Schwarzian derivatives for pluriharmonic mappings

doi:10.1016/j.jmaa.2020.124716

Journal of Mathematical Analysis and Applications

Volume 495, Issue 1, 1 March 2021, 124716

https://doi.org/10.1016/j.jmaa.2020.124716 Get rights and content

Abstract

A pre-Schwarzian and a Schwarzian derivative for locally univalent pluriharmonic mappings in $C^{n}$ are introduced. Basic properties such as the chain rule, multiplicative invariance and affine invariance are proved for these operators. It is shown that the pre-Schwarzian is stable only with respect to rotations of the identity. A characterization is given for the case when the pre-Schwarzian derivative is holomorphic. Furthermore, it is shown that if the Schwarzian derivative of a pluriharmonic mapping vanishes then the analytic part of this mapping is a Möbius transformation. Some observations are made related to the dilatation of pluriharmonic mappings and to the dilatation of their affine transformations, revealing differences between the theories in the plane and in higher dimensions. An example is given that rules out the possibility for a shear construction theorem to hold in $C^{n}$ , for $n \geq 2$ .

Section snippets

Previous definitions of the pre-Schwarzian

For a locally biholomorphic mapping $f : Ω \to C^{n}$ , where Ω is some domain in $C^{n}$ , we adopt the definition of the pre-Schwarzian derivative as a bilinear mapping given by $P f (z) 〈 \cdot, \cdot 〉 = D f {(z)}^{- 1} D^{2} f (z) 〈 \cdot, \cdot 〉, z \in Ω .$ This was introduced by Pfaltzgraff in [21], who mostly considered the linear mapping $P f (z) 〈 z, \cdot 〉$ .

On the other hand, for a locally univalent harmonic mapping $f = h + \overline{g}$ on a planar domain Ω, with dilatation $ω = g^{'} / h^{'} : Ω \to D$ , the definition $P_{f} (z) = {(\log J_{f} (z))}_{z} = P h (z) - \frac{\overline{ω (z)} ω^{'} (z)}{1 - | ω (z) |^{2}}, z \in Ω,$ was introduced in [13] by

The Schwarzian derivative for holomorphic mappings in $C^{n}$

For a locally biholomorphic mapping f in a domain $Ω \subset C^{n}$ , Oda [18] defined the Schwarzian derivatives $S_{i j}^{k} f = \sum_{ℓ = 1}^{n} \frac{\partial^{2} f_{ℓ}}{\partial z_{i} \partial z_{j}} \frac{\partial z_{k}}{\partial f_{ℓ}} - \frac{1}{n + 1} (δ_{i k} \frac{\partial}{\partial z_{j}} + δ_{j k} \frac{\partial}{\partial z_{i}}) \log J_{f},$ for $i, j, k = 1, \dots, n$ ; here $δ_{i j}$ is Kronecker's delta. These differential operators satisfy a certain chain rule formula and, also, in dimension $n \geq 2$ they all vanish only for Möbius transformations, i.e., for the mappings $T (z) = (\frac{ℓ_{1} (z)}{ℓ_{0} (z)}, \dots, \frac{ℓ_{n} (z)}{ℓ_{0} (z)}), z \in C^{n},$ where $ℓ_{i} (z) = a_{i 0} + a_{i 1} z_{1} + \dots + a_{i n} z_{n}, i = 0, \dots, n$ , with $\det {(a_{i j})}_{i j} \neq 0$ . Note that, in contrast to the

Miscellaneous

In this section we give examples which reveal some interesting differences between harmonic mappings in the plane and pluriharmonic mappings in higher complex dimensions.

Acknowledgments

We wish to thank professor Martin Chuaqui for many fruitful discussions and, in particular, for an observation which significantly reduced the length of the proof of Theorem 5.

References (23)

H. Hamada et al.
Growth and distortion theorems for linearly invariant families on homogeneous unit balls in $C^{n}$
J. Math. Anal. Appl.
(2013)
H. Hamada et al.
Pluriharmonic mappings in $C^{n}$ and complex Banach spaces
J. Math. Anal. Appl.
(2015)
E. Cartan
Leçons sur la Théorie des Espaces à Connexion Projective
(1937)
H. Chen et al.
The Landau theorem and Bloch theorem for planar harmonic and pluriharmonic mappings
Proc. Am. Math. Soc.
(2011)
M. Chuaqui et al.
The Schwarzian derivative for harmonic mappings
J. Anal. Math.
(2003)
M. Chuaqui et al.
Pluriharmonic mappings and linearly connected domains in $C^{n}$
Isr. J. Math.
(2014)
J. Clunie et al.
Harmonic univalent functions
Ann. Acad. Sci. Fenn., Ser. A 1
(1984)
P.L. Duren
Harmonic Mappings in the Plane
(2004)
P. Duren et al.
Two-point distortion theorems for harmonic and pluriharmonic mappings
Trans. Am. Math. Soc.
(2011)
M.A. Golberg
The derivative of a determinant
Am. Math. Mon.
(1972)

R. Hernández

Schwarzian derivatives and a linearly invariant family in $C^{n}$

Pac. J. Math.

(2006)

Cited by (1)

NON-LINEAR MODEL OF THE MACROECONOMIC SYSTEM DYNAMICS: MULTIPLIER-ACCELERATOR
2023, Bulletin of the Transilvania University of Brasov, Series III: Mathematics and Computer Science

View full text

Schwarzian derivatives for pluriharmonic mappings

Abstract

Section snippets

Previous definitions of the pre-Schwarzian

The Schwarzian derivative for holomorphic mappings in Cn

Miscellaneous

Acknowledgments

J. Math. Anal. Appl.

J. Math. Anal. Appl.

Leçons sur la Théorie des Espaces à Connexion Projective

The Landau theorem and Bloch theorem for planar harmonic and pluriharmonic mappings

Proc. Am. Math. Soc.

The Schwarzian derivative for harmonic mappings

J. Anal. Math.

Pluriharmonic mappings and linearly connected domains in Cn

Isr. J. Math.

Harmonic univalent functions

Ann. Acad. Sci. Fenn., Ser. A 1

Harmonic Mappings in the Plane

Two-point distortion theorems for harmonic and pluriharmonic mappings

Trans. Am. Math. Soc.

The derivative of a determinant

Am. Math. Mon.

Schwarzian derivatives and a linearly invariant family in Cn

Pac. J. Math.

The Schwarzian derivative for holomorphic mappings in $C^{n}$

Pluriharmonic mappings and linearly connected domains in $C^{n}$

Schwarzian derivatives and a linearly invariant family in $C^{n}$