappings”, Probl. Anal. Issues Anal., 5(23):2 (2016), 20–32 | DOI | MR | Zbl

[15] Hernández R., Martín M. J., “Pre-Schwarzian and Schwarzian derivatives of harmonic mappings”, J. Geom. Anal., 2015:1, 64–91 | DOI | MR | Zbl

[16] Hernández R., Martín M. J., “Criteria for univalence and quasiconformal extension of harmonic mappings in terms of the Schwarzian derivative”, Arch. Math., 104 (2015), 53–59 | DOI | MR | Zbl

[17] Hinton D., “Sturms 1836 oscillation results evolution of the theory”, Sturm–Liouville Theory, 2005, 1–27 | DOI | MR | Zbl

[18] Nehari Z., “The Schwarzian derivatives and schlicht functions”, Bull. Amer. Math. Soc., 55:6 (1949), 545–551 | DOI | MR | Zbl

[19] Nehari Z., “Some criteria of univalence”, Proc. Amer. Math. Soc., 5 (1954), 700–704 | DOI | MR | Zbl

[20] Pommerenke Ch., “Linear-invariante Familien analytischer Functionen. I”, Math. Ann., 155:2 (1964), 108–154 | DOI | MR | Zbl

[21] Sheil-Small T., “Constants for planar harmonic mappings”, J. Lond. Math. Soc., s2-42 (1990), 237–248 | DOI | MR | Zbl

[22] Sobczak-Knec M., Starkov V. V., Szynal J., “Old and new order of linear invariant family of harmonic mappings and the bound for Jacobian”, Ann. Univ. Mariae Curie-Sklodowska. Sect. A, 65:2 (2011), 191–202 | MR | Zbl