# Defining maximum acceptable latency of AI-enhanced CAI tools

Claudio Fantinuoli1,2, Maddalena Montecchio1
1 Johannes Gutenberg University Mainz
2 KUDO

##  1 Abstract

Recent years have seen an increasing number of studies around the design of
computer-assisted interpreting tools with integrated automatic speech
processing and their use by trainees and professional interpreters. This paper
discusses the role of system latency of such tools and presents the results of
an experiment designed to investigate the maximum system latency that is
cognitively acceptable for interpreters working in the simultaneous modality.
The results show that interpreters can cope with a system latency of 3 seconds
without any major impact in the rendition of the original text, both in terms
of accuracy and fluency. This value is above the typical latency of available
AI-based CAI tools and paves the way to experiment with larger context-based
language models and higher latencies.

##  2 Introduction

Automatic speech recognition (ASR) has been regarded as a technology “with
considerable potential for changing the way interpreting is practiced” [23].
In particular, ASR has been proposed as a means to overcome the shortcomings
in current implementations of computer-assisted interpreting (CAI) tools, such
as the inherent difficulty to manually lookup terms while interpreting. By
integrating AI-based features in the interpreter workstation, such as the
real-time automatic suggestion of numbers and other problem triggers [13], and
by integrating advancements in extractive and predictive algorithms based on
machine learning [28], the cognitive load required by the use of CAI tools may
be reduced, offering new opportunities for interpreters to improve their
performance. First empirical research seems to point in this direction [8,
14].

AI-enhanced CAI tools are generally based on the concatenation of several
modules that may comprise a speech-to-text transcription engine, a parsing
module to identify the units of interest, for example numbers, terminology or
proper names, and a visualisation component for the human-machine interaction.
Other implementations may adopt an end-to-end approach, making the cascading
architecture obsolete. Such tools can be deployed on the edge, running
completely on the user’s device, or on the web, hosted on servers and accessed
through a Web Browser. Generally speaking, CAI tools present suggestions with
a certain amount of delay with respect to the original speech. This delay is
due to the architectural design of the tool and its components, for example
the ASR latency or the computation time needed to inference a language model,
but also to the intrinsic nature of the task, which may require a certain
amount of linguistic context in order to make an informed decision on what to
show to the final user. In a time-sensitive activity such as simultaneous
interpreting, system latency may require interpreters to adopt new
interpreting strategies to successfully integrate the suggestions into their
rendition. If latency, however, exceeds a certain threshold, interpreters may
not be able to integrate the suggestions, or may do this at the expenses of
fluency, cohesion, or even accuracy. In this case, the use of such high-
latency tool would become detrimental for the user-machine interaction, for
the interpreter’s cognitive load, and finally for the quality of the
rendition.

Not much is known about this maximum acceptable latency threshold, i.e. the
maximum ear-voice-span (EVS) that interpreters can cope with in order to
successfully integrate external suggestions without impacting the overall
interpreting performance. Not only this value may vary among subjects, but it
may also be different from the one reported in experimental analysis of
interpretation performed without the use of a CAI tool.

Our hypothesis is that displaying suggestions within the average EVS reported
in literature should not negatively impact the interpreter rendition, even
when this is ‘forced’ upon the interpreter by the latency with which the tool
generates such suggestions. To our knowledge this is the first study that
tries to empirically answer this question. On the one hand, knowledge about
this threshold is crucial to understand if current AI-based CAI
implementations can already meet interpreters expectations or if major efforts
should be placed in further reducing system latency. On the other hand, an
acceptable higher latency could allow the integration of more time-demanding
NLP features, such as the automatic prediction of difficult parts of the
speech [[, eg.]]vogler_lost_2019.

The reminder of this paper is organized as follows. Section 3 introduces the
related work in the area of CAI tools and the empirical experiments conducted
on them so far. Section 4 describes the data and the methodology adopted in
this experiment. Section 5 introduces the evaluation framework. Section 6
presents the results. Finally, section 7 discusses the conclusion and the
outlook.

##  3 Related work

Recent years have seen an increasing number of studies around the design of
AI-based computer-assisted interpreting tools and their use by trainees and
professional interpreters.

Computer-assisted interpreting tools are digital devices designed to support
the interpreter in different steps of their work, from preparation to the very
act of interpreting. They have been proposed by several researchers in the
past 20 years or so [29, 24, 27], but it is only recently that there has been
a surge in interest among the community’s members.

Thanks to new advances in artificial intelligence, ASR has reached a quality
level that makes it suitable for integration into supportive technologies. By
automating and extending the query system of CAI tools, this integration may
solve the shortcomings of traditional tools [[,
eg.]]fantinuoli_interpretbank._2016, fantinuoli_computer-assisted_2017,
hansen-schirra_nutzbarkeit_2012 and extend the features available in an
interpreters’ workstation, for example automatically suggesting translations
of specific terms as well as transcribing numbers and proper names in real-
time.

Over the years, a handful of empirical studies have been carried out to test
the feasibility of the human-machine interaction in the simultaneous modality.
They have focused in particular on the effectiveness of ASR-support during the
interpretation of numbers [9, 8, 14], one of the problem triggers of
simultaneous interpreting identified in literature [4, 15, 17, 26]. In order
to measure the impact on the quality of the rendition, these studies have used
either mock-up systems with a very short latency [9, 6] or real-life tools
with a reported latency of under 2 seconds [8, 14].

From an interpreter perspective, the system latency impacts the ear-voice span
(EVS) of interpreters since it forces the interpreter to wait a certain amount
of time before being able to integrate the suggestions into their rendition.
The EVS is the amount of time that separates the words uttered by the speaker
in the source speech and the equivalent rendition in the target speech uttered
by the interpreter [23]. Research in this field has a “long-standing
tradition” [7] and EVS is considered an essential variable that can
potentially impact interpreters’ performance [2, 16, 17, 20] such that it
continues to be an object for analysis and assessment.

Previous research has found that the variable limits of interpreters’ EVS are
generally attributable to two main factors: input-related factors, which might
also enable interpreters to consciously regulate their lag, and personal
factors, e.g. short term memory-related factors, that influence the maximum
capacity of the memory [26, 1, 19]. Interpreters are free to regulate - at
least to a certain degree - their EVS depending on the features of the speech
segment they are translating, increasing or decreasing it consciously as a
strategy [1, 10].

Measuring EVS is not immediate and easy, due to the fact that grammar, word
order and syntactic structure might be different in the two languages [26].
Average EVS reported in literature is between 2 and 3 seconds [2, 19, 20],
with a peak of 10 seconds [22]; if measured in words, a mean ranging from 5 to
10 words has been reported [25] .

##  4 Data and methodology

###  4.1 Dataset

For this experiment we used numbers as unit of interest to be suggested by the
system, as this is a problem trigger that has been largely studied so far in
empirical experiments related to CAI tools. We choose and edited therefore a
speech particularly dense with numbers. The speech had a duration of 7 minutes
and 15 seconds and did not pose any particular difficulties in terms of
terminology.

The speech was delivered in English by a non-native speaker and participants
were asked to translate it simultaneously into their native language
(Italian). The average speech pace was 105.5 words per minute, which
corresponds to an ideal speech rate for the simultaneous interpretation of
improvised speeches and which is close to the ideal read-aloud speech rate of
100 words per minute. The speech had 25 stimuli (numbers); 10 of them were
accompanied by a referent.

The speech was prerecorded and a video simulating an ASR system was edited ad-
hoc by the authors of the experiment in order to retain full control over the
variable ‘latency’. In the video the transcription of the numbers (without the
relevant referent) was shown on screen after a variable delay compared to the
moment the participants received the acoustic signal in the headset. The
acoustic numerical input was removed from the recorded speech and replaced
with a neutral acoustic signal (/beep/). This choice was made in order to
force the participants to use the visualization of the numbers on screen, thus
relying exclusively on the ASR simulation. Every visual input (number) was
displayed based on a preset latency, which gradually increased during the
course of the speech. Numbers were shown in isolation with no embedded
transcription of the speech. The text was divided into five different
sections. In every section the visualization of the stimulus occurred with a
different latency – in the first section, the numbers appeared on the screen 1
second after the acoustic signal, in the second section after 2 seconds, and
so forth. During their performance, participants received the visual numerical
input on a maxi screen situated inside the classroom or on the monitors inside
the booths.

###  4.2 Participants

A total of eight participants took part in this study. All participants were
students. They were Italian native speakers with German and English in their
language combination enrolled in their final year of a Master in Conference
Interpreting program and had at least 1 year experience in simultaneous
interpretation. The participants did not have past experience of interpreting
with CAI tools or ASR-enhanced CAI tools. This condition is in line with
experimental setups adopted in similar studies [[,
eg.]]defrancq_automatic_2020, fantinuoli_measuring_2021.

Some weeks before the experiment, the participants were provided with three
short videos containing a simulation of the ASR system in order to be
minimally exposed to the experimental format. These videos were edited along
the lines of the video used for the actual experiment, but all numerical
visual inputs were displayed with the exact same latency, in contrast with the
variable latency with which the numerical visual inputs were displayed in the
experiment. The goal was to help the participants to get accustomed to the
experiment format, i.e., the way numbers were transcribed, their font size,
the color and the way the order of magnitude was shown [14].

##  5 Evaluation framework and procedure

The analysis of the collected data has been performed on two different levels,
namely on a stimuli-based and on a segment-based level.

The aim of the stimuli-based evaluation is to assess the accuracy level
achieved by the interpreters in rendering the units of interest in the target
language, depending on the variable latency of the mockup system. Within this
level of assessment, the accurate rendition of the units of interest is
evaluated considering the following components: the numerical information
comprising its order of magnitude, and, if present, the relevant referent. A
number of parameters were collected for the evaluation: presence of the number
in the rendition, accuracy of the number, presence of the referent, accuracy
of the referent, pronunciation disfluency. Numbers and referents that were
approximated, generalized or omitted were considered to be errors. Moreover,
lexical, syntactic, phonological or articulation mistakes were also classified
as errors.

The segment-based evaluation assesses the quality of the entire segments in
which the numbers are embedded. This evaluation focuses on two different
parameters. The first is the accuracy of the target segment, measured in terms
of the following linguistic aspects: faithfulness, grammatical correctness,
completeness, logical cohesion, consistency, plausibility. The second
parameter is the listener’s perception in terms of delivery flow. This
evaluation level focuses on the following paralinguistic aspects: perception
of the interpreters’ voice and rhythm, eloquence, presentation, prosody and
communicative effectiveness. The segment-based evaluation has been performed
using a Likert scale (1 to 5) by three different evaluators.

There are three major limitations in this study that could be addressed in
future research. Firstly, the level of interpretation proficiency of the
participants selected for this experiment is biased towards the lower end. It
is reasonable to assume that a randomized sample in terms of proficiency, for
example including also professional interpreters, may increase the maximum
acceptable latency measured in the experiment. Secondly, the experiment
focused only on a single language pair. Because of intrinsic variations among
languages, EVS and different strategies adopted by interpreters on the basis
of the language combination may influence the threshold under scrutiny.
Finally, variations in speech complexity, for example in terms of syntactic
structures, delivery speed etc. have not been taken into consideration in our
study. However, they may have a considerable impact on interpreters cognitive
load and consequently on their ability to successfully integrate suggestions
and may require a dynamic adaptation of latency according to this variable.

##  6 Results

###  6.1 Stimuli-based evaluation

As introduced in the previous section, the stimuli-based evaluation focuses on
the stimuli (numbers) and their referents and not on the whole sentence in
which they are embedded.

No number and no referent has been omitted at any latency by any of the
candidates. The precision, however, varies according to the latency. Figure 1
presents the accuracy of the rendition of the numbers displayed by the ASR
simulation system. The best results were achieved with the first three
latencies: 97.14% at 1 and 2 seconds and 98.85% at 3 seconds. The better score
at 3 seconds may be explained with the interpreters acclimatisation to the
dynamic of the experiment, thus enabling them to adapt their interpreting
approach after having generated a learning effect [21]. The worst results were
observed when the latency increased above 3 seconds, and in particular 93.14%
at 4 and 94.86% at 5 seconds.

$1$$2$$3$$4$$5$$94$$96$$98$ Figure 1: Accuracy of number rendition

In Figure 5 the accuracy of the referents of the displayed numbers is
presented. The highest score was reached with a latency of 3 seconds (100%).
The accuracy decreases with the two higher latencies: 94.28% at 4 seconds and
85.71% at five seconds.

$1$$2$$3$$4$$5$$85$$90$$95$$100$ Figure 2: Accuracy of reference rendition

The number of pronunciation disfluencies (Figure 3) reached a pick with a
latency of 4 seconds (25.71% of pronounced numbers present a disfluency). With
lower latencies the number of disfluencies is low: 2.85% of numbers were
disfluent at 1 second and 5.71% at 2 and 3 seconds.

$1$$2$$3$$4$$5$$5$$10$$15$$20$$25$ Figure 3: Disfluency in number rendition

###  6.2 Segment-based evaluation

The segment-based evaluation aims at extending the assessment of the quality
of the rendition from the unit of interest to the entire segment in which this
unit appears. The segment-level accuracy was higher with a latency of 1 and 2
seconds. In particular, the best score was reached with 2 seconds. As in the
stimuli-based evaluation, this leads us to conclude that the participants,
although able to successfully integrate the accurate interpretation of the
numbers displayed in their own EVS, needed a settling period in order to gain
familiarity with the speech and the experimental format.

$1$$2$$3$$4$$5$$4$$4.2$$4.4$ Figure 4: Accuracy at segment level

A minimal decline in segment-level accuracy was observed from latency 3
onward, while with 5 seconds latency the accuracy degrades quite considerably.
It can be assumed that with this high latency, participants experienced
difficulties in allocating the correct amount of cognitive resources, since
they were required to focus on several tasks – on the screen with the number
displayed with a specific delay, on retaining the information they had just
heard in working memory and on processing the new segments of the speech that
continued to be uttered by the speaker [[, eg.]]gile_basic_2009.

The fluency of the delivery was evaluated high in the latency range from 1 to
3 seconds. Paralinguistic aspects, e.g., the voice and the fluency of the
rendition have been rated more agreeable in the first three sections. With 4
seconds, the delivery flow deteriorated significantly, and it reached its
lowest score in the 5 seconds latency section. It is plausible to assume that
not only the increasing cognitive load triggered by a longer EVS also had
significant repercussions on the manner in which the interpreters delivered
the target speech, as it adversely affected paralinguistic aspects, such as
the rhythm, the intonation, the voice and the appearance [[,
eg.]]buhler_linguistic_1986, becerra_first_2016.

$1$$2$$3$$4$$5$$3.2$$3.4$$3.6$$3.8$$4$ Figure 5: Fluency at segment level

##  7 Conclusions and outlook

In this paper we presented the results of an empirical experiment aimed at
measuring the maximum acceptable latency of an automatic suggestion feature
for simultaneous interpretation. The results seem to suggest that interpreters
are able to integrate suggestions by ad-hoc extending their ear-voice-span to
2 seconds without compromising the quality of their rendition and to 3 seconds
without any major disruption. A further extension of the system latency seems
to induce a consistent reduction of precision in the use of such suggestions
and in the emergence of information losses in the overall rendition. This is
in line with our original hypothesis that the system latency should not exceed
the average interpreter’s EVS. Within the latency threshold outlined in the
experiment, next generation AI-enhanced CAI tools could be able to accommodate
more complex and context-based NLP-features without a significant risk of
impairing the usability of the tool.

## References

  * [1] Linda Anderson, Sylvie Lambert and Barbara Moser-Mercer  “Simultaneous Interpretation: Contextual and translation aspects”  In _Bridging the Gap: Empirical Research in Simultaneous Interpretation_ Benjamins Translation Library, 1994, pp. 101–120
  * [2] Henri Barik  “Simultaneous Interpretation: Qualitative and Linguistic Data”  In _Language and Speech_ 18.3, 1975, pp. 272–297
  * [3] Garcia Becerra  “Do first impressions matter? The effect of first impressions on the assessment of the quality of simultaneous interpreting”  In _Across Languages and Cultures_ 17.1, 2016, pp. 77–98
  * [4] Sabine Braun and Andrea Clarici  “Inaccuracy for Numerals in Simultaneous Interpretation: Neurolinguistic and Neuropsychological Perspectives”  In _The Interpreters’ Newsletter_ 7, 1996, pp. 85–102
  * [5] Hildegrund Bühler  “Linguistic (semantic) and extra-linguistic (pragmatic) criteria for the evaluation of conference interpretation and interpreters”  In _Multilingua_ , 1986, pp. 231–235
  * [6] Sara Canali  “Technologie und Zahlen beim Simultandolmetschen: utilizzo del riconoscimento vocale come supporto durante l’interpretazione simultanea dei numeri”, 2019
  * [7] Bart Defrancq  “Corpus-based research into the presumed effects of short EVS”  In _Interpreting. International Journal of Research and Practice in Interpreting_ 17.1, 2015, pp. 26–45
  * [8] Bart Defrancq and Claudio Fantinuoli  “Automatic speech recognition in the booth: Assessment of system performance, interpreters’ performances and interactions in the context of numbers”  In _Target. International Journal of Translation Studies_ , 2020  DOI: 10.1075/target.19166.def
  * [9] Bart Desmet, Mieke Vandierendonck and Bart Defrancq  “Simultaneous interpretation of numbers and the impact of technological support”  In _Interpreting and technology, Language Science Press_ Language Science PRess, 2018, pp. 13–27
  * [10] Valentina Donato  “Strategies adopted by student interpreters in SI: a comparison between the English-Italian and the German-Italian language-pairs”  In _Interpreter’s Newsletter_ 12, 2003
  * [11] Claudio Fantinuoli  “Computer-assisted preparation in conference interpreting”  In _Translation & Interpreting_ 9.2, 2017, pp. 24–37
  * [12] Claudio Fantinuoli  “InterpretBank. Redefining computer-assisted interpreting tools”  In _Proceedings of the Translating and the Computer 38 Conference_ London: Editions Tradulex, 2016, pp. 42–52
  * [13] Claudio Fantinuoli  “Speech Recognition in the Interpreter Workstation”  In _Proceedings of the Translating and the Computer 39_ London: London, 2017
  * [14] Claudio Fantinuoli and Elisabetta Pisani  “Measuring the impact of automatic speech recognition on interpreter’s performances in simultaneous interpreting”  In _Empirical studies of translation and interpreting: the post-structuralist apporach_ Routledge, 2021
  * [15] Francesca Frittella  ““70.6 Billion World Citizens”: Investigating the difficulty of interpreting number”  In _Translation & Interpreting_ 11.1, 2019, pp. 79–99
  * [16] David Gerver  “Effects of Grammaticalness, Presentation Rate, and Message Length on Auditory Short-Term Memory”  In _Quarterly Journal of Experimental Psychology_ 21.3, 1969, pp. 203–208
  * [17] Daniel Gile  “Basic Concepts and Models for Interpreter and Translator Training: Revised edition”  Amsterdam: John Benjamins Publishing Company, 2009
  * [18] Silvia Hansen-Schirra  “Nutzbarkeit von Sprachtechnologien für die Translation”  In _trans-kom_ 5.2, 2012, pp. 211–226
  * [19] Marianne Lederer  “Simultaneous Interpretation—Units of Meaning and other Features”  In _Language Interpretation and Communication_ Springer, 1978, pp. 323–332
  * [20] Taehyung Lee  “Ear Voice Span in English into Korean Simultaneous Interpretation”  In _Meta_ 47.4, 2004, pp. 596–606
  * [21] Christopher D. Mellinger and Thomas A. Hanson  “Interpreter traits and the relationship with technology and visibility”  In _Translation and Interpreting Studies_ 13.3, 2018, pp. 366–392  DOI: 10.1075/tis.00021.mel
  * [22] P. Oléron and H. Nanpon  “Research into simultaneous translation”  In _The interpreting studies reader_ , 2001, pp. 69–76
  * [23] Franz Pöchhacker  “Introducing Interpreting Studies”  Routledge, 2016
  * [24] Anja Rütten  “Terminology Management Tools for Conference Interpreters – Current Tools and How They Address the Specific Needs of Interpreters”  In _Proceedings of the 39th Conference Translating and the Computer_ Geneva: Tradulex, 2017, pp. 98–103
  * [25] Nicholson Schweda  “Linguistic and Extralinguistic Aspects of Simultaneous Interpretation”  In _Applied Linguistics_ 8, 1987, pp. 194–205
  * [26] Robin Setton and Andrew Dawrant  “Conference interpreting: a complete course”, Benjamins translation library (BTL) volume 120  Amsterdam & Philadelphia: John Benjamins Publishing Company, 2016
  * [27] Christoph Stoll  “Jenseits simultanfähiger Terminologiesysteme” bibtex: stoll_jenseits_2009  Trier: Wvt Wissenschaftlicher Verlag, 2009
  * [28] Nikolai Vogler, Craig Stewart and Graham Neubig  “Lost in Interpretation: Predicting Untranslated Terminology in Simultaneous Interpretation” arXiv: 1904.00930  In _arXiv:1904.00930 [cs]_ , 2019  URL: http://arxiv.org/abs/1904.00930
  * [29] Martin Will  “Zur Eignung simultanfähiger Terminologiesysteme für das Konferenzdolmetschen”  In _trans-kom_ 8.1, 2015, pp. 179–201


# Families of Homomorphic Mappings in the Polydisk

Martin Chuaqui and Rodrigo Hernández<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study classes of locally biholomorphic mappings defined in the polydisk
${\mathbb{P}}^{n}$ that have bounded Schwarzian operator in the Bergman
metric. We establish important properties of specific solutions of the
associated system of differential equations, and show a geometric connection
between the order of the classes and a covering property. We show for modified
and slightly larger classes that the order is Lipschitz continuous with
respect to the bound on the Schwarzian, and use this to estimate the order of
the original classes.

The authors were partially supported by Fondecyt Grants 1150115, 1150284,
1190380, and 1190756\. Key words: Schwarzian operator, polydisk, locally
biholomorphic, norm, covering. MCS 2020. Primary: 32H02, 32A17; Secondary:
30C45.

##  1\. Introduction

In this paper, we investigate families of locally biholomorphic mappings in
the polydisk ${\mathbb{P}}^{n}$ defined by the Schwarzian operator. This is a
classical theme in one complex variable, where the simply-connected case
corresponds to mapping in the unit disk ${\mathbb{D}}$ with bounded Schwarzian
in the hyperbolic norm. The nature of the group of automorphisms in
${\mathbb{D}}$ yields families that are linearly invariant, with results that
are often invariant under the group. The original papers by Pommerenke have
become a landmark in this [19]. In several variables, linear invariance is
present for the families in the ball ${\mathbb{B}}^{n}$, and allows for a
fairly detailed analysis that includes a generalization of the Ahlfors-Weill
extension and examples of locally linearly convex domains [4],[1, 10]. Our aim
here is to study the case for the polydisk, a domain for which the explicit
nature of the automorphisms and the Bergman metric partially compensates for
the absence of linear invariance. Nevertheless, we were unable reproduce a
density function associated to the mappings treated that was both invariant
and convex relative to the Bergman metric, missing thus a crucial step for the
aforementioned Ahlfors-Weill extension.

In Section 2 we present preliminary facts about the Schwarzian operator and
specific bounds that are satisfied in ${\mathbb{P}}^{n}$. In Section 3 we
present the important notion of normalization that gives a geometric insight
for the norm of the families. We also develop techniques based on comparison
to study specific solutions of the linear system related to the Schwarzian
operator. In particular, for sufficiently small values of the parameter in the
family, we show that certain normalized solutions at the origin do vanish in
${\mathbb{P}}^{n}$. This can be thought as having the Ahlfors-Weill extension
at one point, and implies that for $f$ in the family, the image
$f({\mathbb{P}}^{n})$ misses the complex hyperplane
$1-(a_{1}w_{1}+\cdots+a_{n}w_{n})=0$, where $(n+1)(a_{1},\ldots,a_{n})=(\nabla
J_{f})(0)$, $J_{f}$ the complex jacobian of $f$. In one complex variable, this
corresponds to the fact that $-2/f^{\prime\prime}(0)\notin f({\mathbb{D}})$
when $(1-|z|^{2})^{2}|Sf(z)|\leq 2$.

In the final section we estimate the order of the families treated. In [3], a
generalization of Pommerenke’s variational methods was employed to estimate
the order of the families in ${\mathbb{B}}^{n}$, but linear invariance was
required. Here instead, we appeal to a dilation argument that yields Lipschitz
continuity for the order of slightly larger families. Some of our methods can
be considered extensions of techniques that in one variable yield optimal
results [5, 6, 16], and although sharp constants in ${\mathbb{C}}^{n}$ are in
many cases still far out of reach, we have made an effort to be as precise as
possible. The comparison techniques we develop are likely applicable on more
general domains, and our results fit into a broader class of mappings in
several variables studied by different authors, see for example [8, 11, 18,
20].

##  2\. Preliminaries

Let $f:\Omega\subset{\mathbb{C}}^{n}\rightarrow{\mathbb{C}}^{n}$ be a locally
biholomorphic mapping defined on some domain $\Omega$. There are several
equivalent definitions of a Schwarzian derivative in several complex variables
[13, 14, 15], and we will follow the notion offered in [17] because of its
explicit connection with differential equations. The author defines a family
of Schwarzian derivatives of $f=(f_{1},\ldots,f_{n})$ as

${S^{k}_{ij}}f=\displaystyle\sum^{n}_{l=1}\frac{\partial^{2}f_{l}}{\partial
z_{i}\partial z_{j}}\frac{\partial z_{k}}{\partial
f_{l}}-\frac{1}{n+1}\left(\delta^{k}_{i}\frac{\partial}{\partial
z_{j}}+\delta^{k}_{j}\frac{\partial}{\partial z_{i}}\right)\log(J_{f})\,,$

where $i,j,k=1,2,\ldots,n,$ and $\delta^{k}_{i}$ are the Kronecker symbols.
For $n>1$ the Schwarzian derivatives have the properties ${S^{k}_{ij}}f=0$ for
all $i,j,k=1,2,\ldots,n$ iff $f(z)=M(z),$ for some Möbius transformation

$M(z)=\left(\frac{l_{1}(z)}{l_{0}(z)},\ldots,\frac{l_{n}(z)}{l_{0}(z)}\right)\,,$

where $l_{i}(z)=a_{i0}+a_{i1}z_{1}+\cdots+a_{in}z_{n}$ with $\det(a_{ij})\neq
0$. Furthermore, for a composition one has

(2.1) ${S^{k}_{ij}}(g\circ
f)(z)={S^{k}_{ij}}f(z)+\sum^{n}_{l,m,r=1}S^{r}_{lm}g(w)\frac{\partial
w_{l}}{\partial z_{i}}\frac{\partial w_{m}}{\partial z_{j}}\frac{\partial
z_{k}}{\partial w_{r}}\;,\;w=f(z)\,.$

Thus, if $g$ is a Möbius transformation then ${S^{k}_{ij}}(g\circ
f)={S^{k}_{ij}}f$. The coefficients $S^{0}_{ij}f$ are given by

$S^{0}_{ij}f(z)=(J_{f})^{\frac{1}{n+1}}\left(\frac{\partial^{2}}{\partial
z_{i}\partial
z_{j}}(J_{f})^{-\frac{1}{n+1}}-\sum_{k=1}^{n}\,\frac{\partial}{\partial
z_{k}}(J_{f})^{-\frac{1}{n+1}}S^{k}_{ij}f(z)\right)\,.$

In [17] we can also find a description of the functions with prescribed
Schwarzian derivatives ${S^{k}_{ij}}f$. Consider the following over-determined
system of partial differential equations,

(2.2) $\frac{\partial^{2}u}{\partial z_{i}\partial
z_{j}}=\sum^{n}_{k=1}P^{k}_{ij}(z)\frac{\partial u}{\partial
z_{k}}+P^{0}_{ij}(z)u\;,\quad i,j=1,2,\ldots,n\,,$

where $z=(z_{1},z_{2},...,z_{n})\in\Omega$ and $P^{k}_{ij}(z)$ are holomorphic
functions in $\Omega$, for $i,j,k=0,\ldots,n$. The system (2.2) is called
completely integrable if there are $n+1$ (maximun) linearly independent
solutions, and is said to be in canonical form (see [21]) if the coefficients
satisfy

$\sum_{j=1}^{n}P_{ij}^{j}(z)=0\;,\quad i=1,2,\ldots,n\,.$

It was shown that (2.2) is a completely integrable system in canonical form if
and only if $P^{k}_{ij}={S^{k}_{ij}}f$ for a locally biholomorphic mapping
$f=(f_{1},\ldots,f_{n})$, where $f_{i}=u_{i}/u_{0}$ for $1\leq i\leq n$ and
$u_{0},u_{1},\ldots,u_{n}$ is a set of linearly independent solutions of the
system. For a given mapping $f$, $u_{0}=(J_{f})^{-\frac{1}{n+1}}$ is always a
solution of (2.2) with $P_{ij}^{k}={S^{k}_{ij}}f$.

We recall the following definitions from [9], where the individual Schwarzians
$S^{k}f$ are conformed to define the following operators.

######  Definition 2.1.

For each $k=1,\ldots,n$ we let $S^{k}_{f}$ be the $n\times n$ matrix

$S^{k}_{f}=({S^{k}_{ij}}f)\,,\quad i,j=1,\ldots,n\,.$

######  Definition 2.2.

Let $f:\Omega\rightarrow{\mathbb{C}}^{n}$ be locally holomorphic, and let
$T_{z}\Omega$ stand for the tangent space at $z\in\Omega$. We define the
Schwarzian derivative operator as the mapping $S_{f}(z):T_{z}\Omega\to
T_{z}\Omega$ given by

$S_{f}(z)(\vec{v})=\left(S^{1}_{f}(z)(\vec{v})\,,\ldots,S^{n}_{f}(z)\vec{v}\right)\,,$

where $\vec{v}\in T_{z}\Omega$ and
$S^{k}_{f}(z)(\vec{v})=\vec{v}^{\,t}S^{k}_{f}(z)\vec{v}$.

With this, we can rewrite (2.2) with $P^{k}_{ij}={S^{k}_{ij}}f$ as

(2.3) $\mbox{Hess}(u)(z)(\vec{v},\vec{v})=S_{f}(z)(\vec{v})\cdot\nabla
u(z)+S^{0}_{f}(z)(\vec{v})u\,,$

where $S^{0}_{f}(z)(\vec{v})=\vec{v}\,^{t}S^{0}_{f}(z)\vec{v}$.

The Bergman metric in the polydisk
${\mathbb{P}}^{n}=\\{(z_{1},\ldots,z_{n}):|z_{i}|<1,\,i=1,\ldots,n\\}$ is the
hermitian product defined by the diagonal matrix

$g_{ii}(z)=\frac{2}{(1-|z_{i}|^{2})^{2}}\;,$

see, e.g., [12]. The norm of the Schwarzian operator is defined by

$\|S_{f}(z)\|=\displaystyle\sup_{\|\vec{v}\|=1}\|S_{f}(z)(\vec{v}\,)\|\,,$

where

$\|\,\vec{v}\,\|=\left[\,2\sum_{i=1}^{n}\frac{|v_{i}|^{2}}{(1-|z_{i}|^{2})^{2}}\,\right]^{1/2}$

is the Bergman norm of $\vec{v}\in T_{z}{\mathbb{P}}^{n}$. Finally, let

$\|S_{f}\|=\sup_{z\in{\mathbb{P}}^{n}}\|S_{f}(z)\|\,.$

For a system of the form (2.2) that is completely integrable and in canonical
form, we define $\|P\|$ to be $\|S_{f}\|$ of any mapping with Schwarzian
derivatives given by the coefficients $P^{k}_{ij}$. Let now

${\mathcal{F}}_{\alpha}=\\{f:{\mathbb{P}}^{n}\rightarrow{\mathbb{C}}^{n}\,\mid\;f(0)=0\;,\;Df(0)=\textsl{I}\;,\;\|S_{f}\|\leq\alpha\\}\,.$

It is well known that up to multiplication by a diagonal unitary
transformation and a permutation of the coordinates, the automorphisms of the
polydisk are given by

(2.4)
$\psi(z)=\psi_{a}(z)=(\psi_{a_{1}}(z_{1}),\ldots,\psi_{a_{n}}(z_{n}))\,,\;\;z=(z_{1},\ldots,z_{n})\in{\mathbb{P}}^{n}\,,$

where $a=(a_{1},\ldots,a_{n})\in{\mathbb{P}}^{n}$ and
$\psi_{a_{j}}(z_{j})=\displaystyle\frac{z_{j}-a_{j}}{1-\overline{a_{j}}z_{j}}\;,\,1\leq
j\leq n\,.$ Because the automorphisms of ${\mathbb{P}}^{n}$ are not Möbius
transformations unless the origin is fixed, the bound on $\|S_{f}\|$ in
${\mathcal{F}}_{\alpha}$ will generally not be preserved when composing with
them. This poses a significant difference regarding the analysis of the
corresponding family in ${\mathbb{B}}^{n}$, and it will be useful to estimate
the norm of $\|S_{\psi}\|$ for an automorphism of ${\mathbb{P}}^{n}$.

The components $\psi_{j}=\psi_{a_{j}}$ of the mapping $\psi=\psi_{a}$ satisfy

$\begin{array}[]{lll}S^{k}_{kk}\psi(z)&=&\displaystyle\frac{n-1}{n+1}\,\frac{\psi^{\prime\prime}_{k}}{\psi^{\prime}_{k}}(z_{k})\,,\\\\[14.22636pt]
S^{k}_{jj}\psi(z)&=&0\;,\;\;j\neq k\,,\\\\[8.5359pt]
S^{k}_{kj}\psi(z)&=&-\displaystyle\frac{1}{n+1}\frac{\psi^{\prime\prime}_{j}}{\psi^{\prime}_{j}}(z_{j})\;,\;\;j\neq
k\,,\\\ S^{k}_{ij}\psi(z)&=&0\;,\;i\neq j\,,\,k\neq j\,.\end{array}$

It follows that for $\vec{v}=(v_{1},\ldots,v_{n})$ we have

$\vec{v}^{\,t}S^{k}_{\psi}(z)\vec{v}=\frac{n-1}{n+1}\cdot\frac{\psi^{\prime\prime}_{k}}{\psi^{\prime}_{k}}(z_{k})v_{k}^{2}-\frac{2}{n+1}\sum_{j=1,j\neq
k}^{n}\frac{\psi^{\prime\prime}_{j}}{\psi^{\prime}_{j}}(z_{j})v_{k}v_{j}=\frac{v_{k}}{n+1}\sum_{j=1}^{n}d_{kj}\,,$

where
$d_{kj}=-2\displaystyle\frac{\psi^{\prime\prime}_{j}}{\psi^{\prime}_{j}}(z_{j})v_{j}$
for $j\neq k$ and
$d_{kk}=(n-1)\displaystyle\frac{\psi^{\prime\prime}_{k}}{\psi^{\prime}_{k}}(z_{k})v_{k}$.
Therefore

(2.5)
$S_{\psi}(z)(\vec{v})=\frac{1}{n+1}\left(v_{1}D_{1},\ldots,v_{n}D_{n}\right)\,,$

where

$D_{k}=\sum_{j=1}^{n}d_{kj}=(n-1)\frac{\psi^{\prime\prime}_{k}}{\psi^{\prime}_{k}}(z_{k})v_{k}-2\sum_{j\neq
k}\frac{\psi^{\prime\prime}_{j}}{\psi^{\prime}_{j}}(z_{j})v_{j}\,.$

The square of the Bergman norm of $S_{\psi}(z)(\vec{v})$ is given by

$A=\frac{1}{(n+1)^{2}}\sum_{k=1}^{n}\frac{2|v_{k}|^{2}}{(1-|z_{k}|^{2})^{2}}|D_{k}|^{2}\,,$

and if we assume now that $||\vec{v}||=1$, we see that the sum on the right
hand-side is a convex combination of the $|D_{k}|^{2}$. Hence

$A\leq\frac{1}{(n+1)^{2}}\max_{i}|D_{i}|^{2}\,.$

Since $|\psi_{i}^{\prime\prime}/\psi_{i}^{\prime}(z_{i})|\leq
2|a_{i}|/|1-\overline{a_{i}}z_{i}|$ we see from (2.4) that

$|D_{k}|\,\leq\,2(n-1)\frac{|a_{k}||v_{k}|}{|1-\overline{a_{k}}z_{k}|}+4\sum_{j\neq
k}\frac{|a_{j}||v_{j}|}{|1-\overline{a_{j}}z_{j}|}\,,$

and using that
$\displaystyle\frac{1}{|1-\overline{a_{k}}z_{k}|}\leq\frac{2}{1-|z_{k}|^{2}}$
together with the Cauchy-Schwarz inequality, we obtain

$\begin{array}[]{lll}|D_{k}|&\leq&4\left(2|a_{1}|,\dots,(n-1)|a_{k}|,\ldots,2|a_{n}|\right)\cdot\displaystyle\left(\frac{|v_{1}|}{1-|z_{1}|^{2}},\ldots,\frac{|v_{n}|}{1-|z_{n}|^{2}}\right)\\\
&&\\\
&\leq&2\sqrt{2}\sqrt{4|a_{1}|^{2}+\cdots+(n-1)^{2}|a_{k}|^{2}+\cdots+4|a_{n}|^{2}}\\\
&&\\\
&\leq&2\sqrt{2}\sqrt{(n+3)(n-1)}\max_{i}|a_{i}|\;\,\leq\;\,2\sqrt{2}(n+1)\max_{i}|a_{i}|\,.\end{array}$

With $|z|_{\infty}=\max_{i}|z_{i}|$ denoting the polydisk norm, we see that
$A\leq 8|\psi(0)|_{\infty}^{2}$, hence

$\|S_{\psi}\|\leq 2\sqrt{2}|\psi(0)|_{\infty}\,.$

If $f\in{\mathcal{F}}_{\alpha}$ and $\psi$ is an automorphism of
${\mathbb{P}}^{n}$, then we see from (2.1) and the invariance of the Bergman
metric under $\psi$ that

(2.6)
$\|S_{f\circ\psi}\|\leq\|S_{f}\|+\|S_{\psi}\|\leq\alpha+2\sqrt{2}|\psi(0)|_{\infty}\,.$

The notation $|z|$ will be kept to refer to the Euclidean norm. The following
result on estimates for the derivative of a holomorphic function in
${\mathbb{D}}$ will be used in the paper, as well as two lemmas on specific
bounds for the Schwarzians of a mapping $f$ with bounded norm $\|Sf\|$.

######  Lemma 2.3.

Let $g$ be holomorphic in ${\mathbb{D}}$.

(i) If $\,\displaystyle|g(z)|\leq C$ then
$\,\displaystyle|g^{\prime}(z)|\leq\frac{C}{1-|z|^{2}}\,.$

(ii) If $\,\displaystyle|g(z)|\leq\frac{C}{1-|z|^{2}}$ then
$\,\displaystyle|g^{\prime}(z)|\leq\frac{4C}{(1-|z|^{2})^{2}}\,.$

###### Proof.

For the proof, we may assume that $C=1$. The first part then follows at once
from the Schwarz-Pick lemma.

For the second part, fix $a\in{\mathbb{D}}$, which after rotation may be
assumed to real and non-negative. The Cauchy integral formula applied on the
circle $C_{r}:\;|\zeta-a|=r<1-a$ gives

$|g^{\prime}(a)|\leq\frac{1}{2\pi}\int_{C_{r}}\frac{|d\zeta|}{(1-|\zeta|^{2})|\zeta-a|^{2}}=\frac{1}{2\pi
r}\int_{0}^{2\pi}\frac{d\theta}{1-a^{2}-r^{2}-2ar\cos(\theta)}\,.$

The last integral can be computed using the Weierstrass substitution
$\theta=2\arctan(x)$, and gives

$|g^{\prime}(a)|\leq\frac{1}{r\sqrt{U-V}}\,,$

where $U=(1-a^{2}-r^{2})^{2},\,V=4a^{2}r^{2}$. Evaluating the expression
$r\sqrt{U-V}$ at the value $r=r(a)$ where it attains its maximum is rather
cumbersome, but the simple expression $r=(1-a)/\sqrt{2}$ is close enough. At
this value

$U-V=\frac{(1-a)^{2}}{4}\left(a^{2}+6a+1\right)\,,$

and thus

$|g^{\prime}(a)|\leq\frac{2\sqrt{2}}{(1-a)^{2}\sqrt{a^{2}+6a+1}}=\frac{2\sqrt{2}(1+a)^{2}}{(1-a^{2})^{2}\sqrt{a^{2}+6a+1}}\,.$

The quantity $(1+a)^{2}/\sqrt{a^{2}+6a+1}$ decreases to its minimum value at
$a=\sqrt{17}-4$, and is increasing thereafter. From this we can see that the
maximum value is attained at $a=1$ and is equal to $\sqrt{2}$. This finishes
the proof of the lemma. ∎

######  Lemma 2.4.

If $f\in{\mathcal{F}}_{\alpha}$ and $|\vec{v}|_{\infty}=1$ then

(2.7)
$\left|S^{k}_{f}(z)(\vec{v})\right|\leq\frac{3n\alpha}{1-|z|_{\infty}^{2}}\,,$
(2.8)
$\left|S^{0}_{f}(z)(\vec{v})\right|\leq\frac{5n^{2}\alpha+2n(n+1)\alpha^{2}}{(1-|z|_{\infty}^{2})^{2}}\,.$

###### Proof.

For the first part, let $\vec{v}=(v_{1},\ldots,v_{n})$ be a vector with
$|\vec{v}|_{\infty}=1$. It was shown in [2] that when $\|S_{f}\|\leq\alpha$ in
${\mathbb{P}}^{n}$, one has ${S^{k}_{ij}}f\equiv 0$ for $k\neq i,j$ and
$(1-|z_{i}|^{2})|S^{k}_{ik}f(z)|\leq\sqrt{2}\alpha$. Hence

$\hskip
28.45274pt\left|S^{k}_{f}(z)(\vec{v})\right|=|v_{k}|\displaystyle\left|\left(2\sum_{i=1}^{n}S^{k}_{ik}f(z)v_{i}-S^{k}_{kk}f(z)v_{k}\right)\right|$

$\hskip
86.72377pt\leq\displaystyle{\frac{2\sqrt{2}\alpha}{(1-|z_{1}|^{2})}+\cdots\frac{\sqrt{2}\alpha}{(1-|z_{k}|^{2})}+\cdots\frac{2\sqrt{2}\alpha}{(1-|z_{n}|^{2})}}$

$\hskip
86.72377pt=\displaystyle{\frac{2\sqrt{2}(n-1)\alpha+\sqrt{2}\alpha}{1-|z|_{\infty}^{2}}\,=\,\frac{\sqrt{2}(2n-1)\alpha}{1-|z|_{\infty}^{2}}\,\leq\,\frac{3n\alpha}{1-|z|_{\infty}^{2}}}\,.$

For the second part and the abbreviated notation
$\partial_{i}=\partial/\partial z_{i}$, it was shown in [9] that

$S^{0}_{ii}f(z)=-\frac{1}{n-1}\sum_{k=1}^{n}\partial_{k}S^{k}_{ii}f(z)+\frac{1}{n-1}\sum_{k,j=1}^{n}S^{k}_{ij}f(z)S^{j}_{ki}f(z)\,,$

and for $i\neq j$

$S^{0}_{ij}f(z)=\partial_{j}S^{i}_{ii}f(z)-\partial_{i}S^{i}_{ij}f(z)+\sum_{k=1}^{n}S^{k}_{ii}f(z)S^{i}_{kj}f(z)-\sum_{k=1}^{n}S^{k}_{ij}f(z)S^{i}_{ki}f(z)\,.$

Since the ${S^{k}_{ij}}f$ vanish unless $k=i$ or $k=j$, we find that

$S^{0}_{ii}f(z)=-\frac{1}{n-1}\partial_{i}S^{i}_{ii}f(z)+\frac{1}{n-1}\sum_{k=1}^{n}\left(S^{k}_{ki}f(z)\right)^{2}\,.$

and for $i\neq j$

$S^{0}_{ij}f(z)=\partial_{j}S^{i}_{ii}f(z)-\partial_{i}S^{i}_{ji}f(z)-S^{j}_{ij}f(z)S^{i}_{ij}f(z)\,.$

From Lemma 2.3 we see that

(2.9)
$|S^{0}_{ii}f(z)|\leq\frac{1}{n-1}\frac{4\sqrt{2}\alpha+2n\alpha^{2}}{(1-|z|_{\infty}^{2})^{2}}\,,$

and for $i\neq j$

(2.10)
$|S^{0}_{ij}f(z)|\leq\frac{2\sqrt{2}\alpha+2\alpha^{2}}{(1-|z|_{\infty}^{2})^{2}}\,,$

because the bounds for $S^{i}_{ii}$ and $S^{i}_{ji}$ do not depend on $z_{j}$
and $z_{i}$, respectively. Therefore

$\hskip
19.91684pt\left|S^{0}_{f}(z)(\vec{v})\right|=\displaystyle\left|v_{1}\sum_{k=1}^{n}S^{0}_{1k}f(z)v_{k}+\cdots+v_{n}\sum_{k=1}^{n}S^{0}_{nk}f(z)v_{k}\right|$

$\hskip 76.82234pt\leq\displaystyle
n\left(\frac{1}{n-1}\frac{4\sqrt{2}\alpha+2n\alpha^{2}}{(1-|z|_{\infty}^{2})^{2}}+(n-1)\frac{2\sqrt{2}\alpha+2\alpha^{2}}{(1-|z|_{\infty}^{2})^{2}}\right)$

$\hskip
76.82234pt=\displaystyle\frac{2n}{n-1}\frac{\sqrt{2}\left[2+(n-1)^{2}\right]\alpha+\left[n+(n-1)^{2}\right]\alpha^{2}}{(1-|z|_{\infty}^{2})^{2}}\,\leq\,\displaystyle\frac{5n^{2}\alpha+2n(n+1)\alpha^{2}}{(1-|z|_{\infty}^{2})^{2}}\;\,.$

For the last inequality we have used that $2\left[2+(n-1)^{2}\right]\leq
3n(n-1)$ and $n+(n-1)^{2}\leq(n+1)(n-1)$ because $n\geq 2$, and that
$3\sqrt{2}\leq 5$.

∎

######  Lemma 2.5.

Let (2.2) be a completely integrable system in canonical form with
$\|P\|\leq\alpha$. For fixed $\zeta_{1},\ldots,\zeta_{n}\in{\mathbb{D}}$ let
$A=(P^{k}_{j})_{k,j}$ be the matrix with components

$P^{k}_{j}=\zeta_{1}P^{k}_{1j}+\cdots+\zeta_{n}P^{k}_{nj}\,,$

and let $B=(P^{0}_{j})$ be the vector with components

$P^{0}_{j}=\zeta_{1}P^{0}_{1j}+\cdots+\zeta_{n}P^{0}_{nj}\,.$

Then with $|A|=\max_{|v|=1}|Av|$ we have

(2.11) $(1-|z|_{\infty}^{2})|A|\leq 1.6\,n\sqrt{n}\alpha\,,$ (2.12)
$(1-|z|_{\infty}^{2})^{2}|B|\leq c_{1}(n)\alpha+c_{2}(n)\alpha^{2}\,,$

where

(2.13)
$c_{1}(n)=\frac{2\sqrt{2n}}{n-1}\left[2+(n-1)^{2}\right]\;\,,\;\,c_{2}(n)=\frac{2\sqrt{n}}{n-1}\left[n+(n-1)^{2}\right]\,.$

Furthermore, if $n\sqrt{n}\alpha\leq 3\sqrt{2}-4=0.2426...$ then

(2.14) $(1-|z|_{\infty}^{2})^{2}|B|\leq 4.5\,n\sqrt{n}\alpha\,.$

###### Proof.

We first estimate $|P^{k}_{j}|$ for $P^{k}_{ij}={S^{k}_{ij}}f$. For $k\neq j$
the only term contributing in the sum is $\zeta_{k}P^{k}_{kj}$, and thus
$(1-|z|_{\infty}^{2})|P^{k}_{j}|\leq\sqrt{2}\alpha$. When $k=j$ all summands
may contribute and we see that
$(1-|z|_{\infty}^{2})|P^{k}_{j}|\leq\sqrt{2}n\alpha$. Therefore, for every
column $C$ of $A$ we have

$(1-|z|_{\infty}^{2})|C|\leq\sqrt{2}\sqrt{n^{2}+(n-1)}\,\alpha\,.$

The bound (2.11) follows from the estimate $\sqrt{2}\sqrt{n^{2}+n-1}\leq
1.6n$.

For the second part, we use (2.9) and (2.10) to find that

$(1-|z|_{\infty}^{2})^{2}|P^{0}_{j}|\leq\frac{2}{n-1}\left(\sqrt{2}\left[2+(n-1)^{2}\right]\alpha+\left[n+(n-1)^{2}\right]\alpha^{2}\right)\,,$

which establishes (2.12)-(2.13) because $|B|\leq\sqrt{n}\max_{j}|P^{0}_{j}|$.

For the last claim, fix $\epsilon>0$ and suppose that
$n\sqrt{n}\alpha\leq\epsilon$. Then

$c_{1}(n)\alpha+c_{2}(n)\alpha^{2}\leq\left(c_{1}(n)+\frac{c_{2}(n)}{n\sqrt{n}}\epsilon\right)\alpha\,.$

The inequality (2.14) will then hold if

$2\sqrt{2}\left[2+(n-1)^{2}\right]+2\frac{n+(n-1)^{2}}{n\sqrt{n}}\epsilon\leq
4.5\,n(n-1)\,,$

or equivalently, after dividing by $2$ and shifting the term
$\sqrt{2}(n-1)^{2}$ to the right, if

$2\sqrt{2}+\frac{n+(n-1)^{2}}{n\sqrt{n}}\epsilon\leq(n-1)\left[2.25n-\sqrt{2}(n-1)\right]\,,$

that is, if

$\frac{2\sqrt{2}}{n-1}+\frac{n+(n-1)^{2}}{n(n-1)\sqrt{n}}\epsilon\leq(2.25-\sqrt{2})n+\sqrt{2}\,.$

Because the left hand-side is decreasing in $n$ while the right hand-side is
increasing, we see that (2.14) will hold provided it holds for $n=2$, that is,
if

$3\sqrt{2}+\frac{3}{4}\sqrt{2}\epsilon\leq 4.5\,,$

which gives the constant $3\sqrt{2}-4$ as the maximum possible value for
$\epsilon$. This finishes the proof. ∎

It will become useful to compare the constant
$c_{1}(n)\alpha+c_{2}(n)\alpha^{2}$ with the bound $\tau=1.6n\sqrt{n}\alpha$
in simple terms. We have that

$c_{1}(n)\alpha=5\sqrt{2}\,\frac{2+(n-1)^{2}}{4n(n-1)}\tau\leq\frac{15}{8}\sqrt{2}\tau\leq
3\tau\,,$

while

$c_{2}(n)\alpha^{2}=5\,\frac{n+(n-1)^{2}}{4n(n-1)}\tau\alpha\leq\frac{15}{8}\tau\alpha\leq
2\tau\alpha\,,$

hence

(2.15) $c_{1}(n)\alpha+c_{2}(n)\alpha^{2}\leq(3+2\alpha)\tau\,.$

##  3\. Normalization and Finite Norm

In this section we will establish important bounds for mappings in
${\mathcal{F}}_{\alpha}$. In particular, we will show that the family
${\mathcal{F}}_{\alpha}$ is compact and will exhibit a connection between the
order

${\lambda}_{\alpha}=\sup_{f\in{\mathcal{F}}_{\alpha}}|\nabla J_{f}(0)|$

and the size of the largest Euclidean ball centered at the origin that is
contained in every image $f({\mathbb{P}}^{n})$,
$\,f\in{\mathcal{F}}_{\alpha}$. To this end, we will consider the associated
family ${\mathcal{F}}_{\alpha}^{0}$ of mappings $g=T(f)$ for
$f\in{\mathcal{F}}_{\alpha}$ and $T$ a specific Möbius transformation that
ensures the additional normalization $\nabla J_{g}(0)=0$. Any such mapping
will remain locally biholomorphic away from a possibly existing singular set
in ${\mathbb{P}}^{n}$ that depends on $g$ itself. In any case, such a set will
stay away some fixed distance $r_{0}=r_{0}(n,\alpha)$ from the origin, and
will not exist in ${\mathbb{P}}^{n}$ altogether when $\alpha$ is sufficiently
small. The procedure of normalizing is described in the following lemma.

######  Lemma 3.1.

Let $f:{\mathbb{P}}^{n}\rightarrow{\mathbb{C}}^{n}$ be locally biholomorphic,
with $f(0)=0$ and $Df(0)=\textsl{I}$. Then there exists a Möbius map $T$ such
$g=T\circ f$ satisfies that $g(0)=0$, $Dg(0)=\rm{Id}$ and $\nabla J_{g}(0)=0.$

###### Proof.

For $a=(a_{1},\ldots,a_{n})$ let $T$ be a Möbius map of the form

(3.1)
$T_{a}(w)=\left(\frac{w_{1}}{1+a_{1}w_{1}+\cdots+a_{n}w_{n}},\ldots,\frac{w_{n}}{1+a_{1}w_{1}+\cdots+a_{n}w_{n}}\right)\,.$

The mapping $g=T(f)$ has $g(0)=0$ and $Dg(0)=\textsl{I}$. Also,
$Dg(z)=DT(f(z))\cdot Df(z)$ hence $J_{g}(z)=J_{T}(f(z))\cdot J_{f}(z)$ and

$\nabla J_{g}(z)=\nabla J_{T}(f(z))\cdot Df(z)\cdot
J_{f}(z)+J_{T}(f(z))\cdot\nabla J_{f}(z)\,.$

Evaluating at the origin we obtain that

$\nabla J_{g}(0)=\nabla J_{T}(0)+\nabla
J_{f}(0)=-(n+1)(a_{1},\ldots,a_{n})+\nabla J_{f}(0)\,,$

so that the lemma holds if $(n+1)(a_{1},\ldots,a_{n})=\nabla J_{f}(0)$. ∎

It is important to observe that
$u=(J_{g})^{-\frac{1}{n+1}}=(1+a_{1}f_{1}+\cdots+a_{n}f_{n})(J_{f})^{-\frac{1}{n+1}}$
is a solution of (2.2) with $P^{k}_{ij}={S^{k}_{ij}}f$ and the initial
conditions $u(0)=1,\nabla u(0)=0$. It vanishes precisely when
$1+a_{1}f_{1}+\cdots+a_{n}f_{n}=0$, that is, when the image
$f({\mathbb{P}}^{n})$ meets the hyperplane $1+a_{1}w_{1}+\cdots+a_{n}w_{n}=0$,
or $g$ becomes infinite.

In order to analyze normalization, we will require estimates obtained from
comparison techniques for differential equation.

######  Lemma 3.2.

Let $h:[0,1)\rightarrow\mathbb{R}$ be a non-negative function satisfying

$h^{\prime\prime}\leq\frac{2a}{1-x^{2}}h^{\prime}+\frac{b}{(1-x^{2})^{2}}h\;\,,\;h(0)=1\,,\,h^{\prime}(0)=0\,,$

where $a,b>0$ are constants. Then

(3.2) $h^{\prime}\leq\frac{a+c}{1-x^{2}}\,h\,,$

and

(3.3) $h\leq 2\left(\frac{1+x}{1-x}\right)^{\frac{a+c-1}{2}}\,,$

where $c=\sqrt{1+b+a^{2}}$.

###### Proof.

We will analyze the function $h$ as a function of the variable
$s=\displaystyle\frac{1}{2}\log\frac{1+x}{1-x}$ defined on $[0,\infty)$. The
change of variable $x=x(s)$ has $x^{\prime}=1-x^{2}$. Let

$w(s)=\frac{h}{u}(x(s))\,,$

where $u(x)=\sqrt{1-x^{2}}$. Then $w(0)=1,w^{\prime}(0)=0$ and
$w^{\prime}=uh^{\prime}-u^{\prime}h$, with the agreement that derivatives of
$w$ are with respect to $s$, while those of $u,h$ are with respect to $x$. One
further derivative yields
$w^{\prime\prime}=(1-x^{2})(uh^{\prime\prime}-hu^{\prime\prime})$, and because
$u^{\prime\prime}=-u/(1-x^{2})^{2}$ we see that

$w^{\prime\prime}\leq 2a(uh^{\prime})+\frac{1+b}{1-x^{2}}\,uh\,.$

We replace the term $uh^{\prime}$ in the right hand side with
$w^{\prime}+u^{\prime}h=w^{\prime}-(x/u)h=w^{\prime}-xw$, and because $h$, and
thus $w$, are non-negative, we obtain

$w^{\prime\prime}\leq 2a(w^{\prime}-xw)+(1+b)w\leq 2aw^{\prime}+(1+b)w\,.$

Let ${\lambda}_{1,2}=-a\pm\sqrt{1+b+a^{2}}=-a\pm c$ be the roots of the
equation ${\lambda}({\lambda}+2a)=1+b$, where ${\lambda}_{1}>0>{\lambda}_{2}$.
Then

$w^{\prime\prime}+{\lambda}_{1}w^{\prime}\leq(2a+{\lambda}_{1})w^{\prime}+(1+b)w=(2a+{\lambda}_{1})(w^{\prime}+{\lambda}_{1}w)=-{\lambda}_{2}(w^{\prime}+{\lambda}_{1}w)\,,$

which upon amplification by $e^{{\lambda}_{1}s}$ yields

$\left(e^{{\lambda}_{1}s}w^{\prime}\right)^{\prime}\leq-{\lambda}_{2}\left(e^{{\lambda}_{1}s}w\right)^{\prime}\,.$

The initial conditions $w(0)=1,w^{\prime}(0)=0$ imply that

$e^{{\lambda}_{1}s}w^{\prime}\leq-{\lambda}_{2}\left(e^{{\lambda}_{1}s}w-1\right)\,,$

hence

(3.4) $e^{{\lambda}_{1}s}(w^{\prime}+{\lambda}_{2}w)\leq{\lambda}_{2}\,.$

Multiplication now by $e^{({\lambda}_{2}-{\lambda}_{1})s}$ and integration
leads to

$w\,\leq\,\frac{{\lambda}_{2}e^{-{\lambda}_{1}s}-{\lambda}_{1}e^{-{\lambda}_{2}s}}{{\lambda}_{2}-{\lambda}_{1}}\,=\,\frac{1}{2c}\left((c-a)e^{(a+c)s}+(c+a)e^{(a-c)s}\right)$
$=\,\frac{e^{(a+c)s}}{2c}\,\left(c-a+(c+a)e^{-2cs}\right)\,\leq\,e^{(a+c)s}\,.$

In terms of the function $h$ we see that

$h\leq\sqrt{1-x^{2}}\left(\frac{1+x}{1-x}\right)^{\frac{a+c}{2}}=(1+x)\left(\frac{1+x}{1-x}\right)^{\frac{a+c-1}{2}}\leq
2\left(\frac{1+x}{1-x}\right)^{\frac{a+c-1}{2}}\,.$

Finally, we see from (3.4) that $w^{\prime}+{\lambda}_{2}w\leq 0$, which
yields

$h^{\prime}u+\frac{x}{u}h\leq(a+c)\frac{h}{u}\,,$

hence

$h^{\prime}\leq\frac{a+c}{1-x^{2}}\,h\,.$

This proves (3.2) and (3.3). ∎

######  Lemma 3.3.

Let (2.2) be a completely integrable system in canonical form with
$\|P\|\leq\alpha$. Then the solution $u$ with $u(0)=1,\nabla u(0)=0$ satisfies

(3.5) $|u|\leq
2\left(\frac{1+|z|_{\infty}}{1-|z|_{\infty}}\right)^{\gamma}\,,$ (3.6)
$|\nabla u|\leq
2\,\frac{1+2\gamma}{1-|z|_{\infty}^{2}}\left(\frac{1+|z|_{\infty}}{1-|z|_{\infty}}\right)^{\gamma}\,,$

where $\displaystyle 2\gamma=\sqrt{1+b+a^{2}}-1$, $a=0.8n\sqrt{n}\alpha$ and
$b=2(3+2\alpha)\sqrt{n}a$.

###### Proof.

For $\zeta\in\partial{\mathbb{P}}^{n}$ we let $v(t)=u(t\zeta)$, $t\in[0,1]$.
We seek to estimate the functions $w_{j}(t)=\partial u/\partial
z_{j}(t\zeta)$. The $t$-derivatives of these quantities are given by

$\begin{array}[]{lll}w_{j}^{\prime}&=&\displaystyle\frac{\partial^{2}u}{\partial
z_{1}\partial z_{j}}\zeta_{1}+\cdots+\frac{\partial^{2}u}{\partial
z_{n}\partial z_{j}}\zeta_{n}\\\\[14.22636pt]
&=&\displaystyle\left(\sum_{k=1}^{n}P^{k}_{1j}\frac{\partial u}{\partial
z_{k}}+P^{0}_{1j}u\right)\zeta_{1}+\cdots+\displaystyle\left(\sum_{k=1}^{n}P^{k}_{nj}\frac{\partial
u}{\partial z_{k}}+P^{0}_{nj}u\right)\zeta_{n}\\\\[14.22636pt]
&=&\displaystyle\sum_{k=1}^{n}\left(P^{k}_{1j}\zeta_{1}+\cdots
P^{k}_{nj}\zeta_{n}\right)\frac{\partial u}{\partial
z_{k}}+\left(P^{0}_{1j}\zeta_{1}+\cdots
P^{0}_{nj}\zeta_{n}\right)u\\\\[14.22636pt]
&=&\displaystyle\sum_{k=1}^{n}\left(P^{k}_{1j}\zeta_{1}+\cdots
P^{k}_{nj}\zeta_{n}\right)w_{k}+\left(P^{0}_{1j}\zeta_{1}+\cdots
P^{0}_{nj}\zeta_{n}\right)u=\sum_{k=1}^{n}P^{k}_{j}w_{k}+P^{0}_{j}u\,.\end{array}$

Let $\varphi(t)$ the vector function in ${\mathbb{C}}^{n}$ with coordinates
$\varphi_{i}=w_{i}$ for $i=1,\ldots,n$. Since $v^{\prime}=\nabla u\cdot\zeta$
we can see that

$\varphi^{\prime}=A\cdot\varphi+uB\,,$

where $A=(P^{k}_{j})_{k,j}$ is an $n\times n$ matrix and $B$ is the vector
with components $P^{0}_{j}$. Since $|\varphi|^{\prime}\leq|\varphi^{\prime}|$
we see that

$|\varphi|^{\prime}\leq|A||\varphi|+|B||u|\,,$

where $|A|=\max_{|v|=1}|Av|$. The function
$\phi(t)=\displaystyle\sqrt{n}\int_{0}^{t}|\varphi(s)|ds+1$ is an upper bound
for $|u|$ with $\phi(0)=1,\phi^{\prime}(0)=0$ and satisfies

$\phi^{\prime\prime}\leq|A|\phi^{\prime}+\sqrt{n}|B|\phi\,.$

The inequalities (2.11) and (2.15) show that $(1-t^{2})|A|\leq 2a$ and
$(1-t^{2})^{2}|B|\leq b=2\sqrt{n}(3+2\alpha)a$, where $a=0.8n\sqrt{n}\alpha$.
It follows now from Lemma 3.2 that

$\hskip 22.76228pt\phi\leq
2\left(\frac{1+t}{1-t}\right)^{\gamma}\;\;,\quad\;\phi^{\prime}\leq\,2\,\frac{1+2\gamma}{1-t^{2}}\left(\frac{1+t}{1-t}\right)^{\gamma}\,,$

for $2\gamma=a+\sqrt{1+b+a^{2}}-1$. This finishes the proof.

∎

######  Corollary 3.4.

Let $\alpha>0$ be fixed. The set of solutions $u$ with $u(0)=1,\nabla u(0)=0$
of completely integrable systems (2.2) in canonical form with
$\|P\|\leq\alpha$ is compact.

######  Lemma 3.5.

Let $g\in{\mathcal{F}}_{\alpha}^{0}$. Then there exists
$r_{0}=r_{0}(n,\alpha)>0$ such that $g$ is holomorphic in
${\mathbb{B}}^{n}(0,r_{0})$.

###### Proof.

Let $g=T(f)\in{\mathcal{F}}_{\alpha}^{0}$ with $f\in{\mathcal{F}}_{\alpha}$ as
before. Then
$u=(J_{g})^{-\frac{1}{n+1}}=(1+a_{1}w_{1}+\cdots+a_{n}w_{n})(J_{f})^{-\frac{1}{n+1}}$
is a solution of (2.2) with $P^{k}_{ij}={S^{k}_{ij}}f,P^{0}_{ij}=S^{0}_{ij}$
that vanishes precisely when $g$ ceases to be holomorphic. We will show that
$u\neq 0$ in some neighborhood of the origin.

For $\zeta\in\partial{\mathbb{P}}^{n}$ we define $v(t)=|u(t\zeta)|$,
$t\in[0,1)$. Therefore $vv^{\prime}=\mbox{Re}\\{\overline{u}(\nabla
u\cdot\zeta)\\}$ and

$vv^{\prime\prime}+(v^{\prime})^{2}=|\nabla
u\cdot\zeta|^{2}+\mbox{Re}\\{\overline{u}\,\mbox{Hess}(u)(\zeta,\zeta)\\}\,.$

Since $|\nabla u\cdot\zeta|^{2}-(v^{\prime})^{2}\geq 0$ we conclude from (2.3)
that

$v(t)v^{\prime\prime}(t)\geq\mbox{Re}\left\\{\overline{u}S_{f}(t\zeta)(\zeta)\cdot\nabla
u(t\zeta)+|u|^{2}S^{0}_{f}(t\zeta)(\zeta)\right\\}\,,$

and hence

(3.7) $v^{\prime\prime}(t)\geq-|S_{f}(t\zeta)(\zeta)||\nabla
u(t\zeta)|-|S^{0}_{f}(t\zeta)(\zeta)|v(t)\,.\hskip 28.45274pt$

The bound (3.6) and Lemma 2.4 give that

$v^{\prime\prime}+\frac{\varepsilon}{(1-t^{2})^{2}}v\geq-\frac{\delta}{(1-t^{2})^{2}}\left(\frac{1+t}{1-t}\right)^{\gamma}\,,$

with $v(0)=1,v^{\prime}(0)=0$, and
$\varepsilon=\varepsilon(n,\alpha),\delta=\delta(n,\alpha)$ are positive
constants. We rewrite this inequality as

$v^{\prime\prime}+\frac{\varepsilon}{(1-t^{2})^{2}}v=-\frac{\delta}{(1-t^{2})^{2}}\left(\frac{1+t}{1-t}\right)^{\gamma}+\eta(t)\,,$

for some $\eta(t)\geq 0$, and claim that $v\geq y$ for $t\in[0,t_{0}]$, where
$t_{0}$ is the first zero of the solution $y=y(t)$ of

$y^{\prime\prime}+\frac{\varepsilon}{(1-t^{2})^{2}}y=-\frac{\delta}{1-t^{2}}\left(\frac{1+t}{1-t}\right)^{\gamma},\quad
y(0)=1\,,\;y^{\prime}(0)=0\,.$

For this, observe first that $y^{\prime\prime}+\varepsilon(1-t^{2})^{-2}y\leq
0$, hence from Sturm comparison we have that $0\leq y\leq h$ on $[0,t_{0}]$
for the solution of

$h^{\prime\prime}+\frac{\varepsilon}{(1-t^{2})^{2}}h=0\;\,,\;h(0)=1\,,\,h^{\prime}(0)=0\,.$

For $s>0$ we consider the solution

$v_{s}^{\prime\prime}+\frac{\varepsilon}{(1-t^{2})^{2}}v_{s}=-\frac{\delta}{(1-t^{2})^{2}}\left(\frac{1+t}{1-t}\right)^{\gamma}+\eta(t)\;,\;v_{s}(0)=1+s\,,\,v_{s}^{\prime}(0)=0\,.$

The functions $\\{v_{s}\\}_{s>0}$ will converge to $v$ locally uniformly on
$[0,1)$ as $s\rightarrow 0$. Since difference $w=v_{s}-y$ satisfies

$w^{\prime\prime}+\frac{\varepsilon}{(1-t^{2})^{2}}w=\eta(t)\geq
0\;\,,\;w(0)=s\,,\,w^{\prime}(0)=0\,,$

we see, once more, from Sturm comparison that $w\geq sh\geq 0$ on $[0,t_{0}]$.
Therefore, $v_{s}\geq y$ on $[0,t_{0}]$, which proves the claim by letting
$s\rightarrow 0$. We conclude finally that $u(z)\neq 0$ for all
$z\in{\mathbb{P}}^{n}$ with $|z|<r_{0}=t_{0}$, which completes the proof. ∎

######  Theorem 3.6.

The family ${\mathcal{F}}_{\alpha}$ has finite order bounded by $1/s_{0}$ for
some $s_{0}=s_{0}(n,\alpha)>0$. The image of every mapping in
${\mathcal{F}}_{\alpha}^{0}$ covers a Euclidean ball of radius $s_{0}$
centered at the origin, and every mapping in ${\mathcal{F}}_{\alpha}$ covers a
corresponding Euclidean ball of radius $s_{0}/2$ centered at the origin.

###### Proof.

Corollary 3.4 and Lemma 3.5 show that the family ${\mathcal{F}}_{\alpha}^{0}$
is compact in any ball $B(0,t_{1})$ for $t_{1}<t_{0}$. We conclude that there
exists $s_{0}>0$ such that $B(0,s_{0})\subset g({\mathbb{P}}^{n})$ for every
$g\in{\mathcal{F}}_{\alpha}^{0}$. An estimate for $s_{0}$ in terms of
$n,\alpha$ will be given in the final section. The mapping

$f=T^{-1}(g)=\left(\frac{g_{1}}{L(g)},\cdots,\frac{g_{n}}{L(g)}\right)\;\,,\;L(g)=1-(a_{1}g_{1}+\cdots+a_{n}g_{n})$

is holomorphic in ${\mathbb{P}}^{n}$, hence
$L(g)=1-(a_{1}g_{1}+\cdots+a_{n}g_{n})$ cannot vanish in ${\mathbb{P}}^{n}$.
Since $g$ covers a Euclidean ball of radius $s_{0}$ it follows that

$\frac{1}{(n+1)^{2}}|\nabla
J_{f}(0)|^{2}=|a_{1}|^{2}+\cdots+|a_{n}|^{2}\leq\frac{1}{s_{0}^{2}}\,.$

This implies that

$\sup_{f\in{\mathcal{F}}_{\alpha}}|\nabla
J_{f}(0)|\leq\frac{n+1}{s_{0}}<\infty\,.$

Finally, let $f\in{\mathcal{F}}_{\alpha}$ and let $g=T(f)$ be the
corresponding mapping in ${\mathcal{F}}_{\alpha}^{0}$ as before. For $z\in
U=g^{-1}(B(0,s_{0}))$ we see from the triangle and the Cauchy-Schwarz
inequalities that

$|L(g)(z)|=|1-a_{1}g_{1}-\cdots-a_{n}g_{n}|\leq
1+|a_{1}g_{1}|+\cdots+|a_{n}g_{n}|\leq 2\,.$

It follows that

$|f(z)|=\frac{|g(z)|}{|L(g)|}\geq\frac{|g(z)|}{2}\,,$

therefore

$\liminf_{z\rightarrow\partial U}|f(z)|\geq\frac{s_{0}}{2}\,.$

∎

As it turns out, for $\alpha\leq\alpha_{0}(n)$ small enough the mappings in
${\mathcal{F}}_{\alpha}^{0}$ will be holomorphic in all of ${\mathbb{P}}^{n}$.
Equivalently, the solutions $u$ of (2.2) with $u(0)=1,\nabla u(0)=0$ do not
vanish in ${\mathbb{P}}^{n}$. To establish this we will require a variation of
Lemma 3.3 focusing on small $\alpha$. The phenomenon of normalizing at the
origin while keeping the mapping regular also occurs in one variable, with
$\mathcal{F}_{\alpha_{0}}$ corresponding to the Nehari class $||Sf||\leq 2$.
Indeed, for a Nehari mapping $f=z+a_{2}z^{2}+\cdots$ it is known that
$-1/a_{2}\notin f({\mathbb{D}})$, and thus the mapping

$g=\frac{f}{1+a_{2}f}$

which is normalized to have $g^{\prime\prime}(0)=0$ remains holomorphic in
${\mathbb{D}}$. It is to be noted that the mappings in
${\mathcal{F}}_{\alpha}$ are univalent if $3\sqrt{2}\alpha\leq 1$ [2].

######  Lemma 3.7.

Let $h$ be the solution of the equation

(3.8)
$h^{\prime}=\frac{1.6c}{1-x^{2}}h+\frac{4.5c}{(1-x^{2})^{2}}+h^{2}\;\,,\;h(0)=0\,,$

where $c>0$ is a constant. If $c\leq\displaystyle\frac{1}{6.1}=0.163...$ then
the solution $h$ exists on the entire interval $[0,1)$ and

(3.9) $h\leq\frac{x}{1-x^{2}}\,.$

The bound for $c$ cannot be improved beyond
$\displaystyle\frac{2}{6.1+\sqrt{(6.1)^{2}-(1.6)^{2}}}=0.166...$.

###### Proof.

Because of the term $h^{2}$ in the differential equation, $h$ will become
infinite before reaching $x=1$ if at any point $x_{0}<1$ the value $h(x_{0})$
is too large. This will happen for large $c$ because of the term
$4.5c/(1-x^{2})^{2}$ on the right-hand side.

The function $y=\displaystyle\frac{x}{1-x^{2}}$ satisfies

(3.10)
$y^{\prime}=\frac{1+x^{2}}{(1-x^{2})^{2}}\geq\frac{1.6c}{1-x^{2}}y+\frac{4.5c}{(1-x^{2})^{2}}+y^{2}$

provided that

$1+x^{2}\geq 1.6cx+4.5c+x^{2}\,.$

This inequality will hold for all $x\in[0,1)$ precisely when $6.1c\leq 1$. It
remains to be shown that $h\leq y$. To that end, we rewrite (3.10) as

$y^{\prime}=\frac{1.6c}{1-x^{2}}y+\frac{4.5c}{(1-x^{2})^{2}}+y^{2}+\eta(x)\,,$

for some $\eta(x)\geq 0$ on $[0,1)$, and we consider the solution $w_{s}$ of
this last equation with initial condition $w_{s}(0)=s>0$. The function $w_{s}$
will exist on some interval $[0,x_{s})$ for some $x_{s}<1$ (or even
$x_{s}$=1), but will converge to $y$ locally uniformly on $[0,1)$ as
$s\rightarrow 0$. Hence $x_{s}$ will converge to $1$. On $[0,x_{s})$ we have
that

$(w_{s}-h)^{\prime}=\frac{1.6c}{1-x^{2}}(w_{s}-h)+(w_{s}^{2}-h^{2})+\eta(x)$
$\geq\frac{1.6c}{1-x^{2}}(w_{s}-h)+(w_{s}+h)(w_{s}-h)\,.$

Since evidently $w_{s},h\geq 0$, we conclude that $w_{s}-h>0$ because
$(w_{s}-h)(0)=s>0$. The inequality $y\geq h$ results from letting $s$ tend to
$0$.

For the last claim, we express (3.8)in terms of the function $\phi=(1-x)h$, as

$(1-x)\phi^{\prime}=-\left(1-\frac{1.6c}{1+x}\right)\phi+\frac{4.5c}{(1+x)^{2}}+\phi^{2}=(\phi-\beta)^{2}+\gamma\,,$

where $\displaystyle\beta=\frac{1}{2}\left(1-\frac{1.6c}{1+x}\right)$ and

$\gamma=\frac{4.5c}{(1+x)^{2}}-\beta^{2}\,.$

The quantity $\gamma$ will be decreasing in $x$ for, say, $c\leq 1$, and thus

$\gamma\geq\frac{4.5c}{4}-\frac{1}{4}(1-0.8c)^{2}:=d\,.$

The number $d$ will be positive provided $0.64c^{2}-6.1c+1<0$, which happens
for $c\in(0.166...,9.364...)$. Thus for any $c\in(0.166...,1)$ we have $d>0$,
so that $\phi$ is strictly increasing. Hence $\phi^{\prime}\geq d(1-x)^{-1}$
and there exists $x_{0}\in(0,1)$ so that $\phi(x_{0})>1/2>\beta$. For $x\geq
x_{0}$ we thus have that $\phi>1/2>\beta$ and

$\frac{\phi^{\prime}}{(\phi-\frac{1}{2})^{2}}\geq\frac{1}{1-x}\,,$

which upon integration shows that $\phi$ becomes infinite at some $x_{1}<1$. ∎

######  Theorem 3.8.

Let (2.2) be a completely integrable system in canonical form with
$\|P\|\leq\alpha$, and let $u$ be the solution with $u(0)=1,\nabla u(0)=0$. If

(3.11) $n\sqrt{n}\alpha\leq\frac{1}{6.1}$

then $u$ does not vanish in ${\mathbb{P}}^{n}$ and

(3.12) $\left|\frac{\nabla
u}{u}\right|\leq\frac{|z|_{\infty}}{1-|z|_{\infty}^{2}}\,.$

###### Proof.

Let $u$ be as stated in the theorem. For fixed
$\zeta\in\partial{\mathbb{P}}^{n}$ we let

$\varphi(t)=\frac{\nabla u}{u}(t\zeta)\,,$

which is defined on some maximal interval $[0,t_{0})$. We claim that $t_{0}=1$
when $\alpha$ is sufficiently small. We see that

$\varphi^{\prime}(t)=\left[\frac{1}{u}\mbox{Hess
}u(\zeta,\cdot)\right](t\zeta)-\left[\frac{\nabla u\cdot\zeta}{u^{2}}\nabla
u\right](t\zeta)\,,$

It follows from (2.2) that

$\mbox{Hess }u(\zeta,\cdot)=A\cdot\nabla u+uB\,,$

where $A=(a_{ij})$ is the $n\times n$ matrix given by
$a_{ij}=\sum_{k=1}^{n}S_{ik}^{j}f\zeta_{k}$, and $B=(b_{i})$ is the vector
with $b_{i}=\sum_{k=1}^{n}S^{0}_{ik}\zeta_{k}$. Hence the function
$g(t)=|\varphi(t)|$ has $g(0)=0$ and satisfies

(3.13)
$g^{\prime}\leq|\varphi^{\prime}|\leq\frac{ag}{1-t^{2}}+\frac{b}{(1-t^{2})^{2}}+g^{2}\,,$

where, as before, we use (2.11) and (2.14) that see that $a\leq
1.6n\sqrt{n}\alpha$ and $b\leq 4.5n\sqrt{n}\alpha$. The theorem now follows
from Lemma 3.7 with $c=n\sqrt{n}\alpha$. ∎

We draw the following immediate corollary.

######  Corollary 3.9.

If $n\sqrt{n}\alpha\leq\displaystyle\frac{1}{6.1}$ then all mappings in
${\mathcal{F}}_{\alpha}^{0}$ are holomorphic in ${\mathbb{P}}^{n}$.

The next result is also a direct consequence of Theorem 3.8.

######  Theorem 3.10.

Let (2.2) be a completely integrable system in canonical form with
$\|P\|\leq\alpha$, and let $u$ be the solution with $u(0)=1,\nabla u(0)=0$. If

$n\sqrt{n}\alpha\leq\frac{1}{6.1}$

then

$\left(1-|z|_{\infty}^{2}\right)^{\frac{\sqrt{n}}{2}}\leq|u|\leq\frac{1}{\left(1-|z|_{\infty}^{2}\right)^{\frac{\sqrt{n}}{2}}}\,.$

###### Proof.

Since $u(0)=1$, we find from integrating the bound in (3.12) that

$|\log u|\leq\frac{\sqrt{n}}{2}\,\log\frac{1}{1-|z|_{\infty}^{2}}\,,$

which gives the result. ∎

The following corollary follows at once from the previous theorem, and mimics
results for mappings in linearly invariant families [7].

######  Corollary 3.11.

Let $f\in{\mathcal{F}}_{\alpha}^{0}$ for
$n\sqrt{n}\alpha\leq\displaystyle\frac{1}{6.1}$. Then

$(1-|z|_{\infty}^{2})^{\frac{\sqrt{n}(n+1)}{2}}\leq|J_{f}|\leq\frac{1}{(1-|z|_{\infty}^{2})^{\frac{\sqrt{n}(n+1)}{2}}}\,.$

######  Theorem 3.12.

Let $f\in{\mathcal{F}}_{\alpha}$. For $z\in{\mathbb{P}}^{n}$ we let
$\eta(z)=\min_{|v|=1}|Df(z)v|$ and $\beta=\alpha+2\sqrt{2}|z|_{\infty}$. Then
the image $f({\mathbb{P}}^{n})$ covers a Euclidean ball centered at $f(z)$ of
radius at least

$\frac{1}{2}(1-|z|_{\infty})\eta(z)s_{0}(n,\beta)\,,$

and

$|\nabla
J_{f}(z)|_{\infty}\leq\frac{{\lambda}_{\beta}+2|z|_{\infty}}{1-|z|_{\infty}^{2}}\left|J_{f}(z)\right|\,,$

where $s_{0}(n,\beta)$ is the constant defined in Theorem 3.6.

###### Proof.

For $z_{0}\in{\mathbb{P}}^{n}$ fixed, let $g$ be defined by

$g(z)=[D\psi(0)]^{-1}[Df(z_{0})]^{-1}(f(\psi(z))-f(z_{0})),\quad\psi(0)=z_{0}\,,$

where $\psi=\psi_{z_{0}}$ is an automorphism as given by (2.4). Then $g(0)=0$,
$Dg(0)=\textsl{I}$ and we have that
$\|S_{g}\|\leq\alpha+2\sqrt{2}|z_{0}|_{\infty}$. Hence by (2.6)
$g\in\mathcal{F}_{\beta}$ for $\beta=\alpha+2\sqrt{2}|z_{0}|_{\infty}$, and by
Theorem 3.6, $g({\mathbb{P}}^{n})$ contains a ball centered at the origin of
Euclidean radius at least $\frac{1}{2}s_{0}(n,\beta)$. Since $D\psi(0)$ is a
diagonal matrix of elements $1-|z^{0}_{k}|^{2}$, we see that
$\min_{|v|=1}|D\psi(0)v|\geq 1-|z_{0}|_{\infty}^{2}$. The first statement of
the theorem follows now from the definition of $g$. On the other hand,

$J_{g}(z)=J_{\psi}(0)^{-1}\cdot J_{f}(z_{0})^{-1}\cdot J_{f}(\psi(z))\cdot
J_{\psi}(z)\,,$

which implies that $\log J_{g}(z)=\log J_{f}(\psi(z))+\log
J_{\psi}(z)+\mbox{constant}$, and

(3.14) $\nabla J_{g}(0)=\frac{\nabla J_{f}}{J_{f}}(z_{0})D\psi(0)+\frac{\nabla
J_{\psi}}{J_{\psi}}(0)\,.$

Because $g\in\mathcal{F}_{\beta}$ we must have that $|\nabla
J_{g}(0)|\leq{\lambda}_{\beta}$. In addition, with
$z_{0}=(a_{1},\ldots,a_{n})$

$J_{\psi}(z)=\prod_{j=1}^{n}\frac{1-|a_{j}|^{2}}{(1-\overline{a_{j}}z_{j})^{2}}\,,$

therefore

$\frac{\nabla
J_{\psi}}{J_{\psi}}(z)=\left(\frac{2\overline{a_{1}}}{1-\overline{a_{1}}z_{1}},\ldots,\frac{2\overline{a}_{n}}{1-\overline{a_{n}}z_{n}}\right)\,,$

so that $\displaystyle\frac{\nabla J_{\psi}}{J_{\psi}}(0)=2\overline{z}_{0}$.
From (3.14) we obtain

$\left|\frac{\nabla
J_{f}}{J_{f}}(z_{0})D\psi(0)\right|_{\infty}\leq\lambda_{\beta}+2|z_{0}|_{\infty}\,.$

Since $D\psi(0)$ is a diagonal matrix with elements $1-|a_{k}|^{2}$, we have

$(1-|z_{0}|_{\infty}^{2})|\nabla
J_{f}(z_{0})|_{\infty}\leq(\lambda_{\beta}+2|z_{0}|_{\infty})|J_{f}(z_{0})|\,,$

which shows that

$|\nabla
J_{f}(z_{0})|_{\infty}\leq\frac{\lambda_{\beta}+2|z_{0}|_{\infty}}{1-|z_{0}|_{\infty}^{2}}\left|J_{f}(z_{0})\right|\,.$

∎

##  4\. Order

The purpose in this section is to establish an explicit upper bound for the
order of the family ${\mathcal{F}}_{\alpha}$. We will consider the families
${\mathcal{F}}_{\alpha,r}$ consisting of all mappings $f$ that are defined and
locally biholomorphic for $|z|_{\infty}<r<1$, which have
$f(0)=0,Df(0)=\textsl{I}$, and $\|{S^{k}_{ij}}f\|\leq\alpha$; here $\|\cdot\|$
stands for the Bergman norm in ${\mathbb{P}}^{n}$ restricted to the set
$|z|_{\infty}<r$. We see that
${\mathcal{F}}_{\alpha}\subset{\mathcal{F}}_{\alpha,r}$, and it follows as
before that ${\mathcal{F}}_{\alpha,r}$ is compact. For fixed $r$ we see that

(4.1) ${\mu_{r}(\alpha)}=\sup_{f\in{\mathcal{F}}_{\alpha,r}}{|\nabla
J_{f}(0)|}<\infty\,,$

which is attained for some mapping in the class. In addition,
${\lambda}_{\alpha}\leq\mu_{r}(\alpha)$.

######  Theorem 4.1.

Suppose $r^{2}<1/5$. Then the function $\mu_{r}={\mu_{r}(\alpha)}$ is locally
Lipschitz for $\alpha>0$, and whenever the derivative exists,

(4.2) $\alpha\mu_{r}^{\prime}(\alpha)\leq C(r)\mu_{r}(\alpha)\,,$

with $\displaystyle C(r)=\frac{1-r^{2}}{1-5r^{2}}$.

###### Proof.

Let $\alpha>0$ be fixed, and let $f\in{\mathcal{F}}_{\alpha,r}$ be extremal
for (4.1). We consider the dilation of $f$ given by $g(z)=\frac{1}{s}f(sz)$
for $0<s<1$. It is easy to see that $|\nabla J_{g}(0)|=s|\nabla
J_{f}(0)|=s\mu_{r}(\alpha)$, and we claim that $g\in\mathcal{F}_{\beta,r}$ for

$\beta=s\left(\frac{1-s^{2}r^{2}}{1-r^{2}}\right)^{2}\alpha<\alpha\,.$

From the composition law (2.1) we see that
${S^{k}_{ij}}g(z)=s{S^{k}_{ij}}f(sz)$, hence in order to estimate
$\|S_{g}(z)\|$ we must consider how the Bergman norm of given vector $\vec{v}$
changes when measured at $z$ and $sz$. Let $\vec{v}=(v_{1},\ldots,v_{n})$ be a
Bergman unit vector at $z=(z_{1},\ldots,z_{n})$. The square of the Bergman
norm $\|\vec{v}\|_{sz}$ at $sz$ is given by

$\sum_{i}\frac{2|v_{i}|^{2}}{(1-s^{2}|z_{i}|^{2})}=\sum_{i}t_{i}\left(\frac{1-|z_{i}|^{2}}{1-s^{2}|z_{i}|^{2}}\right)^{2}\,,$

where $t_{i}=2|v_{i}|^{2}/(1-|z_{i}|^{2})^{2}$ are non-negative numbers with
$\sum_{i}t_{i}=1$. It follows that $\|\vec{v}\|^{2}_{sz}$ is a number between
the maximum and minimum value of the quantities
$\left[(1-|z_{i}|^{2})/(1-s^{2}|z_{i}|^{2})\right]^{2}$. Because the function
$(1-x^{2})/(1-s^{2}x^{2})$ is decreasing for $x\in[0,1)$ we see that

$\left(\frac{1-r^{2}}{1-s^{2}r^{2}}\right)^{2}\leq\|\vec{v}\|_{sz}^{2}\leq
1\,,$

therefore

$\|S_{g}(z)\|\leq s\left(\frac{1-s^{2}r^{2}}{1-r^{2}}\right)^{2}\alpha\,.$

Simple algebra shows that $s\left(\frac{1-s^{2}r^{2}}{1-r^{2}}\right)^{2}<1$
for $r^{2}<1/5$, hence $g\in\mathcal{F}_{\beta,r}$ and $\beta<\alpha$ as
claimed.

Finally, since $|\nabla J_{g}(0)|=s{\mu_{r}(\alpha)}$ we conclude that
$\mu_{r}(\beta)\geq s{\mu_{r}(\alpha)}$, and so

${\mu_{r}(\alpha)}-\mu_{r}(\beta)\leq(1-s){\mu_{r}(\alpha)}\leq
C(r)(\alpha-\beta)\frac{{\mu_{r}(\alpha)}}{\alpha}\,.$

This proves the theorem. ∎

For our next theorem, recall the order
${\lambda}_{\alpha}=\sup_{f\in{\mathcal{F}}_{\alpha}}|\nabla J_{f}(0)|$.

######  Theorem 4.2.

For each $\sigma>0$ there exists $m(\sigma)>0$ such that

${\lambda}_{\alpha}\leq 2(n+1)+m(\sigma,n)\alpha^{1+\sigma}\,,$

for some $m(\sigma,n)$ depending on $\sigma$ and $n$.

###### Proof.

Let $\alpha_{1}>0$ be fixed. It follows from (4.2) that for
$\alpha\geq\alpha_{1}$

${\mu_{r}(\alpha)}\leq\mu_{r}(\alpha_{1})\left(\frac{\alpha}{\alpha_{1}}\right)^{C(r)}\,,$

where $C(r)=(1-r^{2})/(1-5r^{2})>1$ approaches $1$ as $r\rightarrow 0$. Hence,
for $\alpha\geq\alpha_{1}$ we also have

(4.3)
${\lambda}_{\alpha}\leq\frac{\mu_{r}(\alpha_{1})}{\alpha_{1}^{C(r)}}\,\alpha^{C(r)}\,.$

We now estimate ${\lambda}_{0}$, that is, the order of the family
$\mathcal{F}_{0}$ consisting of Möbius mappings of the form

$f(z)=\left(\frac{z_{1}}{l(z)},\ldots,\frac{z_{n}}{l(z)}\right)\;,\;\,l(z)=1-a_{1}z_{1}-\cdots-
a_{n}z_{n}\,,$

with the requirement that $l(z)\neq 0$ in ${\mathbb{P}}^{n}$. This happens iff
$\sum_{i}|a_{i}|\leq 1$. Hence

$|\nabla
J_{f}(0)|=(n+1)\sqrt{|a_{1}|^{2}+\cdots+|a_{n}|^{2}}\leq(n+1)\left(\sum_{i}|a_{i}|\right)^{2}\leq
n+1\,,$

with equality when a single $a_{i}$ has absolute value 1. This shows that
${\lambda}_{0}=n+1$, and the theorem follows now from (4.3) by choosing
$\alpha_{1}=\alpha_{1}(n)$ so that, for example,
${\lambda}_{\alpha_{1}}=2{\lambda}_{0}$.

∎

Acknowledgement: We are grateful to the referee for a very thorough revision
of the original manuscript and the first revision. The detailed reports have
allowed for several improvements of the paper.

## References

  * [1] A. Andersson, M. Passare and R. Sigurdsson.  Complex Convexity and Analytic Functionals. Progress in Mathematics 225, Birhäuser, Basel, 2004.
  * [2] M. Chuaqui and R. Hernández.  A condition for univalent in the polydisc. Comput. Methods Funct. Theory 9 (2009), 443-454.
  * [3] M. Chuaqui and R. Hernández.  The order of a linearly invariant family in ${\mathbb{C}}^{n}$. J. Math. Anal. Appl. 398 (2013) 372-379.
  * [4] M. Chuaqui and R. Hernández.  Ahlfors-Weill extensions in several complex variables. J. reine angew. Math. 698 (2015) 161-179.
  * [5] M. Chuaqui and B. Osgood.  Sharp distortion theorems associated with the Schwarzian derivative. Jour. London Math. Soc. 48 (1993), 289-298.
  * [6] M. Essén and F.R. Keogh.  The Schwarzian derivative and estimates of functions analytic in the unti disc. Math. Proc. Camb. Phil. Soc. 78 (1975), 501-511.
  * [7] I. Graham and G. Kohr.  Geometric function theory in one and higher dimensions. Pure and Applied Math. 255, Marcel Dekker, 2003.
  * [8] H. Hamada and G. Kohr.  The growth theorem and quasiconformal extension of strongly spirallike mappings of type $\alpha$. Complex Variables 44 (2001), 281-297.
  * [9] R. Hernández.  Schwarzian derivatives and a linearly invariant family in ${\mathbb{C}}$. Pacific J. Math 228 (2006), 201-218.
  * [10] L. Hörmander.  Notions of Convexity. Modern Classics, Birkäuser, Boston, 2007.
  * [11] K. Kikuchi.  Starlike and convex mappings in several complex variables. Pacific J. Math. 44 (1973), 569-580.
  * [12] S. Krantz.  Function Theory of several complex variables.  Pure and Applied Math., Wiley, 1982.
  * [13] R. Molzon and K.P. Mortensen.  Differential operators associated with holomorphic mappings. Ann. Global Anal. Geom. 12 (1994), 291-304.
  * [14] R. Molzon and K.P. Mortensen.  The schwarzian derivative for maps between manifolds with complex projective connections. Trans. Amer. Math. Soc. 348 (1996) 3015-3036.
  * [15] R. Molzon and H. Tamanoi.  Generalized schwarzians in several variables and Möbius invariant differential operators. Forum Math. 14 (2002), 165-188.
  * [16] Z. Nehari.  The Schwarzian derivative and schlicht functions. Bull. Amer. Math. Soc. 55 (1949), 545-551.
  * [17] T. Oda.  On Schwarzian derivatives in several variables (in japanese). Kokyuroku of R.I.M., Kioto Univ. 226 (1974).
  * [18] J.A. Pfaltzgraff.  Distortion of locally biholomorphic maps of the $n$-ball. Complex Variables 33 (1997), 239-253.
  * [19] Ch. Pommerenke.  Linear-invariante Familien analytischer Funktionen I, II. Math. Ann. 155, 156 (1964), 108-154, 226-262.
  * [20] T. Suffridge.  Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions. Lecture Notes in Math. 599 (1976), 146-159, Springer-Verlag.
  * [21] Masaaki Yoshida.  Canonical forms of some system of linear partial differential equations. Proc. Japan Acad. 52 (1976), 473-476.

Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla
306, Santiago 22, Chile,

Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Padre
Hurtado 750, Viña del Mar, Chile,