2.1pc0pc plain[] myps[] Titlesec 0pt section[1.8pc] [.]1.8pc
*subsection[1.8pc] . , [.—][.]

# The titlesec, titleps and titletoc Packages111The titlesec package is
currently at version 2.14. © 1998–2021 Javier Bezos.

Javier Bezos222For bug reports, comments and suggestions go to http://www.tex-
tipografia.com/contact.html. English is not my strong point, so contact me
when you find mistakes in the manual. Other packages by the same author: gloss
(with José Luis Díaz), enumitem, accents, tensind, esindex, dotlessi,
babeltools.

(2021/07/05)

###### Contents

  1. 1 Introduction
  2. I —
    1. 2 Quick Reference
      1. 2.1 Format
      2. 2.2 Spacing
      3. 2.3 Uppercase
      4. 2.4 Tools
    2. 3 Advanced Interface
      1. 3.1 Format
        1. 4 —
        2. 0 —
        3. 1 —
          1. 1 —
            1. 1 —
              1. 1 —
                1. — ‣ 1 —
      2. 2 Spacing
    3. 2 —
  3. 2 —
    1. 1 Spacing related tools
    2. 3 —
      1. 1 Rules

###### Note 1

Please, report any issues you find on
https://github.com/jbezos/titlesec/issues, which is better than just
complaining on an e-mail list or a web forum. Forking and pull requests are
welcome.

####  1 Introduction

This package is essentially a replacement—partial or total—for the LaTeX
macros related with sections—namely titles, headers and contents. The goal is
to provide new features unavailable in current LaTeX; if you just want a more
friendly interface than that of standard LaTeX but without changing the way
LaTeX works you may consider using fancyhdr, by Piet van Oostrum, sectsty, by
Rowland McDonnell, and tocloft, by Peter Wilson, which you can make pretty
things with.333Since the sectioning commands are rewritten, their behaviour
could be somewhat different in some cases.

Some of the new features provided are:

  * •

Different classes and “shapes” of titles, with tools for very fancy formats.
You can define different formats for left and right pages, or numbered and
unnumbered titles, measure the width of the title, add a new section level,
use graphics, and many more. The Appendix shows a good deal of examples, so
jump forward right now!

  * •

Headers and footers defined with no —.̇.mark— intermediates, and perhaps
containing top, first _and_ bot marks at the same time. Top marks correctly
synchronized with titles, without incompatibilities with the float mechanism.
Decorative elements easily added, including picture environments.

  * •

Pretty free form contents, with the possibility of grouping entries of
different levels in a paragraph or changing the format of entries in the
middle of a document.

Titlesec works with the standard classes and with many others, including the
AMS ones, and it runs smoothly with hyperref.444However, be aware the AMS
classes reimplement the original internal commands. These changes will be lost
here. The compatibility with hyperref has been tested with dvips, dvipdfm and
pdftex but it is an unsupported feature. Please, check your version of
hyperref is compatible with titlesec. Unfortunately, it is not compatible with
memoir, which provides its own tools with a limited subset of the features
available in titlesec.

As usual, load the package in the standard way with —. Then, redefine the
sectioning commands with the simple, predefined settings (see section “Quick
Reference”) or with the provided commands if you want more elaborate formats
(see section “Advanced Interface.”) In the latter case, you only need to
redefine the commands you’ll use. Both methods are available at the same time,
but because —

##  Part I —

is usually implemented in a non-standard way, it remains untouched by the
simple settings and should be changed with the help of the “Advanced
Interface.”

####  2 Quick Reference

The easiest way to change the format is by means of a set of package options
and a couple of commands. If you feel happy with the functionality provided by
this set of tools, you need not go further in this manual. Just read this
section and ignore the subsequent ones.

#####  2.1 Format

There are three option groups controlling font, size and align. You need not
set all of these groups, since a default is provided for each one; however,
you must use at least an option from them if you want this “easy setup.”  —rm
sf tt md bf up it sl sc—  Select the corresponding family/series/shape.
Default is —bf—.

—big medium small tiny—  Set the size of titles. Default is —big—, which gives
the size of standard classes. With —tiny—, sections (except chapters) are
typed in the text size. —medium— and —small— are intermediate layouts.

—raggedleft center raggedright—

Control the alignment.

#####  2.2 Spacing

—compact—  This option is independent from those above and reduces the spacing
above and below the titles.

#####  2.3 Uppercase

—uppercase—

2.9 Uppercases titles. Depending on the class, it might not work in `\chapter`
and `\part`.

#####  2.4 Tools

——  Changes the label format in sections, subsections, etc. A —— command is
provided which is respectively —2—, —2.4—, etc. The default value in standard
classes is

    \titlelabel{\thetitle\quad}

and you may add a dot after the counter simply with

    \titlelabel{\thetitle.\quad}

That was done in this document.

——

This command allows to change the —¡format¿— of a sectioning command, as for
example:

    \titleformat*{\section}{\itshape}

####  3 Advanced Interface

Two commands are provided to change the title format. The first one is used
for the “internal” format, i. e., shape, font, label…, the second one defines
the “external” format, i. e., spacing before and after, indentation, etc. This
scheme is intended to easy definitions, since in most of cases you will want
to modify either spacing or format.555Information is “extracted” from the
class sectioning commands, except for chapter and part. Standard definitions
with —“@startsection— are presumed—if sections have been defined without that
macro, arbitrary values for the format an the spacing are provided, which you
may change later. (Sadly, there is no way to catch the chapter or part
formats, and one similar to that of standard classes will be used.) That
redefines existing sectioning commands, but does not create _new_ ones. New
sectioning levels can be added with —as described below, and then their format
can be set with the commands described here.

#####  3.1 Format

A set of shapes is provided, which controls the basic distribution of elements
in a title. The available shapes are:

hang

is the default value, with a hanging label. (Like the standard —

####  4 —

.)

block

typesets the whole title in a block (a paragraph) without additional
formatting. Useful in centered titles 666The label will be slightly displaced
to the left if the title is two or more lines long and the hang shape is used,
except with explicit —““—. and special formatting (including graphic tools
such as —picture—, —pspicture—, etc.)

display

puts the label in a separate paragraph. (Like the standard —

###  Chapter 0 —

.)

runin

A run-in title, like the standard —

###### —

.111Well, not quite. The title is first boxed to avoid some unexpected results
if, for example, there is a \color between the title and the text.
Unfortunately, due to an optimization done by TeX discretionaries may be lost.
I have found no solution, except using luatex, which works as one could
expect. Anyway, if the title doesn’t contain hyphen or dashes, this is not
usually a real problem.

leftmargin

puts the title at the left margin. Titles at the very end of a page will be
moved to the next one and will not stick out in the bottom margin, which means
large titles can lead to underfull pages.222However, floats following the
title a couple of lines after will interfere with the page breaking used here
and sometimes the title may stick out. In this case you may increase the
stretchability of the page elements, use —— or use the package option
—nobottomtitles— described below. Since the mechanism used is independent from
that of the margin pars, they can overlap. A deprecated synonymous is
—margin—.

rightmargin

is like —leftmargin— but at the right margin.

drop

wraps the text around the title, provided the first paragraph is longer than
the title (if not, they overlap). The comments in —leftmargin— also apply
here.

wrap

is quite similar to drop. The only difference is while the space reserved in
drop for the title is fixed, in wrap is automatically readjusted to the
longest line. The limitations explained below related to —calcwidth— also
apply here.

frame

Similar to display, but the title will be framed.

Note, however, some shapes do not make sense in chapters and parts.

——

Here

  * •

—¡command¿— is the sectioning command to be redefined, i. e., —

##  Part 1 —

, —

###  Chapter 1 —

, —

####  1 —

, —

#####  1 —

, —

###### —

, —

###### —

or —

###### —

.

  * •

The paragraph shape is set by —¡shape¿—, whose possible values are those
described above.

  * •

—¡format¿— is the format to be applied to the whole title—label and text. This
part can contain vertical material (and horizontal with some shapes) which is
typeset just after the space above the title.

  * •

The label is defined in —¡label¿—. You may leave it empty if there is no
section label at that level, but this is not recommended because by doing so
the number is not suppressed in the table of contents and running heads.

  * •

—¡sep¿— is the horizontal separation between label and title body and must be
a length (it must not be empty). This space is vertical in —display— shape; in
—frame— it is the distance from text to frame. Both —¡label¿— and —¡sep¿— are
ignored in starred versions of sectioning commands. If you are using —picture—
and the like, set this parameter to 0 pt.

  * •

—¡before-code¿— is code preceding the title body. The very last command can
take an argument, which is the title text.111Remember font size can be changed
safely between paragraphs only, and changes in the text should be made local
with a group; otherwise the leading might be wrong—too large or too small.
However, with the package option explicit the title must be given explicitly
with —#1— (see below). Penalties in this argument may lead to unexpected
results.

  * •

—¡after-code¿— is code following the title body. The typeset material is in
vertical mode with —hang—, —block— and —display—; in horizontal mode with
—runin— and —leftmargin— (2.7 with the latter, at the beginning of the
paragraph). Otherwise is ignored. Penalties in this argument may lead to
unexpected results.

Penalties, marks and the like must be properly synchronized with page breaks.
So, —¡before-code¿— and —¡after-code¿— are not the proper places for
penalties. See —— below.

—Chapter—

It defaults to —Chapter— except in appendices where it is —Appendix—. Use it
instead of —Chapter— when defining a chapter.

#####  2 Spacing

——

The starred version kills the indentation of the paragraph following the
title, except in —drop—, —wrap— and —runin— where this possibility does not
make sense.

  * •

—¡left¿— increases the left margin, except in the —…margin—, and —drop— shape,
where this parameter sets the title width, in —wrap—, the maximum width, and
in —runin—, the indentation just before the title. With negative value the
title overhangs.222This parameter is not equal to —¡indent¿— of
—“@startsection—, which doesn’t work correctly. With a negative value in the
latter and if —¡indent¿— is larger than the label width, the first line of the
title will start in the outer margin, as expected, but the subsequent lines
will not; worse, those lines will be shortened at the right margin.

  * •

—¡before-sep¿— is the vertical space before the title.

  * •

—¡after-sep¿— is the separation between title and text—vertical with —hang—,
—block—, and —display—, and horizontal with —runin—, —drop—, —wrap— and
—…margin—. By making the value negative, you may define an effective space of
less than ——.333See Goossens, Mittelbach and Samarin: The LaTeX Companion,
Reading, Addison Wesley, 1993, p. 25.

  * •

The —hang—, —block— and —display— shapes have the possibility of increasing
the —¡right-sep¿— margin with this optional argument.

If you dislike typing the full skip values, including the —plus— and —minus—
parameters, an abbreviation —*—$n$ is provided. In the —¡before-sep¿— argument
this is equivalent to $n$ —ex— with some stretchability and a minute
shrinkability. In the —¡after-sep¿— some stretchability (smaller) and no
shrinkability.444They stand for $n$ times —1ex plus .3ex minus .06ex— and —1ex
plus .1ex—, respectively. Thus, you can write

    \titlespacing{\section}{0pt}{*4}{*1.5}

The lengths —— and —— are used as units in the —*— settings and you can change
them if you do not like the predefined values (or for slight changes in the
makeup, for example).

Notes. —s not work with either —

###  Chapter 2 —

and —

##  Part 2 —

unless you change its title format as well by means of —simple settings, or
—Arguments in —t be dimensions; —0pt plus —fill includes a command and hence
raises an error.

#####  1 Spacing related tools

These commands are provided as tools for —desc ——

Variants of the —…— commands, with slight differences. In particular, the —…—
commands kills the left and right spaces set by ——— justifies the paragraph,
except the last line which is centered.111Admittedly, a weird name, but it is
short. These commands work in the —frame— label, too.

—— and —— are —— or —— depending on the page. Because of the asynchronous TeX
page breaking, these commands can be used in —

###  Chapter 3 —

only. If you want a general tool to set different formats depending on the
page, see “Extended settings” below.

——

The inter-word space for the current font.

—indentafter noindentafter— (Package options)

By-pass the settings for all of sectioning commands.111Formerly —indentfirst—
and —nonindentfirst—, now deprecated.

—rigidchapters rubberchapters— (Package options)

With —rigidchapters— the space for chapter titles is always the same, and
—¡after-sep¿— in —s not mean the space from the bottom of the text title to
the text body as described above, but from the top of the text title, i. e.,
—¡before-sep¿— $+$ —¡after-sep¿— now is a fixed distance from the top of the
page body to the main text. The default is —rubberchapters— where —¡after-
sep¿— is the separation between title and text as usual. Actually, the name is
misleading because it applies not only to the default chapter, but to any
title of top class. (More on classes below.)

—bottomtitles nobottomtitles nobottomtitles*— (Package options)

If —nobottomtitles— is set, titles close to the bottom margin will be moved to
the next page and the margin will be ragged. The minimal space required in the
bottom margin not to move the title is set (approximately) by

    \renewcommand{\bottomtitlespace}{<length>}

whose default value is —.2—. A simple ragged bottom on the page before is
obtained with a value of 0 pt. —bottomtitles— is the default, which simply
sets —— to a negative value.

The —nobottomtitles*— option provides more accurate computations but titles of
—margin—, —wrap— or —drop— shapes could be badly placed. Usually, you should
use the starred version.

—aftersep largestsep— (Package options)

By default, when there are two consecutive titles the —¡after-sep¿— space from
the first one is used between them. Sometimes this is not the desired
behaviour, especially when the —¡before-sep¿— space is much larger than the
—¡after-sep¿— one (otherwise the default seems preferable). With —largestsep—
the largest of them is used. Default is —aftersep—.

—

—
—pageatnewline— (Package option)

2.6 In version 2.6 and later, `\\` does not allow a page break and therefore
is equivalent to `\\*`. Since I presume none wants a page break inside a
title, this has been made the default. If for some extrange reason you want to
allow page breaks inside the titles, use the package option `pageatnewline`,
which is provided for backward compatibility.

——
—nostruts— (Package option)

2.11 The styles defined by titlesec insert some struts at certain places to
make sure the vertical space is the same with relation with the baseline. This
is not always the desired behavior, so the package options —nostruts— is
provided. An alternative is the macro —— when defining a section (note this
macros is defined only within a title).

#####  1 Rules

The package includes some tools for helping in adding rules and other stuff
below or above the title. Since the margins in titles may be modified, these
macros take into account the local settings to place rules properly. They also
take into account the space used by marginal titles.

——
——
——

The — command allows inserting a line, which may contain text and other
“horizontal” material. it is intended mainly for rules and leaders but in fact
is also useful for other purposes. The line has a fixed width and hence must
be filled, i.e., —— produces an underfull box. Here the optional —¡align¿—
(—l—, —r— or —c—) helps, so that you simply type, say, ——.222The default is
the s parameter of the \makebox command.

Using — in places where vertical material is not expected can lead to
anomalous results. In other words, you can use it in the —¡format¿— (always)
and —¡after-code¿— (—hang—, —display— and —block—) arguments; and in the
—display— shape at the very beginning of the —¡before-code¿— and —¡label¿—
argument as well. But try it out, because very likely it works in other
places.

The —— command, which is enclosed automatically in — if necessary, can be used
to build rules and fillers. The unstarred version draws rules of height .4 pt,
or —¡height¿— if present. For example:

    \titlerule[.8pt]%
    \vspace{1pt}%
    \titlerule

draws two rules of different heights with a separation of 1 pt.

The starred version makes leaders with the —¡text¿— repeated in boxes of its
natural width. The width of the boxes can be changed to —¡width¿—, but the
first box remains with its natural width so that the —¡text¿— is aligned to
the left and right edges of the space to be filled.

For instance, with

    \titleformat{\section}[leftmargin]
      {\titlerule*[1pc]{.}%
       \vspace{1ex}%
       \bfseries}
      {... definition follows

leaders spanning over both main text and title precede the section.

—calcwidth— (Package option)

The —wrap— shape has the capability of measuring the lines in the title to
format the paragraph. This capability may be extended to other three
shapes—namely —display—, —block— and —hang——with this package option. The
length of the longest line is returned in ——.333There are two further
parameters, —“titlewidthfirst— and —“titlewidthlast—, which return the length
of the first and last lines. There are not specific tools for using them, but
you can assign their values to —“titlewidth— and then use —“titleline*—.

As far as TeX is concerned, any box is considered typeset material. If the box
has been enlarged with blank space, or if conversely a box with text has been
smashed, the value of —— may be wrong (as far as humans is concerned). The
—hang— shape, for instance, uses internally such a kind of boxes, but in this
case this behaviour is desired when the title is flushed right; otherwise the
—block— shape produces better results. In other words, using boxes whose
natural width has been overridden may be wrong.444Which include justified
lines, whose interword spacing has been enlarged. Further, some commands may
confuse TeX and stop parsing the title. But if you stick to text, —
— and —
— (and it is very unlikely you might want something else), there will be no
problems.

Another important point is the —¡before-code¿—, —¡label¿—, —¡sep¿—, and
—¡title¿— parameters (but not —¡after-code¿—) are evaluated twice at local
scope; if you increase a counter _globally_ , you are increasing it twice. In
most of cases, placing the conflicting assignment in the —¡after-code¿—
parameter will be ok, but alternativey you can use the following macro.

—


# Some Variations on an Extension of Carath$\acute{E}$odory’s Normality
Criterion

Kuldeep Singh Charak  Kuldeep Singh Charak Department of Mathematics
University of Jammu Jammu-180 006 India<EMAIL_ADDRESS>and  Rahul
Kumar  Rahul Kumar Department of Mathematics University of Jammu Jammu-180 006
India<EMAIL_ADDRESS>
###### Abstract.

In this note we give some variations on an extension of
Carath$\acute{e}$odory’s normality criterion due to Grahl and Nevo.

††footnotetext: 2010 Mathematics Subject Classification 30D45,
30D30.††footnotetext: Keywords and phrases. Normal families,
Carath$\acute{e}$odory’s normality criterion, Spherical metric.††footnotetext:
The work of the first author is partially supported by Mathematical Research
Impact Centric Support (MATRICS) grant, File No. MTR/2018/000446, by the
Science and Engineering Research Board (SERB), Department of Science and
Technology (DST), Government of India.
The work of the second author is supported by the CSIR India.

##  1\. Introduction and Main Results

By $\mathcal{M}(D)$ we shall denote the class of all meromorphic functions on
a domain $D$ in $\mathbb{C}_{\infty},$ where $\mathbb{C}_{\infty}$ has two
metrics $\chi$ and $\sigma$ known as the chordal and spherical metrics. $\chi$
and $\sigma$ are equivalent metrics and hence interchangeable. A subfamily
$\mathcal{F}$ of $\mathcal{M}(D)$ is said to be normal on $D$ if we can
extract a subsequence from each given sequence in $\mathcal{F}$ that converges
locally uniformly on $D$ to a limit function which is either meromorphic on
$D$ or identically equal to $\infty.$ The famous Montel’s Fundamental
Normality Test (see [6]) states that if each member of $\mathcal{F}$ omits
three distinct fixed values in $\mathbb{C}_{\infty},$ then $\mathcal{F}$ is
normal in $D.$

Carath$\acute{e}$odory[3] proved the following extension of Montel’s
Fundamental Normality Test:

Theorem A: Let $\mathcal{F}$ be a subfamily of $\mathcal{M}(\mathbb{D})$ and
suppose that there exists an $\epsilon>0$ such that each $f\in\mathcal{F}$
omits three distinct values $a_{f},b_{f},c_{f}\in\mathbb{C}_{\infty}$
satisfying

$min\\{\sigma(a_{f},b_{f}),\sigma(b_{f},c_{f}),\sigma(c_{f},a_{f})\\}\geq\epsilon.$

Then $\mathcal{F}$ is normal in $\mathbb{D}.$

Grahl and Nevo[5] generalized Theorem A as:

Theorem B: Let $\mathcal{F}$ be a subfamily of $\mathcal{M}(\mathbb{D})$ and
$\epsilon>0.$ Suppose that for each $f\in\mathcal{F}$ there exist three
functions $a_{f},b_{f},c_{f}\in\mathcal{M}(\mathbb{D})\cup\\{\infty\\}$ such
that $f$ omits the functions $a_{f},b_{f},c_{f}$ in $\mathbb{D}$ and

$min\\{\sigma(a_{f}(z),b_{f}(z)),\sigma(b_{f}(z),c_{f}(z)),\sigma(c_{f}(z),a_{f}(z))\\}\geq\epsilon$

for all $z\in\mathbb{D}.$ Then $\mathcal{F}$ is normal in $\mathbb{D}.$

The purpose of this paper is to present some variations on Theorem B.
Precisely, we consider the following variants:

  * (i)

$f-a_{jf}$ and $g-a_{jg}$ share $0$ for each pair of functions
$f,g\in\mathcal{F}$;

  * (ii)

$f^{(k)}-a_{jf}$ and $g^{(k)}-a_{jg}$ share $0$ for each pair of functions
$f,g\in\mathcal{F}$;

  * (iii)

For each $f\in\mathcal{F}$, the zeros of $f-a_{1f}$ are of multiplicity at
least four and those of $f-a_{jf}$ for $j=2,3,$ are of multiplicity at least
three.

######  Theorem 1.1.

Let $\mathcal{F}\subseteq\mathcal{M}(D)$ be such that for each
$f\in\mathcal{F}$ there exist three functions
$a_{1f},a_{2f},a_{3f}\in\mathcal{M}(D)\cup\\{\infty\\}$ satisfying

(1.1) $\sigma(a_{jf}(z),\ a_{kf}(z))\geq\epsilon~{}\mbox{for}~{}j\neq
k~{}\mbox{and}~{}z\in D,\mbox{ for some }\epsilon>0.$

If for each pair of functions $f~{}\mbox{and}~{}g$ in $\mathcal{F},\
f-a_{jf}\mbox{ and }g-a_{jg}$ share $0$ in $D,~{}(j=1,2,3).$ Then
$\mathcal{F}$ is normal in $D.$

######  Example 1.2.

Consider the family $\mathcal{F}=\\{f_{n}:n\in\mathbb{N}\\},$ where

$f_{n}(z)=nz$

on the open unit disk $\mathbb{D}$ and let
$a_{1f_{n}}(z)=nz,~{}a_{2f_{n}}(z)=nz^{2},~{}a_{3f_{n}}(z)=nz^{3}.$ Clearly,
for every $f,g\in\mathcal{F},f-a_{jf}~{}\mbox{and}~{}g-a_{jg}~{}(j=1,2,3)$
share the value $0$ in $\mathbb{D}.$ However, the family $\mathcal{F}$ is not
normal in $\mathbb{D}.$ Note that the functions $a_{jf},j=1,2,3$ do not
satisfy $(\ref{eq1}).$

######  Example 1.3.

Consider the family $\mathcal{F}=\\{f_{n}:n\in\mathbb{N}\\},$ where
$f_{n}(z)=\tan{nz}$ on the open unit disk $\mathbb{D}$ and let
$a_{1f_{n}}(z)=i,~{}a_{2f_{n}}(z)=-i.$ Take $\epsilon=\sigma(i,-i).$ Clearly,
for every $f,g\in\mathcal{F},f-a_{jf}~{}\mbox{and}~{}g-a_{jg}~{}(j=1,2)$ share
the value $0$ in $\mathbb{D}.$ Also
$\sigma(a_{1f_{n}},a_{2f_{n}})\geq\epsilon.$ However, the family $\mathcal{F}$
is not normal in $\mathbb{D}.$ Thus the number of functions $a_{jf}$ in
Theorem 1.1 cannot be reduced to two.

######  Theorem 1.4.

Let $\mathcal{F}\subseteq\mathcal{M}(D)$ be such that all zeros of each
$f\in\mathcal{F}$ are of multiplicity at least $k+1$ and that for each
$f\in\mathcal{F}$ there exist three functions
$a_{1f},a_{2f},a_{3f}\in\mathcal{M}(D)\cup\\{\infty\\}$ satisfying

(1.2) $\sigma(a_{jf}(z),\ a_{lf}(z))\geq\epsilon~{}\mbox{for}~{}j\neq
l~{}\mbox{and}~{}z\in D,\mbox{ for some }\epsilon>0.$

If for each pair of functions $f~{}\mbox{and}~{}g$ in $\mathcal{F},\
f^{(k)}-a_{jf}\mbox{ and }g^{(k)}-a_{jg}$ share $0$ in $D,~{}(j=1,2,3).$ Then
$\mathcal{F}$ is normal in $D.$

The condition, “zeros of each $f\in\mathcal{F}$ have multiplicity at least
$k+1$” in Theorem 1.4 cannot be dropped:

######  Example 1.5.

Consider the family $\mathcal{F}=\\{f_{n}:n\in\mathbb{N}\\},$ where

$f_{n}(z)=nz^{k}$

on the open unit disk $\mathbb{D}.$ Let

$a_{1f_{n}}(z)=1/2,~{}a_{2f_{n}}(z)=1/3,~{}a_{3f_{n}}(z)=1/4$

and take

$\epsilon=\min\left\\{\sigma(1/2,1/3),\sigma(1/2,1/4),\sigma(1/3,1/4)\right\\}.$

Clearly, for every
$f,g\in\mathcal{F},f^{(k)}-a_{jf}~{}\mbox{and}~{}g^{(k)}-a_{jg}~{}(j=1,2,3)$
share the value $0$ in $\mathbb{D}.$ Also
$\sigma(a_{jf_{n}},a_{kf_{n}})\geq\epsilon.$ However, the family $\mathcal{F}$
is not normal in $\mathbb{D}.$

######  Theorem 1.6.

Let $\mathcal{F}\subseteq\mathcal{M}(D)$ be such that for each
$f\in\mathcal{F}$ there exist functions
$a_{1f},a_{2f},a_{3f}\in\mathcal{M}(D)\cup\\{\infty\\}$ such that

  * (i)

$\sigma(a_{jf}(z),a_{kf}(z))\geq\epsilon~{}\mbox{for}~{}j\neq
k~{}\mbox{and}~{}z\in D\mbox{ for some }\epsilon>0;$

  * (ii)

each zero of $f-a_{1f}$ has multiplicity at least four;

  * (iii)

each zero of $f-a_{jf},(j=2,3)$ has multiplicity at least three.

Then $\mathcal{F}$ is normal in $D.$

The following example shows that the condition,
“$\sigma(a_{jf}(z),a_{kf}(z))\geq\epsilon~{}\mbox{for}~{}j\neq
k~{}\mbox{and}~{}z\in D$” in Theorem 1.6 cannot be dropped:

######  Example 1.7.

Consider $\mathcal{F}=\\{f_{n}\\},~{}\mbox{where}~{}f_{n}(z)=nz^{4}+nz^{3}$ on
$\mathbb{D}.$
Let
$a_{1f_{n}}(z)=nz^{3},a_{2f_{n}}(z)=nz^{4}~{}\mbox{and}~{}a_{3f_{n}}(z)=n(z-1/2)^{3}+nz^{4}+nz^{3}.$
Then each zero of $f_{n}-a_{1f_{n}}$ has multiplicity at least four and each
zero of $f_{n}-a_{jf_{n}},(j=2,3)$ has multiplicity at least three but
$\mathcal{F}$ is not normal on $\mathbb{D}.$

######  Theorem 1.8.

Let $\mathcal{F}$ be a subfamily of $\mathcal{M}(D)$ and let
$h\in\mathcal{M}(D).$ Then $\mathcal{F}$ is normal in $D$ if and only if each
$z_{0}\in D$ has a neighbourhood $D(z_{0};r)$ such that each $f\in\mathcal{F}$
satisfies:

$\sigma(f(z),h(z))>\epsilon_{1}~{}\forall z\in
D(z_{0};r)~{}\mbox{or}~{}\sigma(f(z),h(z))<\epsilon_{2}~{}\forall z\in
D(z_{0};r)$

for some $\epsilon_{1},\epsilon_{2}\mbox{ with
}0<\epsilon_{1}<\epsilon_{2}<\pi.$

##  2\. Preparation for the proofs of theorems

Let $f$ be meromorphic function in $\mathbb{C}$ and $a\in\mathbb{C}_{\infty}.$
Then $a$ is called totally ramified value of $f$ if $f-a$ has no simple zeros.
Following result known as Nevanlinna’s Theorem  (see [2]) plays a crucial role
in the proof of Theorem 1.6:

Nevanlinna’s Theorem: Let $f$ be a non-constant meromorphic function,
$a_{1},...,a_{q}\in\mathbb{C}_{\infty}\mbox{ and
}m_{1},...,m_{q}\in\mathbb{N}.$ Suppose that all $a_{j}$-points of $f$ have
multiplicity at least $m_{j},\mbox{ for }~{}j=1,...,q.$ Then

$\sum\limits_{j=1}^{q}(1-\frac{1}{m_{j}})\leq 2.$

If $f$ does not assume the value $a_{j}$ at all, then we take $m_{j}=\infty.$

For $p\in\mathbb{N}$ and $j=1,\ldots,p$ we define the projections
$\pi_{j}:(\mathcal{M}(D))^{p}\to\mathcal{M}(D)$ as:

$\pi_{j}(f_{1},\ldots,f_{p}):=f_{j}~{}\mbox{for}~{}(f_{1},\ldots,f_{p})\in(\mathcal{M}(D))^{p}.$

######  Lemma 2.1.

([5]) Let $p\in\mathbb{N}$ and
$\mathcal{F}\subseteq(\mathcal{M}(\mathbb{D}))^{p}.$ Assume that there exists
$j_{0}\in\\{1,2,\ldots,p\\}$ such that the family $\pi_{j_{0}}(\mathcal{F})$
is not normal at $z_{0}\in\mathbb{D}.$ Then there exist sequences
$(f_{n})_{n}=((f_{1,n},\ldots,f_{p,n}))_{n}\subseteq\mathcal{F},(z_{n})_{n}\subseteq\mathbb{D}~{}\mbox{and}~{}(\rho_{n})_{n}\subseteq(0,~{}1)$
such that
$\lim\limits_{n\to\infty}z_{n}=z_{0},\lim\limits_{n\to\infty}\rho_{n}=0$ and
such that for all $j=1,2,\ldots,p$ the sequences $(g_{j,n})_{n}$ defined by

$g_{j,n}(\zeta):=f_{j,n}(z_{n}+\rho_{n}\zeta)$

converges to functions $g_{j}\in\mathcal{M}(\mathbb{C})\cup\\{\infty\\}$
locally uniformly in $\mathbb{C}$ with respect to the spherical metric, where
at least one of the functions $g_{1},g_{2},\ldots,g_{p}$ is non-constant.

######  Lemma 2.2.

([5]) Let $\mathcal{G}\subseteq(\mathcal{M}(\mathbb{D}))^{2}$ be a family of
pairs of meromorphic functions in $\mathbb{D}$ and $\epsilon>0.$ Assume that

$\sigma(a(z),b(z))\geq\epsilon,~{}\mbox{for
all}~{}(a,b)\in\mathcal{G}~{}\mbox{and all}~{}z\in\mathbb{D}.$

Then the families $\\{a:(a,b)\in\mathcal{G}\\}$ and
$\\{b:(a,b)\in\mathcal{G}\\}$ are normal in $\mathbb{D}.$

######  Lemma 2.3.

([4]) Let $\mathcal{F}\subseteq\mathcal{M}(D),$ all of whose zeros have
multiplicity at least $k+1,$ where $k$ is a positive integer. If
$\mathcal{F}^{(k)}=\\{f^{(k)}:f\in\mathcal{F}\\}$ is normal in $D,$ then
$\mathcal{F}$ is also normal in $D.$

######  Lemma 2.4.

If $\mathcal{G}\subseteq(\mathcal{M}(D))^{2}~{}\mbox{and}~{}0<\epsilon<\pi.$
If

$\sigma(a(z),b(z))<\epsilon~{}\mbox{for all}~{}(a,b)\in\mathcal{G}$

and $\\{b:(a,b)\in\mathcal{G}\\}$ is normal in $D.$ Then
$\\{a:(a,b)\in\mathcal{G}\\}$ is also normal in $D.$

Proof of Lemma 2.4: Suppose on the contrary that $\\{a:(a,b)\in\mathcal{G}\\}$
is not normal at $z_{0}\in D.$ That is, the first projection
$\pi_{1}(\mathcal{G})$ is not normal at $z_{0}.$ Then by Lemma 2.1 there exist
sequences $\\{(a_{n},b_{n})\\}\subseteq\mathcal{G},\\{z_{n}\\}\subseteq
D~{}\mbox{and}~{}\\{\rho_{n}\\}\subset(0,1)$ such that $z_{n}\to
z_{0},\rho_{n}\to 0~{}\mbox{as}~{}n\to\infty,$ and the sequences
$\\{g_{n}\\}~{}\mbox{and}~{}\\{h_{n}\\}$ defined as

$g_{n}(\zeta):=a_{n}(z_{n}+\rho_{n}\zeta)~{}\mbox{and}~{}h_{n}(\zeta):=b_{n}(z_{n}+\rho_{n}\zeta)$

converging locally uniformly in $\mathbb{C}$ (with respect to the spherical
metric) to functions $g~{}\mbox{and}~{}h$ respectively, where at least one of
$g$ or $h$ is non-constant. Since $\\{b:(a,b)\in\mathcal{G}\\}$ is normal in
$D,h$ is constant on
$\mathbb{C},~{}\mbox{say},h(\zeta)=w_{0},\forall\zeta\in\mathbb{C}.$ Since

$\sigma(a_{n}(z_{n}+\rho_{n}\zeta),b_{n}(z_{n}+\rho_{n}\zeta))<\epsilon,$

it follows that $\sigma(g(\zeta),w_{0})\leq\epsilon,~{}\mbox{for
all}~{}\zeta\in\mathbb{C}$ which is a contradiction since $g$ is non-constant.
$\Box$

##  3\. Proofs of Theorems

Proof of Theorem 1.1: Let

$\mathcal{G}:=\\{(f,a_{1f},a_{2f},a_{3f}):f\in\mathcal{F}\\}$

Suppose $\mathcal{F}$ is not normal at $z_{0}\in D.$ That is, the first
projection $\pi_{1}(\mathcal{G})$ is not normal at $z_{0}.$ Then by Lemma 2.1
there exist sequences
$\\{(f_{n},a_{1f_{n}},a_{2f_{n}},a_{3f_{n}})\\}\subseteq\mathcal{G},\\{z_{n}\\}\subseteq
D~{}\mbox{and}~{}\\{\rho_{n}\\}\subset(0,1)$ such that $z_{n}\to
z_{0},\rho_{n}\to 0~{}\mbox{as}~{}n\to\infty$ and the sequences
$\\{g_{n}\\},\\{A_{1n}\\},$
$\\{A_{2n}\\},\\{A_{3n}\\}$ defined by

$g_{n}(\zeta):=f_{n}(z_{n}+\rho_{n}\zeta),$
$A_{1n}(\zeta):=a_{1f_{n}}(z_{n}+\rho_{n}\zeta),A_{2n}(\zeta):=a_{2f_{n}}(z_{n}+\rho_{n}\zeta),A_{3n}(\zeta):=a_{3f_{n}}(z_{n}+\rho_{n}\zeta)$

converges spherically locally uniformly in $\mathbb{C}$ to functions
$g,A,B,C\in\mathcal{M}(\mathbb{C})\cup\\{\infty\\}$ respectively, all of which
are not constant.
Since

$\sigma(a_{jf_{n}}(z),a_{kf_{n}}(z))\geq\epsilon~{}\mbox{for}~{}j\neq
k~{}\mbox{and}~{}z\in D~{}\mbox{and}~{}n\in\mathbb{N}$

it follows that $\\{a_{if_{n}}\\},~{}(i=1,2,3)$ forms a normal family on $D$
and hence the limit functions $A,B,C$ are constant and consequently $g$ is
non-constant.
Claim: $g$ omits at least two of the values ${A,B,C.}$
Suppose on the contrary $g$ assumes two values $B~{}\mbox{and}~{}C.$ Let
$g(\zeta_{0})=B~{}\mbox{and}~{}g(\zeta_{0}^{*})=C~{}\mbox{for
some}~{}\zeta_{0},\zeta_{0}^{*}\in\mathbb{C}.$ By Hurwitz’s theorem there
exist sequences $\zeta_{n}\to\zeta_{0}$ and $\zeta_{n}^{*}\to\zeta_{0}^{*}$
such that

$f_{n}(z_{n}+\rho_{n}\zeta_{n})-a_{2f_{n}}(z_{n}+\rho_{n}\zeta_{n})=0~{}\mbox{and}~{}f_{n}(z_{n}+\rho_{n}\zeta_{n}^{*})-a_{3f_{n}}(z_{n}+\rho_{n}\zeta_{n}^{*})=0.$

By hypothesis, for any positive integer $m,$

$f_{m}(z_{n}+\rho_{n}\zeta_{n})-a_{2f_{m}}(z_{n}+\rho_{n}\zeta_{n})=0~{}\mbox{and}~{}f_{m}(z_{n}+\rho_{n}\zeta_{n}^{*})-a_{3f_{m}}(z_{n}+\rho_{n}\zeta_{n}^{*})=0.$

Fix $m$ and take $n\to\infty$, we have

$f_{m}(z_{0})-a_{2f_{m}}(z_{0})=0~{}\mbox{and}~{}f_{m}(z_{0})-a_{3f_{m}}(z_{0})=0.$

which contradicts the fact that
$\sigma(a_{2f_{m}}(z_{0}),a_{3f_{m}}(z_{0}))\geq\epsilon.$

Without loss of generality, we assume $g$ omits $B~{}\mbox{and}~{}C$ and so
$g(\zeta)$ assume $A$ infinitely many times. Let
$\zeta^{(1)}~{}\mbox{and}~{}\zeta^{(2)}$ be two distinct zeros of
$g(\zeta)-A.$ Choose $\delta>0$ small enough such that
$D(\zeta^{(1)},\delta)\cap D(\zeta^{(2)},\delta)=\emptyset.$
Since $g$ is non-constant, by Hurwitz’s theorem there are points
$\zeta_{n}^{(1)}\in D(\zeta^{(1)},\delta)$ and $\zeta_{n}^{(2)}\in
D(\zeta^{(2)},\delta)$ such that for sufficiently large $n$

$f_{n}(z_{n}+\rho_{n}\zeta_{n}^{(1)})-a_{1f_{n}}(z_{n}+\rho_{n}\zeta_{n}^{(1)})=0~{}\mbox{and}~{}f_{n}(z_{n}+\rho_{n}\zeta_{n}^{(2)})-a_{1f_{n}}(z_{n}+\rho_{n}\zeta_{n}^{(2)})=0.$

By hypothesis, for any positive integer $m,$

$f_{m}(z_{n}+\rho_{n}\zeta_{n}^{(1)})-a_{1f_{m}}(z_{n}+\rho_{n}\zeta_{n}^{(1)})=0~{}\mbox{and}~{}f_{m}(z_{n}+\rho_{n}\zeta_{n}^{(2)})-a_{1f_{m}}(z_{n}+\rho_{n}\zeta_{n}^{(2)})=0.$

Fix $m,$ take $n\to\infty,$ and in view of the fact that
$z_{n}+\rho_{n}\zeta_{n}^{(1)}\to
z_{0}~{}\mbox{and}~{}z_{n}+\rho_{n}\zeta_{n}^{(2)}\to z_{0},$ we have

$f_{m}(z_{0})-a_{1f_{m}}(z_{0})=0.$

Since the zeros of $f_{m}-a_{1f_{m}}$ have no accumulation point, we have

$z_{n}+\rho_{n}\zeta_{n}^{(1)}=z_{0}~{}\mbox{and}~{}z_{n}+\rho_{n}\zeta_{n}^{(2)}=z_{0}$

and so $\zeta_{n}^{(1)}=\zeta_{n}^{(2)},$ which is a contradiction. $\Box$

Proof of Theorem 1.4: By the similar arguments as in the proof of Theorem 1.1,
we conclude that $\mathcal{F}^{(k)}=\\{f^{(k)}:f\in\mathcal{F}\\}$ is normal
on $D$ and hence by Lemma 2.3, the normality of $\mathcal{F}$ on $D$ follows.
$\Box$

Proof of Theorem 1.6: Let

$\mathcal{G}:=\\{(f,a_{1f},a_{2f},a_{3f}):f\in\mathcal{F}\\}$

Suppose $\mathcal{F}$ is not normal at $z_{0}\in D.$ That is, the first
projection $\pi_{1}(\mathcal{G})$ is not normal at $z_{0}.$ Then by Lemma 2.1
there exist sequences
$\\{(f_{n},a_{1f_{n}},a_{2f_{n}},a_{3f_{n}})\\}\subseteq\mathcal{G},\\{z_{n}\\}\subseteq
D~{}\mbox{and}~{}\\{\rho_{n}\\}\subset(0,1)$ such that

  * (a)

$z_{n}\to z_{0},\rho_{n}\to 0~{}\mbox{as}~{}n\to\infty;$

  * (b)

all zeros of $f_{n}-a_{1f_{n}}$ has multiplicity at least four ;

  * (b)

all zeros of $f_{n}-a_{jf_{n}},(j=2,3)$ has multiplicity at least three ;

  * (c)

$\sigma(a_{jf_{n}}(z),a_{kf_{n}}(z))\geq\epsilon~{}\mbox{for}~{}j\neq
k~{}\mbox{and}~{}z\in D~{}\mbox{and}~{}n\in\mathbb{N};$ and

  * (d)

the sequences $\\{g_{n}\\},\\{A_{1n}\\},\\{A_{2n}\\},\\{A_{3n}\\}$ defined by

$g_{n}(\zeta):=f_{n}(z_{n}+\rho_{n}\zeta),$
$A_{1n}(\zeta):=a_{1f_{n}}(z_{n}+\rho_{n}\zeta),A_{2n}(\zeta):=a_{2f_{n}}(z_{n}+\rho_{n}\zeta),A_{3n}(\zeta):=a_{3f_{n}}(z_{n}+\rho_{n}\zeta)$

converges spherically locally uniformly in $\mathbb{C}$ to functions
$g,A,B,C\in\mathcal{M}(\mathbb{C})\cup\\{\infty\\}$ respectively, all of which
are not constant.

Now by $(c)$ it follows that $\\{a_{jf_{n}}\\},(j=1,2,3)$ forms a normal
family in $D$ and hence the limit functions $A,B,C$ are constant and
consequently $g$ is non-constant.
Let $h_{n}(\zeta):=g_{n}(\zeta)-A_{1n}(\zeta).$ Then $h_{n}(\zeta)\to
g(\zeta)-A.$ Now by $(b)$ and Argument principle, it follows that each zero of
$g(\zeta)-A$ has multiplicity at least four. That is, A-points of g are of
multiplicity at least four. Similarly, we find that B-points and C-points of
$g$ are of multiplicity at least three. Since $g$ is non-constant, we arrive
at a contradiction by Nevanlinna’s Theorem. $\Box$

Proof of Theorem 1.8: Suppose $\mathcal{F}$ is normal in $D$ and suppose on
the contrary that there is a neighborhood $D(z_{0},r)$ of $z_{0},$ a sequence
$\\{f_{n}\\}~{}\mbox{in}~{}\mathcal{F}$ two sequences
$\\{z_{n}\\}~{}\mbox{and}~{}\\{w_{n}\\}$ of complex numbers with $z_{n}\to
z_{0}~{}\mbox{and}~{}w_{n}\to z_{0}$ such that

$\sigma(f_{n}(z_{n}),h(z_{n}))\leq\epsilon_{1}~{}\mbox{and}~{}\sigma(f_{n}(w_{n}),h(w_{n}))\geq\epsilon_{2}$

Since $\\{f_{n}\\}$ is normal on $D,$ there is a subsequence (we still denote
it by $\\{f_{n}\\}$) such that $\\{f_{n}\\}$ converges locally uniformly to
$f$ with respect to the spherical metric. Thus

$\displaystyle\epsilon_{1}$ $\displaystyle\geq$
$\displaystyle\lim\limits_{n\to\infty}\sigma(f_{n}(z_{n}),h(z_{n}))$
$\displaystyle=$ $\displaystyle\sigma(f(z_{0}),h(z_{0}))$ $\displaystyle=$
$\displaystyle\lim\limits_{n\to\infty}\sigma(f_{n}(w_{n}),h(w_{n}))$
$\displaystyle\geq$ $\displaystyle\epsilon_{2},$

which is a contradiction.

Conversely, suppose that each $z_{0}\in D$ has a neighborhood with the given
property. Define $h_{n}(z):=h(z)~{}\forall n.$ Then $\\{h_{n}\\}$ is a normal
family in $D$ and $h_{n}\to h$ locally uniformly with respect to the spherical
metric. Then for each $n$

$\sigma(f_{n}(z),h_{n}(z))>\epsilon_{1}~{}\forall z\in
D(z_{0},r)~{}\mbox{or}~{}\sigma(f_{n}(z),h_{n}(z))<\epsilon_{2}~{}\forall z\in
D(z_{0},r).$

In the first case $\\{f_{n}\\}$ is normal in $D(z_{0},r)$ by Lemma 2.2 and in
the second case $\\{f_{n}\\}$ is normal in $D(z_{0},r)$ by Lemma 2.4. $\Box$

##  4\. Normality of Families of Rational Maps

Beardon, Minda and Short([1]) proved the following theorem:

######  Theorem 4.1.

For any family $\mathcal{F}$ of M$\ddot{o}$bius maps and any domain $D$ in
$\mathbb{C}_{\infty}$, the following are equivalent:

  * (a)

The family $\mathcal{F}$ is normal in $D.$

  * (b)

For any set $A=\\{u,v,w\\}$ of distinct points in $\mathbb{C}_{\infty},$ each
point $z_{0}$ in $D$ has a neighbourhood in which each $f$ in $\mathcal{F}$
omits at least two of the values in $A.$

  * (c)

There exists a set $A=\\{u,v,w\\}$ of distinct points in $\mathbb{C}_{\infty}$
such that each point $z_{0}$ in $D$ has a neighbourhood in which each $f$ in
$\mathcal{F}$ omits at least two of the values in $A.$

Whether Theorem 4.1 holds for rational functions of higher degree, is the
objective of this section. Let $\mathcal{R}_{m}$ denote a family of rational
maps of degree $d:0\leq d\leq m,\ m\in\mathbb{N}.$ Suppose $\mathcal{R}_{m}$
is normal in $D.$ Then consider the set $B:=\\{u,v,w,x\\}$, where
$x\in\mathbb{C}$ is different from other elements of the set. Then each
$z_{0}\in D$ has a neighbourhood in which each $R\in\mathcal{R}_{m}$ omits at
least three of the four values in $B$ showing that $(a)$ implies $(b)$ holds
for the family $\mathcal{R}_{m}$. Also $(b)\Rightarrow(c)$ holds trivially for
$\mathcal{R}_{m}.$ Now we prove that $(c)\Rightarrow(a)$ also holds for
$\mathcal{R}_{m}:$
Suppose $(c)$ holds. Let $\\{R_{p}\\}$ be any sequence in $\mathcal{R}_{m}$
and $z_{0}\in D.$ Then $z_{0}$ has a neighborhood $U_{0}$ in which each
$R_{p}$ omits at least two of the values in $A.$ We consider the subsets
$\\{R_{p_{j}}\\},\ \\{R_{p_{k}}\\}$ and $\\{R_{p_{l}}\\}$ of $\\{R_{p}\\}$
such that each $R_{p_{j}},\ R_{p_{k}}$ and $R_{p_{l}}$ omits
$u~{}\mbox{and}~{}v,$ $v~{}\mbox{and}~{}w,$ and $u~{}\mbox{and}~{}w,$
respectively. Without loss of generality, we assume $\\{R_{p_{j}}\\}$ to be a
subsequence of $\\{R_{p}\\}$ which we we again denote by $R_{p}$ itself. Thus
each $R_{p}$ omits $u~{}\mbox{and}~{}v.$ Suppose $\\{R_{p}\\}$ is not normal
at $z_{0}.$ Then by Zalcman’s lemma we can find a sequence in $\\{R_{p}\\}$
which we again denote by $\\{R_{p}\\}$, a sequence $\\{z_{p}\\}$ of complex
numbers with $z_{p}\to z_{0}$ and a sequence $\\{\rho_{p}\\}$ of positive real
numbers with $\rho_{p}\to 0$ such that
$S_{p}(\zeta)=R_{p}(z_{p}+\rho_{p}\zeta)$ converges locally uniformly with
respect to the spherical metric to a non-constant meromorphic function $S$ on
$\mathbb{C}.$ Since each $R_{p}$ omits $u~{}\mbox{and}~{}v,$ it follows that
$S$ also omits $u~{}\mbox{and}~{}v$ and so $S$ takes some value $a$ infinitely
many times. Let $\zeta_{1},\zeta_{2},\ldots,\zeta_{m+1}$ be $m+1$ distinct
zeros of $S-a.$ Since $S$ is non-constant, by Hurwitz’s theorem there is a
sequence $\zeta_{p,i}\to\zeta_{i}$ such that
$R_{p}(z_{p}+\rho_{p}\zeta_{p,i})=a~{}(i=1,2,\ldots,m+1).$ Since $R_{p}$ is a
rational function of degree at most $m,$ there is some $~{}i\neq j,1\leq
i,j\leq m+1$ such that $z_{p}+\rho_{p}\zeta_{p,i}=z_{p}+\rho_{p}\zeta_{p,j}$
which implies $\zeta_{i}=\zeta_{j},$ a contradiction. Hence Theorem 4.1 is
valid for $\mathcal{R}_{m}.$
Further, as an immediate consequence of preceding discussion we find that if
each $R\in\mathcal{R}_{m}.$ omits two given fixed values in
$\mathbb{C}_{\infty},$ on a domain $D$, then $\mathcal{R}_{m}$ is normal in
$D.$

## References

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