Yours is an interesting question. I am not aware. Steven Krantz's 2010 paper is one of the most recent reference papers in this respect. Also, maybe this paper will help, as well as its references:

https://www.researchgate.net/publication/264072709_SHARPENED_FORMS_OF_THE_SCHWARZ_LEMMA_ON_THE_BOUNDARY

Article SHARPENED FORMS OF THE SCHWARZ LEMMA ON THE BOUNDARY

Magda thanks.

I add a few comments:

Let $G$ be bounded connected open subset of complex Banach space,

$p\in G$ and $v\in T_p G$. We define $k_G(p,v)= \inf\{ |h|$,

where infimum is taking over all $h\in T_0 \mathbb{C}$ for which

there exists a holomorphic function such that $f: \mathbb{U}

\rightarrow G $ such that $f(0)=p$ and $df(h)=v$.

One can prove

\begin{thm}

Suppose that $G$ and $G_1$ are bounded connected open subset

of complex Banach space and $f: G \rightarrow G_1$ is

holomorphic. Then $k_G(fz,fz_1)\leq k_G(z,z_1)$ for all

$z,z_1\in G$.

\end{thm}

\begin{thm}

Suppose that $G$ is bounded connected open subset of complex

Banach space and $f: G \rightarrow G$ is holomorphic, $s_0=

dist(f(G),G^c)$, $d_0=diam(G)$ and $q_0=

\frac{d_0}{d_0+s_0}$. Then $k_G(fz,fz_1)\leq q_0 k_G(z,z_1)$.

\end{thm}

Perhaps there are applications of this result in the

Teichm\"{u}ller theory.

One can prove for example

\begin{thm}

Suppose that $G$ is bounded connected open subset of complex

Banach space and $f: G \rightarrow G_*$ is holomorphic, $G_*

\subset G$, $s_0= dist(G_*,G^c)$, $d_0=diam(G)$ and $q_0=

\frac{d_0}{d_0+s_0}$. Then $k_{G_*}(fz,fz_1)\leq q_0

k_{G_*}(z,z_1)$ for $z,z_1 \in G_*$.

\end{thm}

Hence we have

\begin{thm}

Let $D \subset C^n$ domain for which Carth\'{e}odory

pseudo-distance is distance and $f : D\rightarrow D$ holomorphic

mapping such that $\overline{f(D)}$ is a compact subset of $D$.

Then $f$ is contraction with respect to Carth\'{e}odory metric on

$D$. In particular $f$ has fixed points in $D$.

\end{thm}

Can we have further application?

Dear colleague,  I  add  two information related Jack's lemma.

Please consider

  1.LOWER SCHWARZ-PICK ESTIMATES AND ANGULAR

DERIVATIVES  by

J. MILNE ANDERSON  AND ALEXANDER VASIL’EV

2. “Some Remarks on Schwarz lemma

at the boundary”, by Tuğba AKYEL and Örnek, [email protected]

If someone  likes to  consider the question the following papers about Schwarz lemma are interesting (for introduction in the subject):

Robert Osserman, From Schwarz to Pick

to Ahlfors and Beyond, Notice AMS 46, 8, 868-873

Harold P. Boas, Julius and Julia:

Mastering the Art of the Schwarz Lemma

American Mathematical Monthly 01/2010; 117(9), 770-785

There is  the theory of holomorphic functions with domain and range contained in a complex Banach space. We then

review some well-known fixed point theorems for holomorphic functions. Perhaps the most basic is the Earle-Hamilton fixed point theorem, which may

be viewed as a holomorphic formulation of Banach’s contraction mapping

theorem.

A set $S$ is said to lie strictly inside a subset $G$ of a Banach space if there

is some $\epsilon > 0$ such that $B=(x,\epsilon) \subset G$ whenever $x \in S$. The following theorem

may be viewed as a holomorphic version of the Banach's contraction mapping theorem. 

[Earle-Hamilton  theorem].

Let $G$ be a nonempty domain in a complex

Banach space $X$

and let $h:G \rightarrow G$ be a bounded holomorphic function. If

$h(G)$ lies strictly inside

$G$, then $h$ has a unique fixed point in $G$.

The Earle-Hamilton theorem still applies in cases where the holomorphic

function does not necessarily map its domain strictly inside itself. In fact, the

following interesting fixed point theorem is a consequence of two applications

of the Earle-Hamilton theorem.

(Khatskevich-Reich-Shoikhet  theorem).

Let $G$

be a nonempty

bounded convex domain in a complex Banach space and let

$h:G \rightarrow G$

be a holomorphic function having a uniformly continuous extension to

$\overline{G}$. If

there exists an

$\epsilon > 0$

such that $|h(x)-x|\geq \epsilon $

whenever

$x \in \partial G$, then

$h$ has a unique fixed point in $G$.

Dear colleague,  I  add  two information related Jack's lemma.

Please consider

  1.LOWER SCHWARZ-PICK ESTIMATES AND ANGULAR

DERIVATIVES  by

J. MILNE ANDERSON  AND ALEXANDER VASIL’EV

2. “Some Remarks on Schwarz lemma

at the boundary”, by Tuğba AKYEL and Örnek, [email protected]

We will give a few   comments  concerning Schwarz and Jack  lemma  including some recent papers.

1.    See  Steven G. Krantz,

arXiv:1001.1805v1 [math.CV] 12 Jan 2010

The Schwarz Lemma at the,

Boundary,  January 13, 2010

Abstract:

The most classical version of the Schwarz lemma involves the behavior at the origin of a bounded, holomorphic function

on the disc. Pick’s version of the Schwarz lemma allows one to move

the origin to other points of the disc.

In the present paper we explore versions of the Schwarz lemma

at a boundary point of a domain (not just the disc).     Estimates on

derivatives of the function, and other types of estimates as well, are considered. We review recent results of several authors, and present

some new theorems as wel.

%%%%%%%

2.       M.Mateljevic,  Ahlfors-Schwarz   lemma   and    curvature, KragujevacJ.Math.    25 (2003),   155–164.

Abstract:

 In this note we give short review of known results and announce new re-

sults ( see below Theorem 8 and Theorem 6 and its generalizations) . In the first part

of this review paper, we focus on ultrahyperbolic metric and Ahlfors lemmas and to the

estimate opposite to Ahlfors-Schwarz lemma proved by the author  (Theorem 5-6). The

second part is devoted to Ahlfors-Schwarz lemma for harmonic-quasiregular maps and

some results obtained in [AMM]

3.       Ornek,  Bull. Koreian   Math Soc 

https://www.researchgate.net/publication/264072709_SHARPENED_FORMS_OF_THE_SCHWARZ_LEMMA_ON_THE_BOUNDARY

%%%%%

Bull. Korean Math. Soc.

50    (2013), No. 6, pp. 2053–2059

http://dx.doi.org/10.4134/BKMS.2013.50.6.2053

SHARPENED FORMS OF THE SCHWARZ LEMMA

ON THE BOUNDARY

Bulent Nafi    Ornek

Abstract.

In this paper, a boundary version of the Schwarz lemma is

investigated. We obtain more general results at the boundary. Also,

new inequalities of the Schwarz lemma at boundary is obtained and the

sharpness of these inequalities is proved.

%%%%

4.    Representation with majorant of the Schwarz lemma at the

boundary

Bulent Nafi ORNEK and Tugba AKYEL

Let f be a holomorphic function in the unit disc D and |f(z) -1|0$, $f$ be analytic function on $\overline{B}$, $f(B)\subset B$ and $f(0)=0$. Then $f'(0)=k$, where $k>0$.

Contrary, if $f'(0)=k e^{i \alpha}$, where $0

We outline an idea   which  may give  a small improvement of Burns-Krantz ,  Chlest and resent  Ornek-Akyel result.

Theorem 1 (Burns/Krantz).    Let  g  be analytic function of the unit disk U   into self   which satisfy   (i)  g(z)= z + O(1-z)^4  when z approaches 1  throughout  U.

Then g=Id.

The result   is the sharpest possible. Indeed, the example  g(z)= z – (z-1)^3 /10

shows that the exponent 4 in the theorem cannot be replaced by 3. The proof

in fact shows that 4      can   be replaced by  o((z-1)^3).

Def    A(z)= (1+z ) (1-z)^{-1},  A_g=  (1+g ) (1-g)^{-1},  B_g =A_g –A    and   h=Re (A_g- A).

Then    A’=2 (1-z)^{-2},  A’_g=2 (1-g)^{-2} g’,  and   B’_g=2 (1-g)^{-2} g’- 2 (1-z)^{-2}.

We consider the codition   (ii)   B’_g=o(1)  when z approaches 1 throughout  U.

 We can  rewrite the condition  (ii)      in the form   (iii)        (1-g)^{-2}(1-z)^2  g’= 1 + o((1-z)^2  ).

Set    (i1)  g(z)= z + o(1-z)^3 when z approaches 1  throughout  U.  We can  rewrite the condition  (i1)      in the form   (i2)  1- g(z)= 1- z + o(1-z)^3=(1- z)( 1 + o(1-z)^2 ).

Hence    (i1)    implies   (ii) .   By    the maximum principle  (ii)    implies  g=Id.

Recall   that   Steven G. Krantz  paper ,

arXiv:1001.1805v1 [math.CV] 12 Jan 2010

The Schwarz Lemma at the, Boundary,  January 13, 2010,  yields a  very interesting review   of the subject.  In particular, Variants in the Several Complex Variables Setting are considered  in Section 7.  We hope that  this paper  will  inspire some

new results.

Recall   also   that  in the previous  communication  we  gave a few comments concerning  connections between  Schwarz and Jack lemma including some recent papers.

Motiveted by Burns-Krantz , Chlest, resent Ornek-Akyel result  and in particulatr Krantz  arXiv review   we outline some further ideas concerning this very interesting subject,  which  may give  a small improvement of these   results.

Theorem 1 (Burns/Krantz).    Let  g  be analytic function of the unit disk U   into self   which satisfy   (i)  g(z)= z + O(1-z)^4  when z approaches 1  throughout  U.

 Then g=Id.

 The result   is the sharpest possible. Indeed, the example  g(z)= z – (z-1)^3 /10

shows that the exponent 4 in the theorem cannot be replaced by 3. The proof

in fact shows that 4      can   be replaced by  o((z-1)^3).

Def    A(z)= (1+z ) (1-z)^{-1},  A_g=  (1+g ) (1-g)^{-1},  B_g =A_g –A    and   h=Re (A_g- A).

Then    A’=2 (1-z)^{-2},  A’_g=2 (1-g)^{-2} g’,  and   B’_g=2 (1-g)^{-2} g’- 2 (1-z)^{-2}.

We consider the codition   (ii)   B’_g=o(1)  when z approaches 1 throughout  U.

 We can  rewrite the condition  (ii)      in the form   (iii)        (1-g)^{-2}(1-z)^2  g’= 1 + o((1-z)^2  ).

Set    (i1)  g(z)= z + o(1-z)^3 when z approaches 1  throughout  U.  We can  rewrite the condition  (i1)      in the form   (i2)  1- g(z)= 1- z + o(1-z)^3=(1- z)( 1 + o(1-z)^2 ).

Hence    (i1)    implies   (ii) .   By    the maximum principle  (ii)    implies  g=Id.

Using   Bg(z_2) -   Bg(z_1)       is the integral    of     Bg’  over [z_1, z_2],    by (ii), we conclude that   B_g      satisfies Caushy    condition    when z approaches 1 throughout  U.

Hence   B_g        has continuous extention  to 1 and   the directional derivative  in direction (-1,0)   at   1 is  D_r B_g(1)=0.

Ornek-Akyel use the following maximum

principle. If u is harmonic on the unit disk and for every w\in T \liminf_{z\to w} u(z)\ge 0, then u\ge 0.

 Hence   h \ge 0    and  by

Hopf  lemma   h=0.

Under hypothesis  of  Proposition 6.3  (Krantz  arXiv),

h(z) |1 – B(z)|^4=O(1-z)^4 .

In summary, we have an h^2 (U)   harmonic function with nonnegative

radial boundary limits almost everywhere. It then follows, from the Poisson

integral formula for instance, that h is positive on the disc U.

Instead of  Hopf  lemma   we can use:

Theorem 1.  Let f: U  to  U be a holomorphic function and have a continuous

limit at some  z_0 \in T, and let  f(z_0)=1.  Then    |1-f(z)|\ge (1 -|z|)  c,     where   c= (1- Re f(0))/2.

Outline. Let  u= Re f  and v =1-u. Then    v=P[v^*]  and therefore   v\ge (1-r)(1

+r)^{-1}  v(0), where r=|z|.   Next  |1-f(z)|\ge v.

For the result of this type   see also   [2] and   Theorem  1.1[3].

1.    See  Steven G. Krantz,

arXiv:1001.1805v1 [math.CV] 12 Jan 2010

The Schwarz Lemma at the,

Boundary,  January 13, 2010

2.    M. Mateljevic:

Distortion of harmonic functions and harmonic quasiconformal quasi-isometry,

 Revue Roum. Math. Pures Appl. Vol.  51  (2006) 5–6, 711–722.

3.    Miodrag Mateljevi ́c, The Lower Bound for the Modulus of the Derivatives and

Jacobian of Harmonic Injective Mappings,

Filomat 29:2 (2015), 221–244

DOI 10.2298/FIL1502221M

Published by Faculty of Sciences and Mathematics,

University of Nis, Serbia

Available at:

http://www.pmf.ni.ac.rs/filomat

4. Ornek-Akyel,    “Representation with majorant of

the Schwarz lemma at the boundary”,

We will give a few   comments  concerning Schwarz and Jack  lemma  including  some recent papers.

4.    See  Steven G. Krantz,

arXiv:1001.1805v1 [math.CV] 12 Jan 2010

The Schwarz Lemma at the,

Boundary,  January 13, 2010

Abstract:

The most classical version of the Schwarz lemma involves the behavior at the origin of a bounded, holomorphic function

on the disc. Pick’s version of the Schwarz lemma allows one to move

the origin to other points of the disc.

In the present paper we explore versions of the Schwarz lemma

at a boundary point of a domain (not just the disc).     Estimates on

derivatives of the function, and other types of estimates as well, are considered. We review recent results of several authors, and present

some new theorems as wel.

%%%%%%%

 M.Mateljevic,  Ahlfors-Schwarz   lemma   and    curvature, KragujevacJ.Math.    25 (2003),   155–164

A version  of manuscript  Schwarz and Jack lemma(Note on Rigidity of holomorphic mappings \& Schwarz and Jack lemma (In preparation)}

is enclosed.

This a very rough version for temporarily use. But in the enclosed manuscript we  give an answer to the posed question.

\begin{abstract}

Motivated by Burns-Krantz, Chlest, resent Ornek-Akyel result and in

particulatr Krantz arXiv review we outline some further ideas

concerning this very interesting subject, which may give a small

improvement of these results. We use non negative harmonic function as a tool.

We also give a short review of the subject.

\end{abstract}

The subject related to Jack's lemma has been discussed by Ornek in a recent paper.

By $S^*(\alpha)$ we denote the family of starlike univalent functions of of order $\alpha$.

$f\in S^*(\alpha)$ if ${\rm Re} \frac{z f'(z)}{f(z)}> \alpha$. Set $S^*=S^*(0)$ and $C^*=C^*(0)$. It seems that the following result is true.

Lemma:

If $f$ belongs $S^*(\alpha)$, $0< \alpha

Recall first   that the subject related to Jack's lemma has been discussed by \"{O}rnek in a recent paper. It seems that the following result is true.

\begin{thm}

If $f$ belongs $S^*(\alpha)$, $0< \alpha

We expect  that there is  Schwarz lemma for J-holomorphic mapping (It is possible that it  is   known).

 Let F be $(j,J)$-holomorphic from the unit ball $B^n$ of

$\mathbb{C}^n$ into self, $a \in B^n$ and $b=f(a)$. Set $F=

 \varphi_b  \circ f\circ \varphi_a$. It seems that then $F$ is  $(j,jJ)$-holomorphic and we can apply Gromov-Schwarz lemma and   use

 the known estimate for $\varphi_a'(0)$ and $\varphi_b'(b)$.

It seems that we can apply  [Gromov-Schwarz   lemma:

\begin{thm}[Gromov-Schwarz]

Let $(M,J,\mu)$ be a compact Hermitian

 manifold. There exist positive constants $\epsilon_0>0$ and $c>0$

 with

 the following properties.

If $g:\mathbb{U}\rightarrow M$ is a J-holomorphic such that

 $g(\mathbb{U})$ is contained in some $\epsilon_0$-ball $B_{\epsilon_0}

 \subseteq M$, then the norm of the differential of $g$ at the origin     is

 bounded by $c$:     $|dg_0|

It seems that as an application of Heinz's inequality, we can get the following estimate.

Define $P(x_1,x_2,x_3)= (x_1,x_2)$ and let let $D$ be a domain in plane and $X:D\rightarrow \mathbb{R}^3$, the projecting mapping $f$ of $X$ is $P\circ X$.

\begin{thm}

Let $X: \mathbb{U} \rightarrow \mathbb{R}^3$ be harmonic such that the projecting mapping $f$ is injective and $f(\mathbb{U})\supset \mathbb{U}$. If $f(0)= 0$, then

$|X_z(0)|^2 + |X_{z}(0)|^2 \geq c_0$, where $c_0 = 27/4\pi^2 = 0.6839 . . . $.

\end{thm}

In particular, let $S$ be a minimal graph (minimal surface) over a domain $G$ which contains the unit disk $\mathbb{U}$ and $X: \mathbb{U} \rightarrow S$ isothermal parametrization. If $P( X(0))=0$,

then the above estimate holds.

The interesting results   are  obtained    in  Kalaj and Vuorinen,

On harmonic functions and the Schwarz lemma. Proc. Amer. Math. Soc. 140 (2012), no. 1, 161-165.

We (in communication with Franc Forstneric) suggest some application  of Kalaj- Vuorinen  results   to minimal surfaces.

It seems that we can get small improvements.

For example, we prove

If $f$ is harmonic map from the unit disk $\D$ into $\mathbb{S}_0$ with $f(0)=0$, then

\[

|\nabla f(z)| \leq \frac{4}{\pi} \cos (\frac{\pi}{2} \Re f(z)) (1- |z|^2)^{-1}, \quad z\in \D.

\]

M. Markovic  (arXiv:1508.00125v1 [math.CV] 1 Aug 2015,

PROOF OF THE KHAVINSON CONJECTURE NEAR THE

BOUNDARY OF THE UNIT BALL) has found the sharp constants in the estimates for the absolute value of the radial derivative and the modulus of the gradient of a bounded harmonic function coincide near the boundary of the unit ball. It is an interesting result.

The following question is related to the paper.

Question. Let $h_b$ denote the family of harmonic mappings $f$ from the unit disk into self such that $f(0)=b$. Find

$\max \{ L_f(0): f\in h_b\}$.

We have the following.

\begin{prop}

$L_f(0)\leq \sqrt{2}\sqrt{1-|f(0)|^2}$.

\end{prop}

Here, for a function $h$, we

use notation $p=\partial h=\frac{1}{2}(h_{x}^{\prime

}-ih_{y}^{\prime })$ and $q=\overline{\partial

}h=\frac{1}{2}(h_{x}^{\prime }+ih_{y}^{\prime })$, $L_h=|p|+ |q|$ and $l_h=|p|- |q|$.

Let $\mathbb{S}= \mathbb{S}_0= \{w : |Re w|< 1\}$ and

let $B^2$ denote the unit ball in $\mathbb{C}^2$.

Let $B^2$ denote the unit ball in $\mathbb{C}^2$.

Question 2. Let $f:B^2\rightarrow \mathbb{S}_0 \times \mathbb{S}_0$ be a holomorphic mapping with $f(0)=0$.

Find estimate for $M(r)= M_f(r) = \{ |f(z)|: |z|=r \}$.

Whether $|f(z)|\leq \sqrt{2}\frac{2}{\pi} \ln \frac{1+r}{1-r}$, where $r= |z|$?

We outline a proof.

Let $\phi$ be conformal of $\mathbb{U}$ onto $\mathbb{S}_0$ with $\phi(0)=0$, $\underline{\phi}(z,w )= (\phi(z), \phi(w))$

and $\varpi= \underline{\phi}^{-1} \circ f$. Then $f=\underline{\phi} \circ \varpi $ and $|\varpi(\mathbf{z})|\leq |\mathbf{z}|$. Hence

$|f(z)|\leq \sqrt{2}\frac{2}{\pi} \ln \frac{1+r}{1-r}$, where $r= |z|$.

We also can derive a version of Schwarz's Lemma for pluriharmonic mappings.

For $a$ we define $s=s_a= \sqrt{1- |a|^2}$.

We outline a geometric proof  of Schwarz lemma  for several variables.

For $\epsilon>0$ set $E(a,\epsilon)=\varphi_a(\epsilon B)$. Then $z \in E(a,\epsilon)$ iff $|\varphi_a(z)|< \epsilon$

$$\frac{|Pz -c|^2}{\epsilon^2 \rho^2} + \frac{|Qz|^2}{\epsilon^2 \rho}

There is a version of Schwarz lemma for  harmonic quasi-conformal

mappings.

Let $\lambda$ denote hyp distance.

In

[MaK]{MaK} {M. Kne\v zevi\' c, M. Mateljevi\' c,}

{\it On the quasi-isometries of harmonic quasi-conformal

mappings} J Math Anal Appl, 2007, 334(1), 404--413,

it is proved

Let $f$ be a $\,k\,$-\,quasiconfo\-rmal euclidean harmonic mapping

from the unit disc $\mathbb{U}$ into itself. Then for all $\,z\in

\mathbb{U}$ we have

$$

|f_z(z)|\leq\frac{1}{1-k}\,\frac{1-|f(z)|^2}{1-|z|^2}.

$$

Using $L_f(z) \leq (1+k) |f_z(z)|$,

we can rewrite it in the form:

(A) $\lambda(fz',fz) \leq K\lambda(z',z)$, $z',z \in \mathbb{U}$.

Question. Whether (A) holds if codomain is convex set?

Outline (A). Define $\,\sigma_f(z)=(1-k)^2\lambda(f(z))|f_z(z)|^2$

and check that the curvature $\mathbf{K}(\sigma)(z)\leq -1$.

In communication with Pavlovic appears the following  question:

Whether (A) holds if $f$ is k-qr?

If $f$ is k-qr, then

$f=F\circ g$, where $F$ is analytic, and $g$ k- qc.

Whether $\mathbf{K}(\sigma_g)(z)\leq -1$ ?

We will add a few things to my previous answer. The metric density    $\,\sigma_f(z)=(1-k)^2\lambda(f(z))|f_z(z)|^2$

has  the curvature $\mathbf{K}(\sigma)(z)\leq -1$ out of the set of zeros of  f_z  and  it is an  ultrahyperbolic metric on the unit disk.  By Ahlfors-Schwarz lemma

(A) holds if $f$ is k-qr.

For the Schwarz lemma for harmonic functions in several variables see for example Theorem 6.16 [1].

Let H^c denote the family of all harmonic (hyperbolic-harmonic) functions on the unit ball B^p with h(0) = c, 0 < h < 1.

We can prove the corresponding version of Theorem 6.16 [1] for $H^c$ (cf. [2]).

[1] Axler, P. Bourdon and W. Ramey}:

Harmonic function theory, Springer-Verlag, New York 1992.}

[2] M. Mateljevic, Schwarz lemma and Kobayashi metrics for harmonic and holomorphic functions, Journal of Mathematical Analysis and Applications,

Available online 4 April 2018, In Press, Accepted Manuscript, https://doi.org/10.1016/j.jmaa.2018.03.069

There is the following version of Schwarz lemma. For $a\in (-1,1)$, let ${\rm Har}^a$ denote the family of all complex valued harmonics maps $f$ from $\mathbb{U}$ into $(-1,1)$ with $f(0)=a$. Set $c=(a+1)/2$, $\overline{c}= 2\pi c$, $\alpha=\alpha(c)= \overline{c}/2$ and $X(r)=\frac{4}{\pi} \arctan\left(\frac{1 + |z|}{1 - |z|} \tan \frac{\alpha(c)}{2}\right) -1$, where $r= |z|$. \bthm If $h\in {\rm Har}^a$, then $h(z)\leq X(|z|)$

and $|(dh)_0| \leq X'(0)=\frac{4}{\pi} \sin \alpha$. \ethm

If $a=0$, this results reduces to classical Schwarz lemma.

My submission (related to Schwarz lemma) submit/2254433 has been assigned the permanent arXiv

identifier 1805.02979 and is available at:

http://arxiv.org/abs/1805.02979

[1]  Schwarz lemma and distortion for harmonic functions via length and area, 8 May 2018.

Authors: Miodrag Mateljevi\'c

See the file:

Is the Schwarz lemma also relevant in matrix domains? Let's say the first type is in the classical domains? The first type of matrix ball?

Is the Schwarz lemma also relevant in matrix domains? Let's say the first type is in the classical domains? The first type of matrix ball?

As far as I know there are versions of Schwarz lemma in classical domains.

Concerning the planar theory see also manuscript

Note on some classes of holomorphic functions related to Jack's and Schwarz's lemma which is uploaded recently on ResearchGate. Here is short content of the manuscript:

The first author has started to consider some project related to Schwarz lemma with collaborators Svetlik, Khalfallah and Ornek [7, 4, 6, 18, 11]. Here we continue this investigation and outline some possibilities for further research. Concerning the content of this paper (with respect to application of Jack's lemma) note that we first obtained results in section 2 (Growth of alpha-starlike function) related to the growth of some classes of functions which include starlike functions and extend some  Ornek's result. Then studying the approach presented in section 2, later we have developed a more general mapping method for estimates based on Jack's lemma (see section 4, Theorem 4.1, and Theorem 4.3), which we call approach via Jack's lemma, to study growth of some classes of holomorphic functions. From Theorem 4.1 we derive Theorem 4.2. Theorems 4.3 and 4.4 are further results related to subordination principle. From these results it became clear that the growth estimate of starlike functions, Theorem 2.1, can be consider as special case of application of our general method. In addition we study growth of holomorphic function in Theorem 5.1. In Section 6 we give a short review of some results related to the subject.