Abstract
We extend the classical Schwarz–Pick inequality to the class of harmonic mappings between the unit disk and a Jordan domain with given perimeter. It is intriguing that the extremals in this case are certain harmonic diffeomorphisms between the unit disk and a convex domain that solve the Beltrami equation of second order.
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1 Introduction
Let \(\mathbf {U}\) be the unit disk in the complex plane \(\mathbf {C}\) and denote by \(\mathbf {T}\) its boundary. A harmonic mapping f of the unit disk into the complex plane can be written by \(f(z)=g(z)+\overline{h(z)}\), where g and h are holomorphic functions defined on the unit disk. Two of essential properties of harmonic mappings are given by Lewy theorem, and Rado–Kneser–Choquet theorem. Lewy theorem states that a injective harmonic mapping f is indeed a diffeomorphism, or what is the same its Jacobian \(J_f:=|\partial f|^2-|\bar{\partial }f|^2=|g'(z)|^2-|h'(z)|^2\ne 0\). Rado–Kneser–Choquet theorem states that a Poisson extension of a homeomorphism of the unit circle \(\mathbf {T}\) onto a convex Jordan curve \(\gamma \) is a diffeomorphism on the unit disk onto the inner part of \(\gamma \). For those and many more important properties of harmonic mappings, we refer to the book of Duren [2].
The standard Schwarz lemma states that if f is a holomorphic mapping of the unit disk \(\mathbf {U}\) into, itself such that \(f(0)=0\) then \(|f(z)|\le |z|\).
Its counter-part for harmonic mappings states the following ([2, Sect. 4.6]). Let f be a complex-valued function harmonic in the unit disk \(\mathbf {U}\) into itself, with \(f (0) = 0\). Then
and this inequality is sharp for each point \(z \in \mathbf {U}\). Furthermore, the bound is sharp everywhere (but is attained only at the origin) for univalent harmonic mappings f of \(\mathbf {U}\) onto itself with \(f (0) = 0.\)
The standard Schwarz–Pick lemma for holomorphic mappings states that every holomorphic mapping f of the unit disk onto itself satisfies the inequality:
If the equality is attained in (1.1) for a fixed \(z=a\in \mathbf {U}\), then f is a Möbius transformation of the unit disk.
It follows from (1.1) the weaker inequality:
with the equality in (1.2) for some fixed \(z=a\) if and only if \(f(z)=e^{it}\frac{z-a}{1-z\bar{a}}\). We will extend this result to harmonic mappings.
2 Main Result
Theorem 2.1
If f is a harmonic orientation preserving diffeomorphism of the unit disk \(\mathbf {U}\) onto a Jordan domain \(\Omega \) with rectifiable boundary of length \(2\pi R\), then the sharp inequality
holds. If the equality in (2.1) is attained for some a, then \(\Omega \) is convex, and there is a holomorphic function \(\mu :\mathbf {U}\rightarrow \mathbf {U}\) and a constant \(\theta \in [0,2\pi ]\), such that
Moreover, every function f defined by (2.2) is a harmonic diffeomorphism and maps the unit disk to a Jordan domain bounded by a convex curve of length \(2\pi R\) and the inequality (2.1) is attained for \(z=a\).
Corollary 2.2
Under the conditions of Theorem 2.1, if \(R=1\) and \(|\mu |_{\infty }=k<1\), then the mapping F is \(K=\frac{1+k}{1-k}\) bi-Lipschitz, and K—quasi-conformal.
Proof
We have that
and
Thus
Thus, F is K—bi-Lipschitz. Furthermore, we have
and so
Therefore, f is K—quasi-conformal. \(\square \)
Corollary 2.3
If \(\Omega =\mathbf {U}\), then the equality is attained in (2.1) for some a if only if f is a Möbius transformation of the unit disk onto a disk.
Proof of Corollary 2.3
Under conditions of Theorem 2.1, the function (2.2) can be written as
where \(h(z)=\sum _{k=0}^\infty a_k z^k\) is defined on the unit disk and satisfies the condition:
Further
Since
and
it follows that
If \(R=1\), this implies that \(\Omega =\mathbf {U}+a_0\) if and only if \(h\equiv a_0\). This concludes the proof. \(\square \)
Using the corresponding result in [1] and Theorem 2.1, we have
Corollary 2.4
If as in (2.3), \(F(z)=g+\overline{h}\), then \(F(z)=g(z)-h(z)\) is univalent and convex in direction of real axis.
Using Theorem 2.1, we obtain
Corollary 2.5
For every positive constant R and every holomorphic function \(\mu \) of the unit disk into itself, there is a unique convex Jordan domain \(\Omega =\Omega _{\mu ,R}\), with the perimeter \(2\pi R\), such that the initial boundary problem (Beltrami equation)
admits a unique univalent harmonic solution \(f=f_{\mu ,R}:\mathbf {U}\xrightarrow []{{}_{\!\! onto \!\!}}\Omega \).
Remark 2.6
If instead of boundary problem (2.5), we observe
then the solution g is given by
and thus, \(g(\mathbf {U})=e^{i\theta } \cdot \Omega _{\mu , R}\). Here, f is a solution of (2.5).
3 Proof of the Main Result
Proof of Theorem 2.1
Assume first that \(f(z)=g(z)+\overline{h(z)}\) has \(C^1\) extension to the boundary and assume without loss of generality that \(R=1\). Then, we have
Therefore, for \(z=e^{it}\)
Thus
As \( |g'(z)-\overline{h'(z) z^2}|\) is subharmonic, it follows that
Thus, we have that \(|g'(0)|\le 1\). Now, if \(m(z)=\frac{z+a}{1+z \bar{a}}\), then \(m(0)=a\), and thus, \(F(z)=f(m(z))\) is a harmonic diffeomorphism of the unit disk onto itself. Furthermore
Therefore, by applying the previous case to F, we obtain
Assume now that the equality is attained for \(z=0\). Then
or what is the same
Thus, for \(0\le r\le 1\), we have
To continue recall the definition of the Riesz measure \(\mu \) of a subharmonic function u. Namely, there exists a unique positive Borel measure \(\mu \), so that
Here, dm is the Lebesgue measure defined on the complex plane \(\mathbf {C}\). If \(u\in C^2\), then
\(\square \)
We need the following proposition.
Proposition 3.1
[5, Theorem 2.6 (Riesz representation theorem)]. If u is a subharmonic function defined on the unit disk then for \(r<1\), we have
where \(\mu \) is the Riesz measure of u.
By applying Proposition 3.1 to the subharmonic function
in view of (3.2), we obtain that
Thus, in particular, we infer that \(\mu =0\), or what is the same \(\Delta u=0\). As \(u=|w|\), where \(w=|u|e^{i\theta }\) is harmonic, it follows that
Therefore, \(\nabla \theta \equiv 0\), and hence, \(\theta =\mathrm {const}\).
Therefore
is a real harmonic function. Here
and
are analytic functions satisfying the condition \(|H(z)|<|G(z)|\) in view of Lewy theorem. Thus
or what is the same
Thus, \(G(z)-H(z)\) is a real holomorphic function, and therefore, it is a constant function. Furthermore
Hence
Assume without loss of the generality that \(\theta =0\) and \(g'(0)=1\). Then
From (2.4), we infer that
Further for \(z=e^{it}\), from (3.1) and (3.3), we have
To get the representation (2.2), by Lewy theorem, we have that the holomorphic mapping \(\mu (z)=\frac{h'(z)}{g'(z)}\) maps the unit disk into itself. By (3.3), we deduce that
and
It follows by (3.5) and (3.4) that \(\partial _t \arg \partial _t f(z)=1>0\), and this implies that the image of \(\mathbf {U}\) under f is a convex domain.
To prove that, every mapping f defined by (2.2) is a diffeomorphism of the unit disk onto a convex Jordan domain, we use Choquet–Kneser–Rado theorem. First of all, we have
Therefore
which means that \(F(\mathbf {T})\) is a convex curve.
As
if \(z_1, z_2\in \mathbf {T}\) with \(f(z_1)= f(z_2)\), then
and so \(z_1=z_2\). Thus by Choquet–Kneser–Rado theorem, F is a diffeomorphism.
If f is not \(C^1\) up to the boundary, then we apply the approximating sequence. Let \(\Omega \) be a fixed Jordan domain and assume that \(\phi \) is a conformal mapping of the unit disk onto \(\Omega \), with \(\phi (0)=0\). For \(r_n=\frac{n}{n+1}\), let \(\Omega _n=\phi (r_n\mathbf {U})\), and let \(U_n=f^{-1}\Omega _n\). Let \(\phi _n:\mathbf {U}\rightarrow U_n\) be a conformal mapping satisfying the condition \(\phi _n(0)=0\). Then, \(f_n=f\circ \phi _n\) is a conformal mapping of the unit disk onto the Jordan domain \(\Omega _n\). Furthermore, by subharmonic property of \(|\phi '(z)|\), we conclude that
Then, we have that
As \(\phi _n\) converges in compacts to the identity mapping, and thus, \(\phi '_n\) converges in compacts to the constant 1, we conclude that the inequality (2.1) is true for non-smooth domains.
It remains to consider the equality statement in this case. However, we know that \(\partial \Omega \) is rectifiable if and only if \(\partial _t f\in h_1(\mathbf {U})\) (see, e.g., [4, Theorem 2.7]). Here, \(h_1\) stands for the Hardy class of harmonic mappings. Now, the proof is just repetition of the previous approach, and we omit the details. \(\square \)
Example 3.2
If \(\mu (z)=z^n\), then F defined in (2.2), maps the unit disk to \(n+2-\)regular polygon of perimeter \(2\pi R\) and centered at 0. Namely, we have that
The rest follows from the similar statement obtained by Duren in [2, p. 62].
Remark 3.3
If \(\mu \) is a holomorphic mapping of the unit disk onto itself and F is defined by (2.2), then \(F(0)=0\) and
Indeed, we have that
Here, \(\rho =\mathrm {dist}(0,\partial \Omega )\). Thus, we have the sharp inequality:
In [3], it is proved that we have the general inequality
for every harmonic diffeomorphism of the unit disk onto a convex domain \(\Omega \) with \(f(0)=0\). Some examples suggest that the best inequality in this context is
The last conjectured inequality is not proved. The gap between \(\frac{\rho ^2}{2}\) and \(\frac{\rho ^2}{8}\) in (3.7) and (3.9) appears as the mappings F are special extremal mappings which for the case of \(\Omega \) being the unit disk are just rotations.
References
Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fenn. Ser. A I Math 9, 3–25 (1984)
Duren, P.: Harmonic Mappings in the Plane, vol. 156. Cambridge University Press, Cambridge (2004)
Kalaj, D.: On harmonic diffeomorphisms of the unit disc onto a convex domain. Complex Var. Theory Appl. 48(2), 175–187 (2003)
Kalaj, D., Marković, M., Mateljević, M.: Carathéodory and Smirnov type theorems for harmonic mappings of the unit disk onto surfaces. Ann. Acad. Sci. Fenn. Math. 38(2), 565–580 (2013)
Pavlović, M.: Function Classes on the Unit Disc. An Introduction, vol. 52. De Gruyter, Berlin (2014)
Acknowledgements
I am grateful to the referee for useful suggestions and corrections.
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Kalaj, D. A Sharp Inequality for Harmonic Diffeomorphisms of the Unit Disk. J Geom Anal 29, 392–401 (2019). https://doi.org/10.1007/s12220-018-9996-3
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DOI: https://doi.org/10.1007/s12220-018-9996-3