Let $f$ be a twice continuously differentiable self-mapping of a unit disk satisfying Poisson differential inequality $|\Delta f(z)|\leq B\cdot|D f(z)|^{2}$ for some $B>0$ and $f(0)=0.$ In this note, we show that $f$ does not always satisfy the Schwarz-Pick type inequality $$\frac{1-|z|^{2}}{1-|f(z)|^{2}}\leq C(B),$$ where $C(B)$ is a constant depending only on $B.$ Moreover, a more general Schwarz-Pick type inequality for mapping that satisfies general Poisson differential inequality is established under certain conditions.
Deguang ZHONG
,
Fanning MENG
,
Wenjun YUAN
. ON SCHWARZ-PICK TYPE INEQUALITY FOR MAPPINGS SATISFYING POISSON DIFFERENTIAL INEQUALITY[J]. Acta mathematica scientia, Series B, 2021
, 41(3)
: 959
-967
.
DOI: 10.1007/s10473-021-0320-0
[1] Ahlfors L V. An extension of Schwarz's lemma. Trans Amer Soc, 1938, 43:359-381
[2] Anderson G D, Vamanamurthy M K and Vuorinen M. Monotonicity rules in Calculus. Amer Math Mon, 2006, 113:805-816
[3] Beardon A F, Minda D. A multi-point Schwarz-Pick lemma. J d' Anal Math, 2004, 92:81-104
[4] Bernstein S. Sur la généralisation du problém de Dirichlet. Math Ann, 1910, 69:82-136
[5] Chen S L, Kalaj D. The Schwarz type Lemmas and the Landau type theorem of mappings satisfying Poisson's equations. Complex Anal Oper Theory, 2019, 13:2049-2068
[6] Dai S Y, Pan Y F. Note on Schwarz-Pick estimates for bounded and positive real part analytic functions. Proc Amer Math Soc, 2008, 613:635-640
[7] Duren P. Harmonic Mappings in the Plane. Cambridge:Cambridge University Press, 2004
[8] Heinz E. On certain nonlinear elliptic differential equations and univalent mappings. J d'Anal Math, 1956/57, 5:197-272
[9] Heinz E. On one-to-one harmonic mappings. Pacific J Math, 1959, 9:101-105
[10] Hethcote H W. Schwarz lemma analogues for harmonic functions. Int J Math Educ Sci Technol, 1977, 8:65-67
[11] Kalaj D. A priori estimate of gradient of a solution to certain differential inequality and quasiconformal mappings. J d' Anal Math, 2013, 119:63-88
[12] Kalaj D. On quasiconformal harmonic maps between surfaces. Int Math Res Notices, 2015, 2:355-380
[13] Kalaj D, Zhu J F. Schwarz Pick type inequalities for harmonic maps between Riemann surfaces. Complex Var Elliptic Equ, 2018, 69:1364-1375
[14] MacCluer B D, Stroethoff K, Zhao R H. Generalized Schwarz-Pick estimates. Proc Amer Math Soc, 2003, 131:593-599
[15] Li P, Ponnusamy S. Lipschitz continuity of quasiconformal mappings and of the solutions to second order elliptic PDE with respect to the distance ratio metric. Complex Anal Oper Theory, 2018, 12:1991-2001
[16] Osserman R A. A new variant of the Schwarz-Pick-Ahlfors lemma. Manuscr Math, 1999, 100:123-129
[17] Pavlović M. Boundary correspondence under harmonic quasiconformal homeomorphisms of the unit disk. Ann Acad Sci Fenn Math, 2002, 27:365-372
[18] Pavlović M. Introduction to function spaces on the disk. Posebna Izdanja[Special Editions]. 20. Belgrade:Matematiki Institut SANU, 2004
[19] Yau S T. A general Schwarz lemma for Kähler manifolds. Amer J Math, 1978, 100:197-203
[20] Wan T. Constant mean curvature surface, harmonic maps, and universal Teichmüller space. J Differential Geom, 1992, 35:643-657
