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Estimates on Bloch constants for planar harmonic mappings. (English) Zbl 1142.31001
The author proves the following “Schwarz lemma” for harmonic mappings (complex harmonic functions): Assume that $f$ is a harmonic mapping on the unit disk $D$ with $f(0)=0$ and $f(D)\subset D$. Then for $z\in D$, $$\Lambda_f(z):=\vert f_z\vert +\vert f_{\bar{z}}\vert \leq \frac{4(1+\vert f(z)\vert }{\pi(1-\vert z\vert ^2)},$$ with equality for $f(z)=\frac{2}{\pi}\arctan\frac{2y}{1-x^2-y^2}.$ The author uses this result to improve various estimates for the injectivity radius and for Bloch-type constants for bounded harmonic mappings. Such estimates were previously obtained by {\it H. Chen, P. M. Gauthier} and {\it W. Hengartner} [Proc. Am. Math. Soc. 128, No. 11, 3231--3240 (2000; Zbl 0956.30012)], and by {\it M. Dorff} and {\it M. Nowak} [Comput. Methods Funct. Theory 4, No.1, 151--158 (2004; Zbl 1060.30033)]. See also the recent paper of {\it A. Grigoryan} [Complex Var. Elliptic Equ. 51, No. 1, 81--87 (2006; Zbl 1114.30024)].

##### MSC:
 31A05 Harmonic, subharmonic, superharmonic functions (two-dimensional) 30C80 Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
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##### References:
 [1] Laudau, E.: Der Picard -- schottysche satz und die blochsche konstanten, (1926) [2] Hayman, W. K.: Multivalent functions, Cambridge tracts in math. 110 (1994) · Zbl 0904.30001 [3] Chen, Huaihui; Gauthier, P. M.; Hengartner, W.: Bloch constants for planar harmonic mappings, Proc. amer. Math. soc. 128, 3231-3240 (2000) · Zbl 0956.30012 · doi:10.1090/S0002-9939-00-05590-8 [4] Dorff, M.; Nowak, M.: Laudau’s theorem for planar harmonic mappings, Comput. methods funct. Theory 4, 151-158 (2004) · Zbl 1060.30033 [5] Lewy, H.: On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. amer. Math. soc. 42, 689-692 (1936) · Zbl 0015.15903 · doi:10.1090/S0002-9904-1936-06397-4