2021

[1]Miodrag Mateljević

[1]Department of Mathematics, Serbian Academy of Science and Arts, Kneza
Mihaila 35, Belgrade, 11000, Serbia, Serbia

2]Mathmatical Institute of Serbian Academy of Science and Arts, Kneza Mihaila
36, Belgrade, 11000, Serbia, Serbia

# The Boundary Schwarz lemma for harmonic and pluriharmonic mappings and some
generalizations

<EMAIL_ADDRESS>Nikola Mutavdžić<EMAIL_ADDRESS>*  [

###### Abstract

We improve and generalize the classical Schwarz lemmas for planar harmonic
mappings into the sharp forms for harmonic mappings between finite dimensional
Euclidean unit ball and the unit ball in Hilbert space, and present some
applications to sharp boundary Schwarz type lemmas for holomorphic and in
particular pluriharmonic map- pings between the unit balls in Hilbert and
Banach spaces. In the second part of this article, using Burget’s estimate we
establish the sharp boundary Schwarz type lemmas for harmonic mappings between
finite dimensional balls. Since here we do not suppose in general that maps
fix the origin this is a generalization of Theorem 2.5 in the Kalaj’s paper
DKal2016HeinzSchwarz . At the end of this section, we derived some interesting
conclusions considering hyperbolic-harmonic function in the unit ball, which
shows that Hopf’s lemma is not applicable for those functions.

###### keywords:

The Boundary Schwarz lemma, Banach space, Harmonic functions in higher
dimensions, Pluriharmonic mappings

###### pacs:

[

MSC Classification]30C80, 31C05, 31C10, 32A30

##  1 Introduction

The classical Schwarz lemma is a result in complex analysis about holomorphic
functions from the open unit disk to itself. The lemma is less celebrated than
deeper theorems, such as the Riemann mapping theorem, which it helps to prove.
Although it is one of the simplest results showing the rigidity of holomorphic
functions Schwarz lemma has been generalized in various directions and it has
become a crucial theme in many branches of research in Mathematics for more
than a hundred years to the present day. There is vast literature related to
the subject, but here we cite mainly recent papers; for a more complete list
of references see RGschMM1 ; CHPRbanach ; MM-JMAA ; BiltenAcad  and the
references therein for more fundamental results.

We only briefly discuss recent results that have affected our work . We draw
the reader’s attention the the result in subsection 1.2 was obtained before
the result in the next subsection.

###  1.1 Schwarz lemma and Hilbert spaces

In Li_MM , the first author of this paper in a joint paper with Li considered
pluriharmonic and harmonic mappings $f$ defined on the unit ball
$\mathbb{B}^{n}$, $n\geq 2$, differentiable at a point $a$ on the boundary of
$\mathbb{B}^{n}$, and $f(\mathbb{B})$ satisfies some convexity hypothesis at
$f(a)$. For those mappings $f$, they obtained versions of boundary Schwarz
lemma and the sharp estimate of the eigenvalue related to its Jacobian at $a$.

After writing the final version of the manuscript Li_MM  Hamada turned
attention 111in communication with M. Mateljevć to the arxiv paper CHPRbanach
. In CHPRbanach , the authors generalize the classical Schwarz lemmas of
planar harmonic mappings into the sharp forms for Banach spaces, and present
some applications to sharp boundary Schwarz type lemmas for pluriharmonic
mappings in Banach spaces. Recently, Hamada and Kohr published paper
HamadaKohr , where authors discussed rigidity theorems on the boundary for
holomorphic mappings. They explained difference of the constants obtained for
unit ball and polydisc and also presented a generalization for other bounded
symmetric regions in $\mathbb{C}^{n}$ and balanced domains in complex Banach
spaces.

In this paper we get further results using approaches from those papers.

The following result was obtained by I. Graham, H. Hamada and G. Kohr in [GHK
, Proposition 1.8] stated here as:

######  Theorem 1.1 (GHK ).

Let $\mathbb{B}_{j}$ be the unit ball of a complex Hilbert space $H_{j}$ for
$j=1,2$, respectively. Let $f:\mathbb{B}_{1}\rightarrow\mathbb{B}_{2}$ be a
pluriharmonic mapping. Assume that $f$ is of class $C^{1}$ at some point
$z_{0}\in\partial\mathbb{B}_{1}$ and
$f(z_{0})=w_{0}\in\partial\mathbb{B}_{2}$. Then there exists a constant
$\lambda\in\mathbb{R}$ such that $Df(z_{0})^{*}w_{0}=\lambda z_{0}$. Moreover,

$\lambda\geqslant\frac{1-\mathop{\mathrm{Re}\,}(\langle
f(0),w_{0}\rangle)}{2}>0.$

In the Section 2 we will improve this estimate. Next S. Chen, H. Hamada, S.
Ponnusamy, R. Vijayakumar in CHPRbanach  observed that using [GHK ,
Proposition 1.8] and the arguments similar to those in the proof of their
Theorem 3.3 CHPRbanach  one can obtain a better estimate:

######  Proposition 1.2.

$\lambda\geq\max\\{\frac{2}{\pi}-|f(0)|,\frac{1-\mathop{\mathrm{Re}\,}(\langle
f(0),w_{0}\rangle)}{2}\\}.$

Note that under condition $f(0)=0$, in Theorem 1.1 (ii) and (iii) in Li_MM ,
it is shown that $\lambda\geq 2/\pi$. (But it also follows from the above
Proposition 1.2).

Next in CHPRbanach  a version of the boundary Schwarz lemma for the complex
Banach spaces was proved:

######  Theorem 1.3 (Theorem 3.3CHPRbanach ).

Suppose that $B_{X}$ and $B_{Y}$ are the unit balls of the complex Banach
spaces $X$ and $Y$, respectively, and $f:B_{X}\rightarrow B_{Y}$ is a
pluriharmonic mapping. In addition, let $f$ be differentiable at $b\in bB_{X}$
with $|f(b)|_{Y}=1$. Then we have

$|Df(b)b|_{Y}\geq\max\\{\frac{2}{\pi}-|f(0)|,\frac{1-|f(0)|}{2}\\}.$ (1)

In Section 3 we proved Theorem 3.1 which yields better estimate. We leave to
the interesting reader to check this claim. In Theorem 2.12, we establish
Schwarz lemma on the boundary for harmonic functions, mapping the unit ball in
$\mathbb{R}^{n}$ into unit ball in some Hilbert space, which maps origin to
origin. This is a generalization of the work in paper DKal2016HeinzSchwarz .

###  1.2 Schwarz lemma for harmonic functions in several variables

For a short discussion about Schwarz lemma for harmonic functions in the
planar case see Section 5. As far as we know the study related to Schwarz
lemma for real valued harmonic functions, defined on the unit ball in
$\mathbb{R}^{n}$ with codomain $(-1,1)$ was initiated by Khavinson, Burget,
Axler at al., for more details see for example BiltenAcad . Generalizations of
Schwarz lemma for functions of several variables were developed in the work of
Burget burgeth1992  (see also the papers by H.A. Schwarz and E.J.P.G. Schmidt
cited there), which were based on the integration of Poisson kernels over the
so called polar caps, using the spherical coordinates222we refer to this
method as Burget’s spherical cap method and we used some formulas from that
paper, which are described in the first part of Section 4. Khavinson Khavinson
, using also spherical caps, indicates an elementary argument that allows one
to obtain sharp estimates of derivatives of bounded harmonic functions in the
unit ball in $\mathbb{R}^{n}$ (explicitly stated for $n=3$); this
3-dimensional result has a physical interpretation. It is worth mentioning
that a similar idea occurs in the book ABR1992  for maps which fix the origin
in which case the spherical cap is reduced to a hemisphere. Note that
researches have often overlooked Burget’s work (for more details see Section
5).

D. Kalaj DKal2016HeinzSchwarz  considered Heinz-Schwarz inequalities for
harmonic mappings in the unit ball, which is a version of Schwarz lemma on the
boundary.

Recently, these ideas were discussed at the Belgrade Analysis Seminar, and
several recent results in this subject were obtained by the first author and
some of his associates: M.Svetlik, A. Khalfallah, M. Mhamdi, B. Purtić, H.P.
Li and the second author of this paper, see (KaMaMedJ , MaSvAADM , MMKhal ).
For more details see the introduction of paper BiltenAcad  by the first author
of this paper.

In particular we will use here [Proposition 4.3 Li_MM ] which is a corollary
of the estimate obtained in MaSvAADM  (cf. also MMKhal ), stated here as
Proposition 2.4.

In Section 4 using Burget’s estimate we establish Theorem 4.4 for harmonic
mappings between finite dimensional unit balls. Since here we do not suppose
that maps fix the origin this is a generalization of Theorem 2.5 in the
mentioned Kalaj’s paper.

At the end of this section, we derived some interesting conclusions
considering hyperbolic-harmonic function in the unit ball, which shows that
Hopf’s lemma is not applicable for those functions.

Chinese mathematicians have made a great contribution to this field but here
we will mention only results that are related to the results presented
here333Z. Chen, Y. Liu and Y. Pan; S. Dai, H. Chen and Y. Pan;X. Tang, T. Liu
and W. Zhang;J.F. Zhu, etc.

##  2 Boundary Schwarz lemma for pluriharmonic mappings in Hilbert spaces

Let $H$ be a complex Hilbert space with inner product
$\langle\cdot,\cdot\rangle$. Then $H$ ca be regarded as a real Hilbert space
with inner product $\mathop{\mathrm{Re}\,}\langle\cdot,\cdot\rangle$. Let
$|\cdot|$ be the induced norm in $H$. Let $\mathbb{B}$ be the unit ball of
$H$. For each $z_{0}\in\partial\mathbb{B}$, the tangent space
$T_{z_{0}}(\partial\mathbb{B})$ is defined by

$T_{z_{0}}(\partial\mathbb{B})=\\{\beta\in H:\mathop{\mathrm{Re}\,}\langle
z_{0},\beta\rangle=0\\}.$

Let $H_{1}$ and $H_{2}$ be complex Hilbert spaces and let $\Omega$ be a domain
in $H_{1}$.

######  Definition 2.1.

A mapping $f:\Omega\rightarrow H_{2}$ is said to be differentiable at
$z\in\Omega$ if there exists a bounded linear map $Df(z)\in
L_{\mathbb{R}}(H_{1},H_{2})$ such that

$f(z+h)=f(z)+Df(z)h+o(|h|),\mbox{ as }h\rightarrow 0.$

If $f$ is differentiable at each point of $\Omega$, then $f$ is said to be
differentiable on $\Omega$. In this case, the mapping

$\mathcal{D}f:\Omega\rightarrow L_{\mathbb{R}}(H_{1},H_{2}),\ z\mapsto Df(z)$

is called the derivative (or differential) of $f$ on $\Omega$. If
$\mathcal{D}f$ is continuous in a neighborhood of $z$, the mapping $f$ is said
to be of class $C^{1}$ at $z$. If $Df(z)$ is bounded complex linear for each
$z\in\Omega$, then $f$ is said to be holomorphic on $\Omega$.

######  Definition 2.2.

A $C^{2}-$mapping $f:\mathbb{B}_{1}\rightarrow H_{2}$ is said to be
pluriharmonic if the restriction of the complex valued function
$f_{w}(z)=\langle f(z),w\rangle$ to every complex line is harmonic for each
$w\in H_{2}$.

Let $\mathbb{B}_{j}$ be the unit ball of a complex Hilbert space $H_{j}$ for
$j=1,2,$ respectively. Note that if $f$ is differentiable at
$z_{0}\in\partial\mathbb{B}_{1}$ with values in $H_{2}$, then the adjoint
operator $Df(z_{0})^{*}$ is defined by

$\mathop{\mathrm{Re}\,}(\langle
Df(z_{0})^{*}w,z\rangle_{H_{1}})=\mathop{\mathrm{Re}\,}(\langle
w,Df(z_{0})z\rangle_{H_{2}})\quad\mbox{for}\quad z\in H_{1},\ w\in H_{2},$

where $\langle\cdot,\cdot\rangle_{H_{j}}$ is the inner product of $H_{j},\
j=1,2$.

For $a\in H_{1}$ and $v\in T_{a}(H_{1})$ (the tangent space at the point $a$),
we define the half space $H(a,v)=\\{y\in H_{1}:\mathop{\mathrm{Re}\,}\langle
y-a,v\rangle<0\\}$. If $v$ is a unit vector, then we use notation $n_{a}$ for
it and write simply $H_{a}$ instead of $H(a,v)$. We also notice that in our
approach the following simple result is useful:

######  Claim 2.3.

Assume that $f$ is differentiable at a point $a\in H_{1}$ and let $b=f(a)\in
H_{2}$. Then by the definition of adjoint operator, we have

$\mathop{\mathrm{Re}\,}\langle
Df(a)Z,n_{b}\rangle=\mathop{\mathrm{Re}\,}\langle Z,Df(a)^{*}n_{b}\rangle,$

for any $Z\in T_{a}(H_{1})$. The following statements hold:

  * (i)

If $Df(a)$ maps $H_{a}$ into $H_{b}$, then $Df(a)^{*}n_{b}=\lambda n_{a}$,
where $\lambda>0$.

  * (ii)

If further, $f$ maps $H_{a}$ into $H_{b}$, then $Df(a)^{*}n_{b}=\lambda
n_{a}$, where $\lambda\geq 0$. In particular if $Df(a)^{*}n_{b}\neq 0$, then
$\lambda>0$.

  * (iii)

In both cases (i) and (ii), we have
$\lambda=|Df(a)^{*}n_{b}|=\mathop{\mathrm{Re}\,}\langle
Df(a)n_{a},n_{b}\rangle$, and $\lambda\leq|Df(a)n_{a}|$.

  * (iv)

Let $|a|=1$. Define $u(x)=\mathop{\mathrm{Re}\,}\langle f(x),n_{b}\rangle$.
Then $\lambda=D_{r}u(a)$.

Proof: Here it is convenient to identify $H_{a}$ and $H_{b}$ with subsets of
$T_{a}(H_{1})$ and $T_{b}(H_{2})$, respectively. By hypothesis $Df(a)$ maps
$H_{a}$ into $H_{b}$, and therefore we have

$0=\mathop{\mathrm{Re}\,}\langle
Df(a)X,n_{b}\rangle=\mathop{\mathrm{Re}\,}\langle X,Df(a)^{*}n_{b}\rangle$

for all $X\in T_{a}(H_{a})$. This shows that $X_{0}=Df(a)^{T}n_{b}$ is
orthogonal on $T_{a}(H_{a})$. In our setting, it means that it equals to
$\lambda n_{b}$. Then by definition of the transpose, one has

$\mathop{\mathrm{Re}\,}\langle
Df(a)n_{a},n_{b}\rangle=\mathop{\mathrm{Re}\,}\langle
n_{a},Df(a)^{*}n_{b}\rangle=\mathop{\mathrm{Re}\,}\langle n_{a},\lambda
n_{a}\rangle=\lambda.$

Since $n_{a}\in H_{a}$, $Df(a)n_{a}\in H_{b}$, by the definition of $H_{b}$,
we first conclude that $\langle Df(a)n_{a},n_{b}\rangle>0$, and hence,
$\lambda>0$. This completes the proof of (i). For the proof of (ii), which is
similar to (i), we leave it to the interested reader by considering two cases:
$X_{0}=0$ and $X_{0}\neq 0$.

(iii) is an immediate corollary of (i) and (ii). (iv) is consequence of the
fact that $D_{r}u(a)=Re\langle Df(a)a,n_{b}\rangle=\lambda$

The proof of Proposition 2.5 and Theorem 3.1 is based on the following result:

######  Proposition 2.4 (Proposition 4.3 Li_MM ).

Let $u:\mathbb{D}\rightarrow\mathbb{D}$ be a harmonic function such that
$u(0)=b$. Assume that $u$ has a continuously extension to the boundary point
$z_{0}\in\mathbb{T}$, $u(z_{0})=c\in\mathbb{T}$ and
$a=\tan\frac{|P_{c}b|\pi}{4}$, where $P_{c}b=\langle b,c\rangle c$. If $u$ is
differentiable at $z_{0}$, then
$|D_{r}u(z_{0})|\geq\frac{2}{\pi}\frac{1-|a|}{1+|a|}$.

Note here that $a=|a|$ and $s^{-}(|P_{c}b|)=\frac{1-|a|}{1+|a|}=e^{-}(|a|)$.

Proof: Consider the function $v=\mbox{Re}(\overline{c}u)$. Then function $v$
is real-valued harmonic and apply real version.

######  Proposition 2.5.

Let $\mathbb{B}_{j}$ be the unit ball of a complex Hilbert space $H_{j}$ for
$j=1,2$, respectively. Let $f:\mathbb{B}_{1}\rightarrow\mathbb{B}_{2}$ be a
pluriharmonic mapping. Assume that $f$ is differentiable at some point
$z_{0}\in\partial\mathbb{B}_{1}$ and
$f(z_{0})=w_{0}\in\partial\mathbb{B}_{2}$. Then there exists a constant
$\lambda\in\mathbb{R}$ such that $Df(z_{0})^{*}w_{0}=\lambda z_{0}$. Moreover,

$\lambda\geqslant s^{-}(b)=\frac{2}{\pi}\cot(\frac{\pi}{4}(1+b))>0,\mbox{
where }b=\mathop{\mathrm{Re}\,}(\langle f(0),w_{0}\rangle).$

We note that $s^{-}(x)\geqslant\frac{1-x}{2},\ x\in(-1,1)$

Proof: Let us consider function $u:\mathbb{U}\rightarrow(-1,1)$, defines with
$u(z)=\mathop{\mathrm{Re}\,}\langle f(zz_{0}),w_{0}\rangle$. Function $u$ will
be harmonic function and we have $u(0)=b$. Function $u$ has continuous
extension an point $z_{0}\in\mathbb{T}$ and we can check that $u(1)=1$.
Applying Proposition 2.4, we get $|D_{r}u(1)|\geqslant s^{-}(b)$. Also, we
have that $D_{r}u(1)=Re\langle Df(z_{0})z_{0},w_{0}\rangle=\lambda$.

Suppose that function $f:\Omega\rightarrow H_{2}$, $\Omega$ is a domain in
$H_{1}$ is holomorphic in $\Omega$ and $z_{0}\in\Omega$ be any point. We
define hermitian adjoint operator $Df(z_{0})^{\dagger}$ in the next manner

$\langle Df(z_{0})^{\dagger}w,z\rangle_{H_{1}}=\langle
w,Df(z_{0})z\rangle_{H_{2}}\quad\mbox{for}\quad z\in H_{1},\ w\in H_{2},$

where $\langle\cdot,\cdot\rangle_{H_{j}}$ is the inner product of $H_{j},\
j=1,2$.

######  Lemma 2.6.

$($ChenLiuPan $)$Let $\varphi_{\xi}(z)=A\frac{\xi-z}{1-\langle z,\xi\rangle}$
be the holomorphic automorphism of $\mathbb{B}_{1}$ where $A:H_{1}\rightarrow
H_{1}$ in the sense that $A(v)=s_{\xi}v+\frac{\xi\langle
v,\xi\rangle}{1+s_{\xi}}$, $s_{\xi}=\sqrt{1-|\xi|^{2}}$ and $v\in H_{1}$. Then
$\varphi_{\xi}$ is biholomorphic in a neighborhood of
$\overline{\mathbb{B}}_{1}$, and

$A^{2}=s_{\xi}^{2}Id+\xi\langle\cdot,\xi\rangle,\ \ \ A\xi=\xi,\ \ \
\varphi_{\xi}^{-1}=\varphi_{\xi},\ \ \
D\varphi_{\xi}(z)=A\left[-\frac{Id}{1-\langle
z,\xi\rangle}+\frac{(\xi-z)\langle\cdot,\xi\rangle}{(1-\langle
z,\xi\rangle)^{2}}\right].$

If we denote $P(v)=\xi\langle v,\xi\rangle$ it can be checked that
$P^{\dagger}=P$. From this $A^{\dagger}=A$ follows. Also, if $Q(v)=z\langle
v,\xi\rangle$ and $R(v)=\xi\langle v,z\rangle$ then $Q^{\dagger}=R$. Now, we
have

$D\varphi_{\xi}(z)^{\dagger}=\left[-\frac{Id}{1-\overline{\langle
z,\xi\rangle}}+\frac{\xi\langle\cdot,\xi-z\rangle}{(1-\overline{\langle
z,\xi\rangle})^{2}}\right]A$

Let us denote $L_{z}=\left[-\frac{Id}{1-\overline{\langle
z,\xi\rangle}}+\frac{\xi\langle\cdot,\xi-z\rangle}{(1-\overline{\langle
z,\xi\rangle})^{2}}\right]$.

######  Lemma 2.7.

$($ChenLiuPan $)$For every $z_{0}\in\mathbb{B}_{1}$ we have
$D\varphi_{\xi}(z_{0})^{\dagger}\varphi_{\xi}(z_{0})=\frac{1-|\xi|^{2}}{|1-\langle
z_{0},\xi\rangle|^{2}}z_{0}$.

Proof: By direct computation
$D\varphi_{\xi}(z_{0})^{\dagger}\varphi_{\xi}(z_{0})=L_{z_{0}}A^{2}\frac{\xi-
z_{0}}{1-\langle z_{0},\xi\rangle}$. We can easily check that $A^{2}\frac{\xi-
z_{0}}{1-\langle z_{0},\xi\rangle}=\xi-\frac{s^{2}z_{0}}{1-\langle
z_{0},\xi\rangle}$. According to this, we have

$\begin{split}D\varphi_{\xi}(z_{0})^{\dagger}\varphi_{\xi}(z_{0})&=L_{z_{0}}(\xi-\frac{s^{2}z_{0}}{1-\langle
z_{0},\xi\rangle})=\\\ &=-\frac{\xi}{1-\overline{\langle
z_{0},\xi\rangle}}+\frac{\xi\langle\xi,\xi-z_{0}\rangle}{(1-\overline{\langle
z_{0},\xi\rangle})^{2}}+\frac{s^{2}z_{0}}{|1-\langle
z_{0},\xi\rangle|^{2}}-\frac{s^{2}\xi\langle z_{0},\xi-
z_{0}\rangle}{|1-\langle z_{0},\xi\rangle|^{2}(1-\overline{\langle
z_{0},\xi\rangle})}=\\\
&=-\frac{\xi(1-\langle\xi,z_{0}\rangle)}{(1-\overline{\langle
z_{0},\xi\rangle})^{2}}+\frac{\xi\langle\xi,\xi-
z_{0}\rangle}{(1-\overline{\langle
z_{0},\xi\rangle})^{2}}+\frac{s^{2}z_{0}}{|1-\langle
z_{0},\xi\rangle|^{2}}-\frac{s^{2}\xi(\langle z_{0},\xi\rangle-1)}{|1-\langle
z_{0},\xi\rangle|^{2}(1-\overline{\langle z_{0},\xi\rangle})}=\\\
&=\frac{-\xi(1-|\xi|^{2})}{(1-\overline{\langle
z_{0},\xi\rangle})^{2}}+\frac{s^{2}z_{0}}{|1-\langle
z_{0},\xi\rangle|^{2}}-\frac{s^{2}\xi(\langle z_{0},\xi\rangle-1)}{|1-\langle
z_{0},\xi\rangle|^{2}(1-\overline{\langle z_{0},\xi\rangle})}=\\\ &=\frac{-\xi
s^{2}(1-\langle z_{0},\xi\rangle))}{(1-\overline{\langle
z_{0},\xi\rangle})^{2}(1-\langle z,\xi\rangle)}+\frac{s^{2}z_{0}}{|1-\langle
z_{0},\xi\rangle|^{2}}-\frac{s^{2}\xi(\langle z_{0},\xi\rangle-1)}{|1-\langle
z_{0},\xi\rangle|^{2}(1-\overline{\langle z_{0},\xi\rangle})}=\\\
&=\frac{1-|\xi|^{2}}{|1-\langle z_{0},\xi\rangle|^{2}}z_{0}.\end{split}$

Let $V_{1}$ and $V_{2}$ be two complex vector space. We define sets or real
linear, complex linear and complex antilinear operators between $V_{1}$ and
$V_{2}$ in the following sense.

If $L:V_{1}\rightarrow V_{2}$ is additive linear operator, then

$\displaystyle
L\in\mathcal{L}_{\mathbb{R}}(V_{1},V_{2})\iff\forall\lambda\in\mathbb{R},\zeta\in
V_{1}:L(\lambda\zeta)=\lambda L(\zeta)$ $\displaystyle
L\in\mathcal{L}_{\mathbb{C}}(V_{1},V_{2})\iff\forall z\in\mathbb{C},\zeta\in
V_{1}:L(z\zeta)=zL(\zeta)$ $\displaystyle
L\in\overline{\mathcal{L}}_{\mathbb{C}}(V_{1},V_{2})\iff\forall
z\in\mathbb{C},\zeta\in V_{1}:L(z\zeta)=\bar{z}L(\zeta)$

It can be shown that the next statement holds:
$\mathcal{L}_{\mathbb{R}}(V_{1},V_{2})=\mathcal{L}_{\mathbb{C}}(V_{1},V_{2})\oplus\overline{\mathcal{L}}_{\mathbb{C}}(V_{1},V_{2}).$

First, we check that
$\mathcal{L}_{\mathbb{C}}(V_{1},V_{2})\cap\overline{\mathcal{L}}_{\mathbb{C}}(V_{1},V_{2})=\\{0\\}$.
If we argue by contradiction, we assume that there exists complex both linear
and antilinear operator $L$ between $V_{1}$ and $V_{2}$ and $\zeta\in V_{1}$,
such that $L(\zeta)\neq 0$. Then $L(i\zeta)=iL(\zeta)=-iL(\zeta)$ so
$L(\zeta)=0$, which is contradiction.

Now, let us perceive arbitrary real linear operator $L$ from $V_{1}$ into
$V_{2}$. We can define operators $L_{1},L_{2}:V_{1}\rightarrow V_{2}$ such
that $L_{1}(\zeta)=\frac{1}{2}(L(\zeta)-iL(i\zeta))$ and
$L_{2}(\zeta)=\frac{1}{2}(L(\zeta)+iL(i\zeta))$. We argue that $L=L_{1}+L_{2}$
where
$L_{1}\in\mathcal{L}_{\mathbb{C}}(V_{1},V_{2}),L_{2}\in\overline{\mathcal{L}}_{\mathbb{C}}(V_{1},V_{2})$.
If we regard $z=x+iy$ as any complex number, and $\zeta\in V_{1}$ arbitrary,
then

$\displaystyle L_{1}(z\zeta)$
$\displaystyle=\frac{1}{2}(L((x+iy)\zeta)-iL(i(x+iy)\zeta))=\frac{1}{2}(L(x\zeta+yi\zeta)-iL(-y\zeta+xi\zeta))=$
$\displaystyle=\frac{1}{2}(xL(\zeta)+yL(i\zeta)+iyL(\zeta)-ixL(i\zeta))=\frac{1}{2}((x+iy)L(\zeta)-(x+iy)iL(i\zeta))=$
$\displaystyle=(x+iy)L_{1}(\zeta)=zL_{1}(\zeta).$

Analogously to this, we can get

$\displaystyle L_{2}(z\zeta)$
$\displaystyle=\frac{1}{2}(L((x+iy)\zeta)+iL(i(x+iy)\zeta))=\frac{1}{2}(L(x\zeta+yi\zeta)+iL(-y\zeta+xi\zeta))=$
$\displaystyle=\frac{1}{2}(xL(\zeta)+yL(i\zeta)-iyL(\zeta)+ixL(i\zeta))=\frac{1}{2}((x-iy)L(\zeta)+(x-iy)iL(i\zeta))=$
$\displaystyle=(x-iy)L_{1}(\zeta)=\bar{z}L_{2}(\zeta).$

######  Claim 2.8.

Let $H_{1}$ and $H_{2}$ be two complex Hilbert spaces and $L:H_{1}\rightarrow
H_{2}$ be a bounded complex linear operator. Then $L^{*}=L^{\dagger}$.

Proof: Now, assume that $L$ is bouned real linear operator from $H_{1}$ to
$H_{2}$. Then, there are unique bounded operators $L_{1}$ and $L_{2}$, complex
linear and complex antilinear, respectively, which satisfies $L=L_{1}+L_{2}$.
For this operators, we can find bounded complex linear operator
$L_{1}^{\dagger}$ such that $\langle L_{1}^{\dagger}(w),z\rangle=\langle
w,L_{1}(z)\rangle$, and bounded, complex antilinear operator
$L_{2}^{\ddagger}$ defined with expression $\langle
L_{2}^{\ddagger}(w),z\rangle=\langle w,L_{2}(z)\rangle$, for all $z\in
H_{1},w\in H_{2}$. We argue that $L_{1}^{*}=L_{1}^{\dagger}$ and
$L_{2}^{*}=L_{2}^{\ddagger}$. First, since both complex linear and complex
antilinear operators are real linear, we can define real adjonit for this
operators. Also, if $\langle L_{1}^{\dagger}(w),z\rangle=\langle
w,L_{1}(z)\rangle$ we get $\mathop{\mathrm{Re}\,}\langle
L_{1}^{\dagger}(w),z\rangle=\mathop{\mathrm{Re}\,}\langle w,L_{1}(z)\rangle$,
for all $z\in H_{1},w\in H_{2}$. The same argument stands for operator
$L_{2}^{{\ddagger}}$.

######  Proposition 2.9.

Let $\mathbb{B}_{j}$ be the unit ball of a complex Hilbert space $H_{j}$ for
$j=1,2$, respectively. Let $f:\mathbb{B}_{1}\rightarrow\mathbb{B}_{2}$ be a
pluriharmonic mapping such that $f(\xi)=0$. Assume that $f$ is differentiable
at some point $z_{0}\in\partial\mathbb{B}_{1}$ and
$f(z_{0})=w_{0}\in\partial\mathbb{B}_{2}$. Then there exists a constant
$\lambda\in\mathbb{R}$ such that

$Df(z_{0})^{*}w_{0}=\lambda\frac{1-|\xi|^{2}}{|1-\langle
z_{0},\xi\rangle|^{2}}z_{0},$

where $\lambda\geq\frac{2}{\pi}$.

Proof: Let $\varphi_{\xi}(z)=A\frac{\xi-z}{1-\langle z,\xi\rangle}$ be the
holomorphic automorphism of $\mathbb{B}_{1}$ where
$A=s_{\xi}Id+\frac{\xi\langle\cdot,\xi\rangle}{1+s_{\xi}}$,
$s_{\xi}=\sqrt{1-|\xi|^{2}}$.

Assume that $\varphi_{\xi}(z_{0})=p\in\partial\mathbb{B}_{2}$. Let
$g(z)=f\circ\varphi_{\xi}(z)$. Then $g$ is a pluriharmonic mapping of
$\mathbb{B}_{1}$ into $\mathbb{B}_{2}$ satisfying

$g(0)=f\circ\varphi_{\xi}(0)=f(\xi)=0,$

and

$g(p)=f\circ\varphi_{\xi}(p)=f(z_{0})=w_{0}\in\partial\mathbb{B}_{2}.$

According to Theorem 2.5 we know that there exists a nonnegative number
$\lambda\in\mathbb{R}$ such that

$D^{*}_{g}(p)w_{0}=\lambda p.$

From $\varphi_{\xi}^{2}=Id$ it follows that
$D{\varphi_{\xi}}(p)D{\varphi_{\xi}}(x_{0})=Id$ and therefore (1)
$D{\varphi_{\xi}}(x_{0})^{*}D{\varphi_{\xi}}(p)^{*}=Id$. Since
$Dg(p)=Df(x_{0})D{\varphi_{\xi}}(p)$, we have
$Dg(p)^{*}=(Df(z_{0})D{\varphi_{\xi}}(p))^{*}=D{\varphi_{\xi}}(p)^{*}Df(z_{0})^{*}$
and therefore (2) $D{\varphi_{\xi}}(p)^{*}D_{f}(z_{0})^{*}w_{0}=\lambda p$. By
(1) and (2) we find $Df(z_{0})^{*}w_{0}=\lambda D{\varphi_{\xi}}(z_{0})^{*}p$.

$g(p)=f\circ\varphi_{\xi}(p)=f(z_{0})=w_{0}\in\partial B_{2}.$

According to Theorem 2.5 we know that there exists a nonnegative number
$\lambda\in\mathbb{R}$ such that

$Dg(p)^{*}w_{0}=\lambda p.$

From $\varphi_{\xi}^{2}=Id$ it follows that
$D_{\varphi_{\xi}}(p)D_{\varphi_{\xi}}(z_{0})=Id$ and therefore (1)
$D\varphi_{\xi}(x_{0})^{*}D\varphi_{\xi}(p)^{*}=Id$. Since
$Dg(p)=Df(x_{0})D\varphi_{\xi}(p)$, we have
$Dg(p)^{*}=(D_{f}(z_{0})D\varphi_{\xi}(p))^{*}=D\varphi_{\xi}(p)^{*}Df(z_{0})^{*}$
and therefore (2) $D\varphi_{\xi}(p)^{*}Df(z_{0})^{*}w_{0}=\lambda p$. By (1)
and (2) we find that

$Df(z_{0})^{*}w_{0}=\lambda D\varphi_{\xi}(z_{0})^{*}p.$

From previous Lemma we concluded that $\langle
z_{0},D\varphi_{\xi}(z_{0})^{\dagger}p\rangle=\mu$, where
$\mu=\frac{1-|\xi|^{2}}{|1-\langle z_{0},\xi\rangle|^{2}}$. Now, we can
conclude that $\langle D\varphi_{\xi}(z_{0})z_{0},p\rangle=\mu$, from which
$\mathop{\mathrm{Re}\,}\langle D\varphi_{\xi}(z_{0})z_{0},p\rangle=\mu$
follows. From Theorem 1.1 we conclude that
$D\varphi_{\xi}(z_{0})^{*}p=\mu_{1}a$, for some $\mu_{1}>0$. From the proof of
Theorem 1.1 we have $\mu_{1}=\mathop{\mathrm{Re}\,}\langle
D\varphi_{\xi}(z_{0})z_{0},p\rangle=\langle
D\varphi_{\xi}(z_{0})z_{0},p\rangle=\mu$.

Suppose that $\Omega\subset\mathbb{R}^{n}$ to be a domain and $H$ is a Hilbert
space. Let $f:\Omega\rightarrow H$ be a function such that $f\in
C^{2}(\Omega)$. We define partial derivatives with respect to coordinates
$x_{i},i=1,\ldots n$ of the base $\\{e_{1},...,e_{n}\\}$ in $\mathbb{R}^{n}$
at the point $a\in\Omega$ with formula:

$\frac{\partial f}{\partial x_{i}}(a)=Df(a)e_{i}$

.

######  Definition 2.10.

Function $f$ is harmonic in domain $\Omega$ if
$\sum\limits_{i=1}^{n}\frac{\partial^{2}f}{\partial x_{i}^{2}}(a)=0$ for every
$a\in\Omega$.

Let us denote with $\mathbb{B}$ an unit ball in the space $H$, and
$\mathbf{B}^{n}$ unit ball in $\mathbb{R}^{n}$.

It is well-known that a harmonic function $u\in L^{\infty}(\mathbf{B}^{n})$
has the following integral representation

$u(x)=\mathcal{P}[f](x)=\int_{\mathbf{S}^{n-1}}P(x,\zeta)f(\zeta)d\sigma(\zeta),$

where $f$ is the boundary function of $\mathbb{S}^{n-1}$, and

$P[x,\zeta]=\frac{1-|x|^{2}}{|x-\zeta|^{n}}\ \ \ ,\ \ \
\zeta\in\mathbf{S}^{n-1}$

is the Poisson kernel and $\sigma$ is the unique normalized rotation invariant
Borel measure on $\mathbf{S}^{n-1}$. According to DKal2016HeinzSchwarz , we
know that if $u$ is a harmonic self-mapping of $\mathbf{B}^{n}$ such that
$u(0)=0$, then

$|u(x)|\leq U(rN),$ (2)

where $r=|x|$, $N=\\{0,\cdots,0,1\\}$ and $U$ is a harmonic function of
$\mathbf{B}^{n}$ into $[-1,1]$ defined by

$U(x)=P[\chi_{S^{+}}-\chi_{S^{-}}](x)$ (3)

where $\chi$ is the indicator function and
$S^{+}=\\{x\in\mathbf{S}^{n-1}:x_{n}\geq 0\\}$,
$S^{-}=\\{x\in\mathbf{S}^{n-1}:x_{n}\leq 0\\}$. We refer to (ABR1992, ,
Chapter 6) for more details.

Recall that the hypergeometric function ${}_{p}F_{q}$ is defined for $|x|<1$
by the power series ((Landrews, , (2.1.2)))

${}_{p}F_{q}[a_{1},a_{2},\dots,a_{p}\mathchar
24635\relax\;b_{1},b_{2},\ldots,b_{q}\mathchar
24635\relax\;x]=\sum_{n=0}^{\infty}\frac{(a_{1})_{n}\cdots(a_{p})_{n}}{(b_{1})_{n}\cdots(b_{q})_{n}}\frac{x^{n}}{n!}.$

Here $(a)_{n}$ is the Pochhammer symbol and given as follows
$(a)_{n}=\frac{\Gamma(n+a)}{\Gamma(a)}$.

The following result is the so-called Heinz-Schwarz inequalities.

######  Lemma 2.11.

(DKal2016HeinzSchwarz, , Lemma 2.3) The function $V(r)=\frac{\partial
U(rN)}{\partial r}$, $0\leq r\leq 1$ is decreasing on the interval $[0,1]$,
and we have

$V(r)\geq
V(1)=C_{m}=:\frac{m!\left(1+m-(m-2)_{2}F_{1}\left[\frac{1}{2},1\mathchar
24635\relax\;\frac{3+m}{2}\mathchar
24635\relax\;-1\right]\right)}{2^{3m/2}\Gamma\left[\frac{1+m}{2}\right]\Gamma\left[\frac{3+m}{2}\right]}.$
(4)

We refer the readers to (DKal2016HeinzSchwarz, , Remark 2.7) for more details
on the constant $C_{m}$ and related functions, when $m=2,3,4$.

A version of Theorem 1.2 Li_MM  holds for harmonic functions, where codomain
is unit ball $\mathbb{B}$ in Hilbert space.

######  Theorem 2.12.

Suppose that $f:\mathbf{B}^{n}\rightarrow\mathbb{B}$ is a harmonic function,
such that $f(0)=0$, and $f$ has continuous extension to the point
$a\in\partial\mathbf{B}^{n}$ such that $f(a)=b\in\partial\mathbb{B}$. Then
$(1)\limsup\limits_{r\rightarrow 1_{-}}|D_{r}f(ra)|\geq C_{n}$.

Suppose in addition that $f$ has differentiable extension to $a$.

  * (i)

Then there exists a positive number $\lambda\in\mathbb{R}$ such that
${Df(a)}^{*}b=\lambda a$ and

  * (ii)

$\lambda\geq C_{n},$

where $C_{n}$ is given by $(\ref{Cn})$.

  * (iii)

In particularly if $n=2$, we have $\lambda\geq\frac{2}{\pi}$. This is sharp.

Proof: (i) follows from Claim 2.3. Set $u=\mathop{\mathrm{Re}\,}\langle
f,b\rangle$. Since $u$ is harmonic and it maps $\mathbf{B}^{n}$ into $(-1,1)$,
$u(0)=0$ and $u(a)=1$, using Theorem 6.16 ABR1992  we have $u(x)\leq U(rN)$
and therefore

$1-u(x)\geq 1-U(rN),\ \ \ \mbox{for}\ \ \ r=|x|<1.$

2.11), and (DKal2016HeinzSchwarz, , Lemma 2.4),we find Hence

$\frac{1-u(x)}{1-|x|}\geq\frac{1-U(rN)}{1-r}.$

Next if we define $u_{0}(t)=u(ta)$ and $U_{0}(t)=U(ta)$, $0<t<1$, for every
$0<t<1$ there are $c_{t},d_{t}\in(t,1)$ such that
$1-u_{0}(t)=u_{0}^{\prime}(c_{t})(1-t)$,
$1-U_{0}(t)=U_{0}^{\prime}(d_{t})(1-t)$ and $u_{0}^{\prime}(c_{t})\geq
U_{0}^{\prime}(d_{t})$. Hence by Lemma 2.11, $u_{0}^{\prime}(c_{t})\geq C_{n}$
and therefore we get (1). If in addition $f$ has differentiable extension to
$a$, then

$D_{r}u(a)=\lim\limits_{|x|\rightarrow
1^{-}}\frac{1-u(x)}{1-|x|}\geq\lim\limits_{r\rightarrow
1^{-}}\frac{1-U(rN)}{1-r}=\left.\frac{\partial U(rN)}{\partial
r}\right|_{r=1}=C_{n}.$

Since by Claim 2.3 (iv), $\lambda=\mathop{\mathrm{Re}\,}\langle
D_{r}f(a),b\rangle=D_{r}u(a)$, (ii) follows. An application of Proposition 2.4
yields (iii).

##  3 Boundary Schwarz lemma and Banach spaces

We will use notation from CHPRbanach . Let $X$ and $Y$ be real or complex
Banach spaces with norm $|\cdot|_{X}$ and $|\cdot|_{Y}$ respectively. We
denote with $\mathcal{L}(X,Y)$ the space of all continuous linear operators
from $X$ into $Y$ with the standard operator norm

$|A|=\sup\limits_{x\in X\setminus\\{0\\}}\frac{|Ax|}{|x|},$

where $A\in\mathcal{L}(X,Y)$. Then $\mathcal{L}(X,Y)$ is a Banach space with
respect to this norm. Denote by $X^{*}$ the dual space of the real or complex
Banach space $X$. For $x\in X\setminus\\{0\\}$, let

$T(x)=\\{l_{x}\in X^{*}:l_{x}(x)=|x|\mbox{ and }|l_{x}|=1\\}.$

Then the well known Han-Banach theorem implies that $T(x)\neq\varnothing.$

Let $f$ be a mapping of a domain $\Omega\subset X$ into real or complex Banach
space $Y$, where $X$ is a complex Banach space. We say that $f$ is
differentiable at $z\in\Omega$ if there exists a bounded real linear operator
$Df(z):X\rightarrow Y$ such that

$\lim\limits_{|h|\rightarrow 0^{+}}\frac{|f(z+h)-f(z)-Df(z)h|}{|h|}=0.$

Here $Df(z)$ is called the Fréchet derivative of $f$ at $z$. If $Y$ is a
complex Banach space and $Df(z)$ is bounded complex linear for each
$z\in\Omega$, then $f$ is said to be holomorphic on $\Omega$. Let $\Omega$ be
a domain in a complex Banach space $X$. A mapping $f$ of $\Omega$ into a real
or complex Banach space $Y$ is said to be pluriharmonic if the restriction of
$l\circ f$ to every holomorphic curve is harmonic for any $l\in Y^{*}$.

######  Theorem 3.1.

Suppose that $B_{1}$ and $B_{2}$ are the unit balls of the complex Banach
spaces $X$ and $Y$, respectively, and $f:B_{1}\rightarrow B_{2}$ is a
pluriharmonic mapping. Assume that the function $f$ is differentiable at
$b\in\partial B_{X}$ with $|f(b)|=1.$ Then we have

$|Df(b)b|\geq s^{-}(|f(0)|).$

Proof: We consider the function
$p(z)=\mathop{\mathrm{Re}\,}(l_{f(b)}(f(zb)))$, for $z\in\mathbb{U}$. Since
$f$ is pluriharmonic we have that the function $p$ is harmonic function on
$\mathbb{U}$. Also, from $|l_{f(b)}|=1$ we get
$|\mathop{\mathrm{Re}\,}(l_{f(b)}(f(zb)))|\leq|l_{f(b)}(f(zb))|\leq|f(zb)|<1$,
so we get that the function $p$ maps the unit disc into interval $(-1,1).$
From definition of $l_{f(b)}$ we can conclude that $p(1)=1$. Also, we have
that
$|p(0)|=|\mathop{\mathrm{Re}\,}l_{f(b)}(f(0))|\leq|l_{f(b)}(f(zb))|\leq|f(0)|$.
Now we can conclude that

$|D_{r}p(1)|\geq s^{-}(|p(0)|)\geq s^{-}(|f(0)|),$

since the function $s^{-}$ is decreasing on $(-1,1)$. On the other hand, we
have that $|D_{r}p(1)|\leq|Df(b)b|$. Indeed, we have that

$|D_{r}p(1)|=\lim\limits_{r\rightarrow
1^{-}}\frac{|p(1)-p(r)|}{1-r}=\lim\limits_{r\rightarrow
1^{-}}\left|\mathop{\mathrm{Re}\,}l_{f(b)}\frac{f(b)-f(rb)}{1-r}\right|=|\mathop{\mathrm{Re}\,}l_{f(b)}Df(b)b|\leq|Df(b)b|,$

which concludes our proof.

##  4 Hyperbolic harmonic functions in higher dimensions

We use notation from burgeth1992 . Let $\mathbf{B}^{n}$ be unit ball in
$\mathbb{R}^{n}$ and $\mathbf{S}^{n-1}$ be the surface of unit ball and
$\Delta$ is Laplacian partial differential operator. Consider next Laplace-
Beltrami operator

$\Delta_{0}=\frac{1-|x|^{2}}{4}(\Delta+\frac{2(n-2)}{1-|x|^{2}}\langle
x,\nabla\rangle).$

Any twice continuously differentiable function $h$ which is defined on
$\mathbf{B}^{n}$ and fulfills $\Delta_{0}h=0$ is called hyperbolic harmonic on
$\mathbf{B}^{n}$.

In the sequel we will use some specific properties of both harmonic and
hyperbolic-harmonic kernel, which are listed below:

  * a)

There exists a Poisson formula for hyperbolic harmonic as well as for harmonic
functions on $\mathbf{B}^{n}$. Let $\sigma$ denote the usual surface measure
on $\mathbf{S}^{n-1}$ and $f$ is $\sigma$-integrabile function on
$\mathbf{S}^{n-1}$. Set $x\in\mathbf{B}^{n}$ and $\eta\in\mathbf{S}^{n-1}$.
Depending on whether we define $P(x,\eta)$ as

$\frac{1}{\sigma(\mathbf{S}^{n-1})}\frac{1-|x|^{2}}{|x-\eta|^{n}}\mbox{ or as
}\frac{1}{\sigma(\mathbf{S}^{n-1})}\frac{(1-|x|^{2})^{n-1}}{|x-\eta|^{2(n-1)}},$

we get a harmonic or a hyperbolic harmonic function on $\mathbf{B}^{n}$ by

$h(x)=P[f](x)=\int_{\mathbf{S}^{n-1}}P(x,\eta)f(\eta)\mathrm{d}\sigma(\eta).$

In the sequel we use the same notation $P$ for both Poisson kernels.

  * b)

$1=P[\mathbf{1}](x)=\int_{\mathbf{S}^{n-1}}P(x,\eta)\mathrm{d}\sigma(\eta).$
(5)

Where $\mathbf{1}(\eta)=1$ for all $\eta\in\mathbf{S}^{n-1}$ is a constant
function.

This is immediate consequence of the fact that constant functions belongs to
classes of hyperbolic and hyperbolic functions, both respectively.

  * c)

Harmonic (resp. hyperbolic-harmonic) functions possess mean value property
with respect to (hyperbolic) spheres.

  * d)

The theorem of Fatou, concerning the $\sigma-$a.e, existence of non-tangential
limits, is valid in both cases.

For convenience we set

$\displaystyle M_{c}^{n}(|x|)=2P[\chi_{S(c,\tilde{x})}](x)-1$ (6)
$\displaystyle m_{c}^{n}(|x|)=2P[\chi_{S(c,-\tilde{x})}](x)-1,$ (7)

where $x\in\mathbf{B}^{n},\tilde{x}=\frac{x}{|x|}$ and $S(c,\tilde{x})$
denotes the polar cap with center $\tilde{x}$ and $\sigma-$measure $c$. Also,
$\chi_{A}$ is an indicator function of the set $A$. It is easy to verify that
the expressions on the right hand side of (6) inherit the rotational
invariance of the measure $\sigma$.

For derivation on the explicit formula (8), we refer to paper of first author
MSS061021Fil , specifically, to the Proposition 5.10. In this proposition we
use the following notation: $\sigma_{n-1}$ is surface area of the sphere
$\mathbf{S}^{n-1}$ and $\varphi$ is an angle between radius vector of point
$\eta\in\mathbf{S}^{n-1}$ and radius vector of the point $\tilde{x}.$

######  Proposition 4.1 (MSS061021Fil , Proposition 5.10).

If $f$ is a function on $\mathbf{S^{n-1}}$ depending only on $\varphi$, then

$\int_{\mathbf{S^{n-1}}}f(\eta)\mathrm{d}\sigma(\eta)=\sigma_{n-2}\int_{0}^{\pi}f(\varphi)\sin^{n-2}\varphi\mathrm{d}\varphi.$

Since $|x-\eta|^{2}=1-2r\cos\varphi+r^{2}$ we have that both our kernels
depend only on $\varphi$. Let us define
$\sigma_{*}(n)=\frac{\sigma_{n-2}}{\sigma_{n-1}}$. Using formula
$\sigma_{n-1}=\frac{2\pi^{n/2}}{\Gamma(\frac{n}{2})}$ we get
$\sigma_{*}(n)=\frac{1}{\sqrt{\pi}}\frac{\Gamma(n/2)}{\Gamma((n-1)/2)}$.

Using Proposition 4.1 we can rewrite (6) as

$\displaystyle
M_{c}^{n}(|x|)=2\sigma_{*}(n)(1-|x|^{2})^{\nu}\int_{0}^{\alpha(c)}\frac{\sin^{n-2}t}{(1-2|x|\cos
t+|x|^{2})^{\mu}}\mathrm{d}t-1$ (8) $\displaystyle
m_{c}^{n}(|x|)=2\sigma_{*}(n)(1-|x|^{2})^{\nu}\int_{\pi-\alpha(c)}^{\pi}\frac{\sin^{n-2}t}{(1-2|x|\cos
t+|x|^{2})^{\mu}}\mathrm{d}t-1,$ (9)

where $(\nu,\mu)=(1,n/2)$ in harmonic case and $(\nu,\mu)=(n-1,n-1)$ in the
hyperbolic-harmonic case and $\alpha(c)$ is the spherical angle of
$S(c,\tilde{x})$.

######  Theorem 4.2.

burgeth1992  Let $h$ be a harmonic or hyperbolic-harmonic function taking
values in $(-1,1)$ and $h(0)=a,-1<a<1$. Then for $c=\frac{a+1}{2}$ and all
$x\in\mathbf{B}^{n}$

$m_{c}^{n}(|x|)\leq h(x)\leq M_{c}^{n}(|x|).$

Equality on the right (resp., left-) hand side for some
$z\in\mathbf{B}^{n}\setminus\\{0\\}$ implies

$\displaystyle h(x)=2P[\chi_{S(c,\tilde{z})}](x)-1,$
$\displaystyle(\mbox{respectively, }h(x)=2P[\chi_{S(c,-\tilde{z})}](x)-1)$

for all $x\in\mathbf{B}^{n}$

######  Lemma 4.3.

Let $(\nu,\mu)=(1,n/2)$ (harmonic case). Then

$\frac{\mathrm{d}M_{c}^{n}}{\mathrm{d}r}(r)\Bigr{\rvert}_{r=1}=\frac{2^{2-n}}{\sqrt{\pi}}\frac{\Gamma(n/2)}{\Gamma((n-1)/2)}\int_{\alpha(c)}^{\pi}\frac{\sin^{n-2}t}{\sin^{n}(t/2)}\mathrm{d}t.$

Proof: We will use the following notation $T(r)=\frac{1-M_{c}^{n}(r)}{1-r}$.
Then:

$\frac{\mathrm{d}M_{c}^{n}}{\mathrm{d}r}(r)\Bigr{\rvert}_{r=1}=\lim\limits_{r\rightarrow
1^{-}}T(r).$

By using formula (6) we have

$\displaystyle T(r)$
$\displaystyle=\frac{1-(2P[\chi_{S(c,\tilde{z})}](x)-1)}{1-r}=\frac{2(1-P[\chi_{S(c,\tilde{z})}](x))}{1-r}.$
$\displaystyle\mbox{If we use formula }(\ref{kernel})\mbox{ we get }$
$\displaystyle T(r)$
$\displaystyle=\frac{2P[\mathbf{1}-\chi_{S(c,\tilde{z})}](x)}{1-r}=\frac{2P[\chi_{\mathbf{S}^{n-1}\setminus
S(c,\tilde{z})}](x)}{1-r}.$ Now, by using version of Proposition 4.1 we get
the folloing important result: $\displaystyle T(r)$
$\displaystyle=2\sigma_{*}(n)(1+r)\int_{\alpha(c)}^{\pi}\frac{\sin^{n-2}t}{(1-2r\cos
t+r^{2})^{n/2}}\mathrm{d}t$

We can reformulate this equation, in the following manner

$T(r)=2\sigma_{*}(n)(1+r)\int_{\alpha(c)}^{\pi}Q(r,t)\mathrm{d}t,$

where $Q(r,t)=\frac{\sin^{n-2}t}{(1-2r\cos t+r^{2})^{n/2}}$. Since we have
limit of proper integral in the last expression, we can derive next formula

$\frac{\mathrm{d}M_{c}^{n}}{\mathrm{d}r}(r)\Bigr{\rvert}_{r=1}=4\sigma_{*}(n)\int_{\alpha(c)}^{\pi}\frac{\sin^{n-2}t}{2^{n}\sin^{n}(t/2)}\mathrm{d}t.$

Let us define

$D_{n}(a)=\frac{2^{2-n}}{\sqrt{\pi}}\frac{\Gamma(n/2)}{\Gamma((n-1)/2)}\int_{\alpha(c)}^{\pi}\frac{\sin^{n-2}t}{\sin^{n}(t/2)}\mathrm{d}t.$

Then the next theorem holds:

######  Theorem 4.4.

Suppose that $f:\mathbf{B}^{n}\rightarrow\mathbf{B}^{m}$ is a harmonic
function, such that $f(0)=a_{0}$, and $f$ has a continuous extension to the
point $x_{0}\in\partial\mathbf{B}^{n}$ such that
$f(x_{0})=y_{0}\in\partial\mathbf{B}^{m}$.

Then $\limsup\limits_{r\rightarrow 1_{-}}|D_{r}f(rx_{0})|\geq D_{n}(a)$.

If, in addition, $f$ has a differentiable continuation at point $x_{0}$, then
there exists a positive number $\lambda\in\mathbb{R}$ such that
${Df(x_{0})}^{*}y_{0}=\lambda x_{0}$ and

$\lambda\geq D_{n}(a),$

where $a=\langle a_{0},y_{0}\rangle$. This is sharp.

Proof: Let us define function $h(x)=\langle f(x),y_{0}\rangle$. This function
is harmonic (respectively hyperbolic harmonic) in $\mathbf{B}^{n}$, with
$h(0)=a$, and $h(x_{0})=1$. Since, by the Theorem of Fotou $M_{c}^{n}(1)=1$ we
have an implication

$\frac{h(x_{0})-h(rx_{0})}{1-r}\geq\frac{1-M_{c}^{n}(r)}{1-r}.$

If $u(r)=h(rx_{0}),r\in[0,1)$ then
$u^{\prime}(r)=Dh(rx_{0})x_{0}=D_{r}h(rx_{0})$. From Lagrange’s theorem we
have that for every $r\in[0,1)$ there exists $r_{0}\in(r,1)$ such that

$\frac{1-u(r)}{1-r}=u^{\prime}(r_{0})=D_{r}h(r_{0}x_{0})\geq\frac{1-M_{c}^{n}(r)}{1-r}.$

This means that $\limsup\limits_{r\rightarrow
1^{-}}D_{r}h(rx_{0})\geq\liminf\limits_{r\rightarrow
1_{-}}\frac{1-u(r)}{1-r}\geq D_{n}(a)$. Cauchy -Schwarz inequality provides us
that $|D_{r}f(x)|\geq D_{r}h(x)$, which gives us

$\limsup\limits_{r\rightarrow 1^{-}}|D_{r}f(rx_{0})|\geq D_{n}(a).$

At the end of this section we will investigate whether or not we can formulate
the similar version of Schwarz lemma on the boundary, for hyperbolic harmonic
functions.

######  Lemma 4.5.

Let $(\nu,\mu)=(n-1,n-1)$, where $n>2$ (hyperbolic-harmonic case). Then

$\frac{\mathrm{d}M_{c}^{n}}{\mathrm{d}r}\left(r\right)\Bigr{\rvert}_{r=1}=0.$

Proof: Like in previous lemma, we have

$\frac{\mathrm{d}M_{c}^{n}}{\mathrm{d}r}(r)\Bigr{\rvert}_{r=1}=\lim\limits_{r\rightarrow
1^{-}}T(r),$

Let us define $Q_{hyp}(r,t)=\frac{\sin^{n-2}t}{(1-2r\cos t+r^{2})^{n-1}}$.
Then

$T(r)=2\sigma_{*}(n)(1-r)^{n-2}(1+r)^{n-1}\int_{\alpha(c)}^{\pi}Q_{hyp}(r,t)\mathrm{d}t.$

Also, define $J_{hyp}(r)=\int_{\alpha(c)}^{\pi}Q_{hyp}(r,t)\mathrm{d}t$. We
can pass with the limit, under proper integral sign, to get

$\lim\limits_{r\rightarrow
1^{-}}J_{hyp}(r)=J_{hyp}=\int_{\alpha(c)}^{\pi}q_{hyp}(t)dt,$

where $q_{hyp}(t)=4^{-n+1}\sin^{n-2}t\sin^{-2(n-1)}t/2$.
From this we can draw a conclusion $T(r)\sim d_{n}(1-r)^{n-2},r\rightarrow
1^{-}$. This immediately gives our assertion. By this Lemma we conclude that
we have different situation concerning hyperbolic-harmonic function, mapping
unit ball in $\mathbb{R}^{n}$ into unit disc in $\mathbb{R}^{m}$, in
comparison with the harmonic function in same settings. Namely, we found
explicit hyperbolic-harmonic function, that maps unit ball into interval
$(-1,1)$ such that $u(x_{0})=1$, for some $x_{0}$ on the boundary of the unit
ball, but radial derivative in the point $x_{0}$ is vanishing.

At the first glance, this may look as surprise, having in mind famous Hopf
lemma. Function $u$ is satisfying $L(u)=0$, where $L$ is uniformly elliptical
partial diferential operator of second order, it has global maximum at point
$x_{0}$ on the boundary of unit ball, so we expected that normal derivative in
the point $x_{0}$ must be grater than zero.

Let $\Omega$ be a domain in $\mathbb{R}^{n},n\geq 2$, $x\in\Omega$ be a point
and $u$ belongs to $C^{2}(\Omega)$. We define

$Lu=a^{ij}(x)D_{ij}u+b^{i}(x)D_{i}u+c(x)u,a^{ij}=a^{ji}.$

The summation convention that repeated indices indicate summation from 1 to n
is followed here. We adopt the following definitions: operator $L$ is elliptic
in point $x\in\Omega$ if the coefficient matrix $A(x)=[a^{ij}(x)]$ is positive
definite. If $\Lambda(x),\lambda(x)$ are the greatest and the smallest
eigenvalue of matrix $A(x)$ and $\Lambda/\lambda$ is bounded in $\Omega$ we
say that $L$ is uniformly elliptic in $\Omega$. Also we will need next
condition. Let $k>0$ is a constant and $x\in\Omega$ be an arbitrary

$\frac{|b^{i}(x)|}{\lambda(x)}\leq k,i=1,\ldots,n.$ (10)

Now, we can formulate Hopf Lemma

######  Lemma 4.6 (Hopf lemma,gilbarg_trudinger , Lemma 3.4).

Suppose that $L$ is uniformly elliptic operator, that satisfies condition
(10), $c=0$ and $Lu\geq 0$ in $\Omega$. Let $x_{0}\in\partial\Omega$ be such
that

  * (i)

$u$ is continuous at $x_{0}$;

  * (ii)

$u(x_{0})>u(x)$ for all $x\in\Omega$;

  * (iii)

$\partial\Omega$ satisfies an interior sphere condition at $x_{0}$.

Then the outer normal derivative of $u$ at $x_{0}$, if it exists, satisfies
the strict inequality

$\frac{\partial}{\partial\nu}u(x_{0})>0.$

What turns out to be is that Hopf lemma demands some conditions on the
coefficients standing by the first order derivatives of the elliptic partial
differential operator L, that hyperbolic-harmonic functions does not
satisfies. We have that hyperbolic-harmonic functions satisfies
$Lu=\Delta_{0}u=0$, where $A(x)=Id$ and
$b_{i}(x)=\frac{2(n-2)}{1-|x|^{2}},i=1,\ldots,n$. Since
$\Lambda(x)=\lambda(x)=1,x\in\mathbf{B}^{n}$, we conclude that operator
$\Delta_{0}$ does not satisfies condition (10) in $\mathbf{B}^{n}$, so we can
not apply Hopf lemma in this situation.

A part of our result can be interpreted as a confirmation that condition (10)
can not be excluded from the statement of Hopf lemma.

##  5 Appendix

Motivated by the role of the Schwarz lemma in Complex Analysis and numerous
fundamental results, see for instance MM-JMAA ; Osserman  and references
therein, in 2016, the first author RGschMM1  has posted on ResearchGate the
project “Schwarz lemma, the Carathéodory and Kobayashi Metrics and
Applications in Complex Analysis”.444Various discussions regarding the subject
can also be found in the Q&A section on ResearchGate under the question “What
are the most recent versions of the Schwarz lemma?” RGschMM2 .

In this project and in MM-JMAA , cf. also kavu , we developed the method
related to holomorphic mappings with strip codomain (we refer to this method
as the approach via the Schwarz–Pick lemma for holomorphic maps from the unit
disc into a strip; shortly ”planar strip method”). It is worth mentioning that
the Schwarz lemma has been generalized in various directions; see RGschMM1 ;
CHPRbanach  and the references therein.

Even in planar case researches had some difficulties in handling Schwarz lemma
for harmonic maps of the unit disc into self which does not fix the origin. It
seems that the researchers have overlooked Burgeth and H. W. Hethcote results
and they have had some difficulties to handle the case $f(0)\neq 0$ in this
context; see for more details MMKhal ; MaSvAADM ; BiltenAcad .

In joint paper of the first author with M. Svetlik MaSvAADM  using ”planar
strip method” which is a completely different approach than B.
Burgethburgeth1992 , we get a simple proof of an optimal version of the
Schwarz lemma for real valued harmonic functions (without the assumption that
$0$ is mapped to $0$ by the corresponding map), which improves H. W. Hethcote
result.555Note here that Burget’s spherical cap method yield optimal estimate
in both planar and spatial case.

In joint paper of the first author with A. Khalfallah and M. Mhamdi KaMaMedJ ,
some properties of mappings admitting a Poisson-type integral representations
and the boundary Schwarz lemma were considered.

Presently on this project the first author works with some of his associates:
A. Khalfallah, M. Arsenović, M.Svetlik, M. Mhamdi, B. Purtić, H.P. Li, J.
Gajić and the second author of this paper.

Chinese mathematicians have made a great contribution to this field but here
we will mention only some whose results are related to our results. For some
interesting complex $n$-dimensional generalisations of classical Schwarz lemma
type results see Jian-Feng Zhua’s articles JZhu-RIM  and JZhu2016 . In paper
TangLiuZhang  the authors proved Schwarz lemma on the boundary for holomorphic
mappings between unit balls in $\mathbb{C}^{n}$, and some of theirs rigidity
properties. Generalization of this theorem, for separable complex Hilbert
space was given by Z. Chen, Y. Liu and Y. Pan in ChenLiuPan . While proving
Proposition 2.9, we independently proved Lemma 2.7, but later found that
result proven in ChenLiuPan , as it can be seen in corresponding reference. In
DaiChenPan  the authors proved a higher order Schwarz-Pick lemma for
holomorphic mappings between unit balls in complex Hilbert spaces.

For generalizations of Schwarz lemmas for planar harmonic mappings into the
sharp forms of Banach spaces we refer the interested reader to Chen, Hamada et
al. CHfunct ; CHPRbanach  and literature cited there for the background.
Recall the main purpose of the paper CHPRbanach  is to develop some methods to
investigate the Schwarz type lemmas for holomorphic mappings and pluriharmonic
mappings in Banach spaces. Initially, they extend the classical Schwarz lemmas
for holomorphic mappings to Banach spaces. Furthermore, they improve and
generalize the classical Schwarz lemmas for planar harmonic mappings and
obtain sharp versions for Banach spaces, and present some applications to
sharp boundary Schwarz type lemmas for pluriharmonic mappings in Banach
spaces. The obtained results provide improvements and generalizations of the
corresponding results in CHfunct  (cf. alsoLi_MM ).

Acknowledgments The authors are indebted to M. Arsenović for an interesting
discussions on this paper.

## References

  *   * (1) Kalaj, D.: Heinz-Schwarz inequalities for harmonic mappings in the unit ball. Ann. Acad. Sci. Fenn. Math. 41, 457–464 (2016)
  * (2) https://www.researchgate.net/project/Schwarz-lemma-the-Caratheodory-and-Kobayashi-Metrics-and-Applications-in-Complex-Analysis
  * (3) Chen, S., Hamada, H., Ponnusamy, S., Vijayakumar, R.: Schwarz type lemmas and their applications in Banach spaces. arXiv:2110.02767v1 [math.CV]
  * (4) Mateljević, M.: Schwarz lemma and Kobayashi metrics for harmonic and holomorphic functions. J. Math. Anal. Appl. 464, 78–100 (2018)
  * (5) Mateljević, M.: The Ahlfors-Schwarz lemma, curvature, distance and distortion. Bulletin T.CLIII de l’Académie serbe des sciences et des arts -2020 Classe des Sciences mathématiques et naturelles Sciences mathématiques 45, 67–117 (2020)
  * (6) Li, H., Mateljević, M.: Boundary Schwarz lemma for harmonic and pluriharmonic mappings in the unit ball. accepted in Journal of Mathematical Inequalities (November 1, 2021)
  * (7) Hamada, H., Kohr, G.: A rigidity theorem at the boundary for holomorphic mappings with values in finite dimensional bounded symmetric domains. Mathematische Nachrichten. (2021;1-9). https://doi.org/%****␣SchNM_MM_Malay.bbl␣Line␣125␣****10.1002/mana.202100023
  * (8) Graham, I., Hamada, H., Kohr, G.: A schwarz lemma at the boundary on complex Hilbert balls and applications to starlike mappings. J. Anal. Math. 140, 31–53 (2020)
  * (9) Burgeth, B.: A Schwarz lemma for harmonic and hyperbolic-harmonic functions in higher dimensions. Manuscripta Math. 77, 289–291 (1992)
  * (10) Khavinson, D.: An Extremal Problem for Harmonic Functions in the Ball. Canadian Mathematical Bulletin 35(2), 218–220 (1992). https://doi.org/10.4153/CMB-1992-031-8
  * (11) Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory. Springer, New York (1992)
  * (12) Khalfallah, A., Mateljević, M., Mhamdi, M.: Some properties of mappings admitting general Poisson representations. accepted in Med. J. Math (Mediterranean Journal of Mathematics) (published on October 2021.)
  * (13) Mateljević, M., Svetlik, M.: Hyperbolic metric on the strip and the Schwarz lemma for HQR mappings. Appl. Anal. Discrete Math. 14, 150–168 (2020)
  * (14) Mateljević, M., Khalfallah, A.: On some Schwarz type inequalities. J. Inequal. Appl. 164 (2020)
  * (15) Chen, Z., Liu, Y., Pan, Y.: A Schwarz lemma at the Boundary of Hilbert Balls. Chinese Annals of Mathematics, Series B 39, 695–704 (2018)
  * (16) Landrews, G., Askey, R., Roy, R.: Special Functions. Cambridge University Press, ??? (1999)
  * (17) Mateljevic, M., Salimov, R., Sevost’yanov, E.: Hölder and Lipschitz continuity in Orlicz-Sobolev classes, distortion and harmonic mappings. accepted in Filomat (October 12, 2021)
  * (18) Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin Heidelberg New York Tokyo (1993)
  * (19) Osserman, R.: A sharp Schwarz inequality on the boundary. Proc. Am. Math. Soc. 128, 3513–3517 (2000)
  * (20) https://www.researchgate.net/post/What_are_the_most_recent_versions_of_The_Schwarz_Lemma
  * (21) Kalaj, D., Vourinen, M.: On harmonic functions and the Schwarz lemma. Proc. Amer. Math. Soc. 140(1), 161–165 (2012)
  * (22) Zhu, J.-F.: Schwarz lemma and boundary schwarz lemma for pluriharmonic mappings. Filomat 32, 5382–5402 (2018)
  * (23) Zhu, J.-F.: A Harnack inequality on the boundary of the unit ball. J. Math. Anal. Appl. 438, 519–522 (2016)
  * (24) Tang, X., Liu, T., Zhang, W.: Schwarz lemma at the boundary and rigidity property for holomorphic mappings on the unit ball of $\mathbb{C}^{n}$. Proc. Amer. Math. Soc. 20, 1709–1716 (2016)
  * (25) Dai, S., Chen, H., Pan, Y.: The high order Schwarz-Pick lemma on complex Hilbert balls. Science China Mathematics (accepted December 28, 2009). https://doi.org/10.1007/s11425-010-0119-2
  * (26) Chen, S., Hamada, H.: Some sharp Schwarz-Pick type estimates and their applications of harmonic and pluriharmonic functions. J. Funct. Anal. (to appear)
  * (27) Leutwiler, H.: Best constant in the harnack inequality for the Weinstein equation. Aequationes Mathematicae 34, 304–315 (1987)
  * (28) Melentijević, P.: A proof of the khavinson conjecture in $\mathbb{R}^{3}$. Advances in Mathematics 352, 1044–1065 (2019). https://doi.org/10.1016/j.aim.2019.06.025
  * (29) Rudin, W.: Function Theorey of the Unit Ball in $\mathbb{C}^{n}$. Springer, New York (1980)
  * (30) https://www.researchgate.net/publication/325430073_Miodrag_Mateljevic_Rigidity_of_holomorphic_mappings_Schwarz_and_Jack_lemma