# Construction of smooth rhythms through a monotone invariant measure

Fumio HAZAMA
Tokyo Denki University
Hatoyama, Hiki-Gun, Saitama JAPAN
e-mail<EMAIL_ADDRESS>
###### Abstract

The present article introduces the notion of quasi-smoothness of marked
rhythms through a certain transformation $Ref$, called reformation map. A
marked rhythm consists of a rhythm together with a marker, and the map $Ref$
modifies the marked onset of the rhythm. It is shown that an iteration of the
map $Ref$ transforms an arbitrary marked rhythm into a quasi-smooth one. A
numerical criterion for a marked rhythm to be quasi-smooth is given in terms
of the difference of its rhythm part. Through this criterion, the rhythm part
of any quasi-smooth marked rhythm is shown to be smooth.

$\mathbf{Keywords}$: marked rhythm, reformation map, deformation map, quasi-
smoothness, finite dynamical system, monotone invariant measure.

##  0 Introduction

The main purpose of this paper is to propose a new method of construction of
smooth rhythms. The notion of smoothness of a rhythm is introduced in (Hazama
2022), and is shown to give us a larger category of rhythms than that of
maximally even ones, which have been widely studied (Clough et a. 2000;
Demaine 2009; Toussaint 2013). In (Hazama 2022), a self-map $Rav$ on the space
$\mathbf{R}_{N}^{n}$ of rhythms of $N$ beats with $n$ onsets is introduced,
and a rhythm is defined to be smooth if it is periodic under $Rav$. One of the
main results of (Hazama 2022) gives us a numerical criterion for a rhythm to
be smooth in terms of its width. The proof of the validity of the criterion,
given there, is quite involved however, and the author of the paper expresses
his hope about a possible simplification of its original proof. Our main
result realizes his hope by introducing a self-map $Ref$, called reformation
map, on the space $m\mathbf{R}_{N}^{n}=\mathbf{Z}_{n}\times\mathbf{R}_{N}^{n}$
of marked rhythms. The first coordinate $k$ of a marked rhythm
$(k,\mbox{\boldmath$a$})$ indicates which entry of the rhythm $a$ is modified
by $Ref$. Since $Ref$ does not change any other entries, the dynamical
behavior of $Ref$ will be seen to be much simpler than that of $Rav$, which
changes all the entries at the same time. Our main result shows that, if a
marked rhythm is periodic under $Ref$, then its rhythm part is smooth in the
sense of (Hazama 2022).

The plan of this paper is as follows. Section one introduces our main tool Ref
on $m\mathbf{R}_{N}^{n}$ for our study. The map leads us to a finite dynamical
system $(m\mathbf{R}_{N}^{n},Ref)$, which we denote by $\mathbf{Ref}_{N}^{n}$.
The notion of quasi-smoothness of a marked rhythm is defined by the property
that it is periodic under $Ref$. We investigate the dynamics of $Ref$ through
its descent to a quotient dynamical system
$\mathbf{Def}_{N}^{n}=(m\mathbf{D}_{N}^{n},Def)$ by an automorphism of
$\mathbf{Ref}_{N}^{n}$. The compatibility of $Ref$ and $Def$ is proved at the
end of this section. Section two axiomatizes a crucial point which emerges in
our treatment of the dynamical system $\mathbf{Ref}_{N}^{n}$ together with its
quotient system $\mathbf{Def}_{N}^{n}$. We introduce the notion of “monotone
invariant measure” on an arbitrary finite dynamical system $\mathbf{F}=(X,F)$
with an automorphism $\varphi$. It is shown to descend to the quotient
dynamical system $\overline{\mathbf{F}}=(\overline{X},\overline{F})$ by the
action of $\varphi$. The measure enables one to relate the periodic behavior
of the self-map $F$ on $X$ with that of the self-map $\overline{F}$ on
$\overline{X}$. Section three constructs a monotone invariant measure
$\mu_{m\mathbf{R}}$ on $\mathbf{Ref}_{N}^{n}$. This is modeled after one of
the measures proposed and investigated in (Rota 2001). A general result in the
previous section allows $\mu_{m}{\mathbf{R}}$ to descend to a measure
$\mu_{m\mathbf{D}}$ on $\mathbf{Def}_{N}^{n}$, and enables us to relate the
periodicity of $Def$ and that of $Ref$. Section four introduces the notion of
$\mu_{m\mathbf{D}}$-stability, which is shown to be a necessary condition for
the periodicity. The final step toward our goal is provided by the fact that
$\mu_{m\mathbf{D}}$-stability of $\mbox{\boldmath$D$}\in m\mathbf{D}_{N}^{n}$
implies that the width of $D$ is less than or equal to 1. Thus we arrive at
our main theorem which shows the equivalence of the quasi-smoothness and the
smoothness.

##  1 Reformation map for marked rhythms

In this section we fix some notation and terminology, and introduce our main
subject, the reformation map.

###  1.1 Basic setup

For any integer $m\geq 3$, let $\mathbf{Z}_{m}=\\{0,1,\cdots,m-1\\}$, and let
$R_{m}:\mathbf{Z}\rightarrow\mathbf{Z}_{m}$ denote the function which maps an
integer to its least nonnegative remainder modulo $m$. The usual addition and
subtraction operations on $\mathbf{Z}/m\mathbf{Z}$ are translated to the ones
on the set $\mathbf{Z}_{m}$, which we denote by “$+_{m}$” and “$-_{m}$”
respectively. Namely we define

$\displaystyle a+_{m}b$ $\displaystyle=$ $\displaystyle R_{m}(a+b),$
$\displaystyle a-_{m}b$ $\displaystyle=$ $\displaystyle R_{m}(a-b),$

for any pair $a,b\in\mathbf{Z}_{m}$. For any pair $a,b$ of distinct elements
in $\mathbf{Z}_{m}$, we define the close interval by
$[a,b]_{m}=\\{a,a+_{m}1,\cdots,b\\}\subset\mathbf{Z}_{m}$.

Let $N$ be an integer $\geq 3$, and let $n$ be an integer with $3\leq n\leq
N$. Throughout the paper the integers $N,n$ are assumed to satisfy these
conditions. An $n$-tuple
$\mbox{\boldmath$a$}=(a_{0},a_{1},\cdots,a_{n-1})\in\mathbf{Z}_{N}^{n}$ of
mutually distinct elements of $\mathbf{Z}_{N}$ is a rhythm of length $N$ with
$n$ onsets, if it satisfies the condition

$\displaystyle\sum_{k=0}^{n-1}(a_{k}-_{N}a_{k-_{n}1})=N.$ (1.1)

Here the summation on the left hand side means the usual addition on
$\mathbf{Z}$. We denote by $\mathbf{R}_{N}^{n}$ the set of rhythms of length
$N$ with $n$ onsets. A marked rhythm is the pair $(k,\mbox{\boldmath$a$})$ of
an element of $k\in\mathbf{Z}_{n}$, called the marker, and a rhythm
$\mbox{\boldmath$a$}\in\mathbf{R}_{N}^{n}$, called the rhythm part. The $k$-th
coordinate $a_{k}$ is called the marked entry of the marked rhythm
$(k,\mbox{\boldmath$a$})$. The set of marked rhythm is denoted by
$m\mathbf{R}_{N}^{n}$:

$\displaystyle m\mathbf{R}_{N}^{n}=\mathbf{Z}_{n}\times\mathbf{R}_{N}^{n}.$

Our main ingredient $Ref$ of this papr will be defined as a self-map on
$m\mathbf{R}_{N}^{n}$.

For our investigation of rhythms and marked rhythms, it is indispensable to
consider their differences, whose effectiveness is also fully illustrated in
the comprehensive book (Toussaint 2013). For any rhythm $a$, its difference
$d(\mbox{\boldmath$a$})$ is defined by

$\displaystyle
d(\mbox{\boldmath$a$})=(a_{0}-_{N}a_{n-1},a_{1}-_{N}a_{0},\cdots,a_{n-1}-_{N}a_{n-2})$

Since a rhythm consists of mutually distinct elements of $\mathbf{Z}_{N}$, it
follows from (1.1) that the difference belongs to the set

$\displaystyle\mathbf{D}_{N}^{n}=\\{(d_{0},d_{1},\cdots,d_{n-1})\in\mathbf{Z}_{N}^{*}|\sum_{k=0}^{n-1}d_{k}=N\\},$

where $\mathbf{Z}_{N}^{*}=\mathbf{Z}_{N}\setminus\\{0\\}$. For any marked
rhythm $(k,\mbox{\boldmath$a$})\in m\mathbf{R}_{N}^{n}$, the difference
$\Delta(k,\mbox{\boldmath$a$})$ is defined by

$\displaystyle\Delta(k,\mbox{\boldmath$a$})=(k,d(\mbox{\boldmath$a$})).$

Hence it belongs to

$\displaystyle m\mathbf{D}_{N}^{n}=\mathbf{Z}_{n}\times\mathbf{D}_{N}^{n}.$

Its elements are called marked differences. The first component is called the
marker and the second is the difference part.

Let $tr$ denote the bijective self-map on $\mathbf{R}_{N}^{n}$ defined by

$\displaystyle tr(\mbox{\boldmath$a$})=(a_{0}+_{N}1,\cdots,a_{n-1}+_{N}1)$
(1.2)

for any $\mbox{\boldmath$a$}=(a_{0},\cdots,a_{n-1})\in\mathbf{R}_{N}^{n}$, and
let $\sim_{tr}$ denote the equivalence relation on $\mathbf{R}_{N}^{n}$
generated by the bijection $tr$. It is not hard to see that the space
$\mathbf{D}_{N}^{n}$ is identified with the quotient space
$\mathbf{R}_{N}^{n}/\sim_{tr}$ of $\mathbf{R}_{N}^{n}$ by the equivalence
relation $\sim_{tr}$. Similarly, if we define the bijective self-map $mtr$ on
$m\mathbf{R}_{N}^{n}$ by

$\displaystyle mtr(k,\mbox{\boldmath$a$})=(k,tr(\mbox{\boldmath$a$})),$ (1.3)

and denote by $\sim_{mtr}$ the equivalence relation generated by the bijection
$mtr$, then we see that the space $m\mathbf{D}_{N}^{n}$ is identified with the
quotient space $m\mathbf{R}_{N}^{n}/\sim_{mtr}$ of $m\mathbf{R}_{N}^{n}$ by
the equivalence relation $\sim_{mtr}$.

###  1.2 Definition of the reformation map $Ref$

In this section, we define a self map “Ref”, called reformation map, on
$m\mathbf{R}_{N}^{n}$. It is based on the discrete average
$rav:\mathbf{Z}_{N}\times\mathbf{Z}_{N}\rightarrow\mathbf{Z}_{N}$ defined as
follows.

###### Definition 1.1

For any pair $(a,b)\in\mathbf{Z}_{N}^{2}$, its discrete average
$rav_{N}(a,b)\in\mathbf{Z}_{N}$ is defined by

$\displaystyle rav_{N}(a,b)=a+_{N}\left\lfloor\frac{b-_{N}a}{2}\right\rfloor.$

Note that the discrete average is compatible with the addition by one modulo
$N$, namely we have

$\displaystyle rav_{N}(a+_{N}1,b+_{N}1)=rav_{N}(a,b)+_{N}1.$ (1.4)

Although this appears to be trivial, the whole our study in this paper is
built on this simple equality. We define a self map
$ref:\mathbf{R}_{N}^{3}\rightarrow\mathbf{R}_{N}^{3}$ as follows.

###### Definition 1.2

For any $(a,b,c)\in\mathbf{R}_{N}^{3}$, we put

$\displaystyle ref(a,b,c)=(a,rav_{N}(a,c),c).$ (1.5)

Note that the map $ref$ transforms only the middle entry into the discrete
average of the first and the third entries. It follows from (1.4) that

$\displaystyle ref\circ tr=tr\circ ref.$ (1.6)

Based on this map, we define the reformation map, denoted by $Ref$, on the
space $m\mathbf{R}_{N}^{n}$ as follows.

###### Definition 1.3

For any $(k,\mbox{\boldmath$a$})\in m\mathbf{R}_{N}^{n}$ with
$\mbox{\boldmath$a$}=(a_{0},a_{1},\cdots,a_{n-1})$, we define
$Ref(k,\mbox{\boldmath$a$})$ to be $(k+_{n}1,\mbox{\boldmath$b$})$, where
$\mbox{\boldmath$b$}=(b_{0},b_{1},\cdots,b_{n-1})$ is given in the following
way: Let $(c_{k-_{n}1},c_{k},c_{k+_{n}1})=ref(a_{k-_{n}1},a_{k},a_{k+_{n}1})$,
and define

$\displaystyle b_{i}=\begin{cases}c_{k-_{n}1},&\mbox{ if }i=k-_{n}1,\\\
c_{k},&\mbox{ if }i=k,\\\ c_{k+_{n}1},&\mbox{ if }i=k+_{n}1,\\\ a_{i},&\mbox{
if }i\notin\\{k-_{n}1,k,k+_{n}1\\}\\\ \end{cases}$

We see below through a few examples how the reformation map acts on marked
rhythms. In order to visualize marked rhythms, we employ their circle graphs,
whose construction follows. When
$\mbox{\boldmath$A$}=(k,\mbox{\boldmath$a$})=(k,(a_{0},\cdots,a_{n-1}))$ is an
element of $m\mathbf{R}_{N}^{n}$, we draw a small disk at
$\chi_{N}(a_{j})\in\mathbf{C}$ for any $j\in\mathbf{Z}_{n}$, where
$\chi_{N}:\mathbf{Z}_{N}\rightarrow\mathbf{C}$ is defined by
$\chi_{N}(a)=e^{2\pi ia/N}$, and connect the $n$ disks by a polygonal line.
The marked entry is enclosed in a circle. The numbers $m=0,1,\cdots,N-1$,
surrounding the unit circle, indicate the spots where the nodes $\chi_{N}(m)$
locate.

Example 1 For the marked rhythm $\mbox{\boldmath$A$}=(0,(0,1,2))\in
m\mathbf{R}_{8}^{3}$, the marked entry “$0$” is moved by $Ref$ to the discrete
average “$5$” of “$2$” and “$1$”, and the marker “$0$” goes to the next “$1$”:

Fig 1: $\mbox{\boldmath$A$}=(0,(0,1,2))$ and
$Ref(\mbox{\boldmath$A$})=\mbox{\boldmath$A$}^{(1)}=(1,(5,1,2))$

By iterating the application of $Ref$, the marked rhythm
$\mbox{\boldmath$A$}^{(1)}=(1,(5,1,2))$ on the right of Figure 1 is mapped to
$\mbox{\boldmath$A$}^{(2)}=(2,(5,7,2))$:

Fig 2: $\mbox{\boldmath$A$}^{(1)}=(1,(5,1,2))$ and
$Ref(\mbox{\boldmath$A$}^{(1)})=\mbox{\boldmath$A$}^{(2)}=(2,(5,7,2))$

We see that the rhythm part of the marked rhythm $\mbox{\boldmath$A$}^{(2)}$
is smooth, since its width is equal to one. The main purpose of this paper is
to show that this is a general phenomenon. Namely, we can make the rhythm part
of an arbitrary marked rhythm smooth by iterated applications of $Ref$.

The equality (1.6) implies that the compatibility relation

$\displaystyle Ref\circ mtr=mtr\circ Ref$ (1.7)

holds. Hence the self-map $Ref$ on $m\mathbf{R}_{N}^{n}$ descends to
$m\mathbf{D}_{N}^{n}=m\mathbf{R}_{N}^{n}/\sim_{mtr}$. A concrete expression of
the descended map is given in the next subsection.

###  1.3 Descent of $Ref$ to $m\mathbf{D}_{N}^{n}$

We introduce a self map $Def$, called deformation map, on
$m\mathbf{D}_{N}^{n}$, and show that it is compatible with the reformation map
$Ref$ through the difference map $\Delta$.

###### Definition 1.4

For any $(k,\mbox{\boldmath$d$})\in m\mathbf{D}_{N}^{n}$ with
$\mbox{\boldmath$d$}=(d_{0},d_{1},\cdots,d_{n-1})$, we define
$Def(k,\mbox{\boldmath$d$})$ to be $(k+_{n}1,\mbox{\boldmath$e$})$, where
$\mbox{\boldmath$e$}=(e_{0},e_{1},\cdots,e_{n-1})$ is given in the following
way: Let

$\displaystyle(f_{k},f_{k+_{n}1})=(\left\lfloor\frac{d_{k}+_{N}d_{k+_{n}1}}{2}\right\rfloor,\left\lceil\frac{d_{k}+_{N}d_{k+_{n}1}}{2}\right\rceil),$

and define

$\displaystyle e_{i}=\begin{cases}f_{k},&\mbox{ if }i=k,\\\
f_{k+_{n}1},&\mbox{ if }i=k+_{n}1,\\\ d_{i},&\mbox{ if
}i\notin\\{k,k+_{n}1\\}\\\ \end{cases}$

###### Proposition 1.1

We have the following commutative diagram:

$\displaystyle\begin{CD}m\mathbf{R}_{N}^{n}@>{\Delta}>{}>m\mathbf{D}_{N}^{n}\\\
@V{Ref}V{}V@V{Def}V{}V\\\
m\mathbf{R}_{N}^{n}@>{\Delta}>{}>m\mathbf{D}_{N}^{n}\\\ \end{CD}$ (1.8)

###### Proof.

Let $(k,\mbox{\boldmath$a$})=(k,(a_{0},\cdots,a_{n-1}))\in
m\mathbf{R}_{N}^{n}$ be an arbitrary marked rhythm, and let
$Ref(k,\mbox{\boldmath$a$})=(k+_{n}1,\mbox{\boldmath$b$})$. It follows from
Definition 1.3 that $b_{i}=a_{i}$ for any
$i\in\mathbf{Z}_{n}\setminus\\{k\\}$. Furthermore, if we put
$\Delta(k,\mbox{\boldmath$a$})=(k,\mbox{\boldmath$d$})$ and
$Def(k,\mbox{\boldmath$d$})=(k+_{n}1,\mbox{\boldmath$e$})$, then it follows
from Definition 1.4 that $e_{i}=d_{i}$ for any
$i\in\mathbf{Z}_{n}\setminus\\{k,k+_{n}1\\}$.Therefore, for the proof of the
commutativity, we have only to focus on the behavior of the subtriple
$(a_{k-_{n}1},a_{k},a_{k+_{n}1})$ of the rhythm part $a$. It follows that we
are reduced to proving the assertion for the case when $n=3$. Let
$(1,(a_{0},a_{1},a_{2}))\in m\mathbf{R}_{N}^{3}$ be an arbitrary marked rhythm
with marker equal to 1. It is mapped by $\Delta\circ Ref$ to

$\displaystyle\Delta(Ref(1,(a_{0},a_{1},a_{2})))$
$\displaystyle=\Delta(2,(a_{0},rav_{N}(a_{0},a_{2}),a_{2}))$
$\displaystyle=(2,(a_{0}-_{N}a_{2},rav_{N}(a_{0},a_{2})-_{N}a_{0},a_{2}-_{N}rav_{N}(a_{0},a_{2})))$
$\displaystyle=(2,(a_{0}-_{N}a_{2},\left\lfloor\frac{a_{2}-_{N}a_{0}}{2}\right\rfloor,(a_{2}-_{N}a_{0})-_{N}\left\lfloor\frac{a_{2}-_{N}a_{0}}{2}\right\rfloor))$
$\displaystyle=(2,(a_{0}-_{N}a_{2},\left\lfloor\frac{a_{2}-_{N}a_{0}}{2}\right\rfloor,\left\lceil\frac{a_{2}-_{N}a_{0}}{2}\right\rceil)),$
(1.9)

the last equality being a consequence of the equality
$x=\left\lfloor\frac{x}{2}\right\rfloor+_{N}\left\lceil\frac{x}{2}\right\rceil$,
which holds for any $x\in\mathbf{Z}_{N}$. On the other hand, the value of
$Def\circ\Delta$ at $(1,(a_{0},a_{1},a_{2}))$ is equal to

$\displaystyle Def(\Delta(1,(a_{0},a_{1},a_{2})))$
$\displaystyle=Def(1,(a_{0}-_{N}a_{2},a_{1}-_{N}a_{0},a_{2}-_{N}a_{1}))$
$\displaystyle=(1,(a_{0}-_{N}a_{2},\left\lfloor\frac{(a_{1}-_{N}a_{0})+_{N}(a_{2}-_{N}a_{1})}{2}\right\rfloor,$
$\displaystyle\hskip
142.26378pt\left\lceil\frac{(a_{1}-_{N}a_{0})+_{N}(a_{2}-_{N}a_{1})}{2}\right\rceil)$
$\displaystyle=(1,(a_{0}-_{N}a_{2},\left\lfloor\frac{a_{2}-_{N}a_{0}}{2}\right\rfloor,\left\lceil\frac{a_{2}-_{N}a_{0}}{2}\right\rceil)).$
(1.10)

Since the rightmost sides of (1.9) and (1.10) coincide, we see that the two
maps $Def\circ\Delta$ and $\Delta\circ Def$ give one and the same value at
$(1,(a_{0},a_{1},a_{2}))$. Since even if the marker is not equal to 1, a
similar proof can be given, this completes the proof. ∎

###  1.4 Finite dynamical systems $\mathbf{Ref}_{N}^{n}$ and
$\mathbf{Def}_{N}^{n}$

We introduce two finite dynamical systems employing the maps $Ref$ and $Def$.

###### Definition 1.5

(1) Let $\mathbf{Ref}_{N}^{n}$ denote the dynamical system
$(m\mathbf{R}_{N}^{n},Ref)$.
(2) Let $\mathbf{Def}_{N}^{n}$ denote the dynamical system
$(m\mathbf{D}_{N}^{n},Def)$.

The notion of quasi-smoothness is introduced through the periodicity of the
self map $Ref$. For any finite dynamical system $\mathbf{F}=(X,F)$, let
$Per(\mathbf{F})$ denote the set of periodic points:

$\displaystyle Per(\mathbf{F})=\\{x\in X|\mbox{ There exists a positive
integer }m\mbox{ such that }F^{m}(x)=x\\}.$

###### Definition 1.6

A marked rhythm $\mbox{\boldmath$A$}\in m\mathbf{R}_{N}^{n}$ is quasi-smooth
if $\mbox{\boldmath$A$}\in Per(\mathbf{Ref}_{N}^{n})$. A rhythm
$\mbox{\boldmath$a$}\in\mathbf{R}_{N}^{n}$ is quasi-smooth if there exists a
$k\in\mathbf{Z}_{n}$ such that $\iota_{k}(\mbox{\boldmath$a$})$ is quasi-
smooth as a marked rhythm, where $\iota_{k}:\mathbf{R}_{N}^{n}\rightarrow
m\mathbf{R}_{N}^{n}$ denotes an inclusion defined by
$\iota_{k}(\mbox{\boldmath$a$})=(k,\mbox{\boldmath$a$})\in
m\mathbf{R}_{N}^{n}$.

The main theorem in this article is summarized as follows:

A rhythm is smooth if and only if it is quasi-smooth.

Thus we obtain another iterative method of construction for smooth rhythms,
which is much simpler than the one employed in (Hazama 2022). Furthermore, as
the reader will see in Section four, the long-term behavior of the self map
$Ref$ is far more easier for us to understand than that of $Rav$.

##  2 Monotone invariant measure

We axiomatize the method of construction of quasi-smooth rhythms. This will
enable us to prove the validity not only of our iterative method, but also of
other possible methods which produce certain target objects with a desired
property in a given dynamical system.

###  2.1 Automorphism, invariance, and monotonicity

###### Definition 2.1

Let $\mathbf{F}=(X,F)$ be a finite dynamical system.
(1) A bijection $\varphi:X\rightarrow X$ is called an automorphism of
$\mathbf{F}$ if it commutes with $F$, namely the equality

$\displaystyle F\circ\varphi=\varphi\circ F$ (2.1)

holds. We denote by “$\sim_{\varphi}$” the equivalence relation on $X$
generated by $\varphi$, and put $\overline{X}=X/\sim_{\varphi}$, the quotient
space of $X$ by $\sim_{\varphi}$.
(2) A real valued function $\mu$ on $X$ is called an invariant measure on
$\mathbf{F}$ with respect to an automorphism $\varphi$ of $\mathbf{F}$, if

$\displaystyle\mu(\varphi(x))=\mu(x)$ (2.2)

holds for any $x\in X$.
(3) A real valued function $\mu$ on $X$ is called a monotone measure on
$\mathbf{F}$, if

$\displaystyle\mu(F(x))\geq\mu(x)$ (2.3)

holds for any $x\in X$.

Under the conditions (2.1) and (2.2), the self-map $F$ as well as the function
$\mu$ on $X$ descends to the quotient space $\overline{X}$. We record this
fact for later use.

###### Proposition 2.1

(1) There exists a unique self map $\overline{F}$ on the quotient
$\overline{X}$ such that the following diagram commutes:

$\displaystyle\begin{CD}X@>{F}>{}>X\\\ @V{\pi}V{}V@V{{\pi}}V{}V\\\
\overline{X}@>{\overline{F}}>{}>\overline{X}\\\ \end{CD}$ (2.4)

where $\pi:X\rightarrow\overline{X}$ denotes the quotient map.
(2) The function $\mu$ descends to a unique function $\overline{\mu}$ on the
quotient $\overline{X}$ such that the following diagram commutes:

$\displaystyle\begin{CD}X@>{\mu}>{}>\mathbf{R}\\\
@V{\pi}V{}V@V{{id_{\mathbf{R}}}}V{}V\\\
\overline{X}@>{\overline{\mu}}>{}>\mathbf{R}\\\ \end{CD}$ (2.5)

The following proposition relates the monotonicity of $\mu$ with that of
$\overline{\mu}$.

###### Proposition 2.2

Suppose that $\mu$ is an invariant measure on $\mathbf{F}$ with respect to an
automorphism $\varphi$ of $\mathbf{F}$. Let $\overline{F}$ and
$\overline{\mu}$ denote the induced maps defined as above. Then $\mu$ is a
monotone measure on $\mathbf{F}$ if and only if $\overline{\mu}$ is a monotone
measure on $\overline{\mathbf{F}}$.

###### Proof.

For any $\overline{x}\in\overline{X}$, take an arbitrary
$x\in\pi^{-1}(\\{\overline{x}\\})$. Then we have

$\displaystyle\overline{\mu}(\overline{F}(\overline{x}))$ $\displaystyle=$
$\displaystyle\overline{\mu}(\overline{F}(\pi(x)))$ $\displaystyle=$
$\displaystyle\overline{\mu}(\pi(F(x)))\hskip 42.67912pt(\mbox{by (2.4)})$
$\displaystyle=$ $\displaystyle\mu(F(x))\hskip 56.9055pt(\mbox{by (2.5)})$

Since $\overline{\mu}(\overline{x})=\overline{\mu}(\pi(x))=\mu(x)$, the above
computation implies that the inequality
$\overline{\mu}(\overline{F}(\overline{x}))\geq\overline{\mu}(\overline{x})$
holds if and only if $\mu(F(x))\geq\mu(x)$. This completes the proof. ∎

For any dynamical system $(X,F)$, let $Per(X,F)$ denote the set of periodic
points, namely

$\displaystyle Per(X,F)=\\{x\in X|\mbox{ there exists a positive integer $m$
such that $F^{m}(x)=x$}\\}.$

A monotone measure provides us with a useful necessary condition for an
element $x\in X$ to be periodic.

###### Proposition 2.3

Let $\mu$ be a monotone measure on $(X,F)$. If $x\in Per(X,F)$, then we have

$\displaystyle\mu(F(x))=\mu(x).$ (2.6)

Furthermore, for any positive integer $k$, we have

$\displaystyle\mu(F^{k+1}(x))=\mu(F^{k}(x)).$ (2.7)

###### Proof.

Contrary to (2.6), suppose that we have

$\displaystyle\mu(F(x))>\mu(x).$

Let $m$ be a period of $x$ with respect to $F$. By the monotonicity (2.3) we
have the following chain of inequalities.

$\displaystyle\mu(x)<\mu(F(x))\leq\mu(F^{2}(x))\leq\mu(F^{m}(x))=\mu(x).$

This contradiction shows that we must have $\mu(F(x))=\mu(x)$. The equality
(2.7) follows from (2.6), since $F^{k}(x)$ is also a periodic point. This
completes the proof. ∎

In view of the importance of the condition (2.6) and (2.7) for the study of
the periodic points of a dynamical system, we introduce the following.

###### Definition 2.2

${\rm(1)}$ A point $x\in X$ is said to be $(\mu,F)$-invariant if the equality
$\mu(F(x))=\mu(x)$ holds.
${\rm(2)}$ A point $x\in X$ is said to be $\mu$-stable if it is
$(\mu,F^{\ell})$-invariant for any positive integer $\ell$. The set of
$\mu$-stable points is denoted by $St(\mu)$.

Through this terminology, Proposition 2.3 can be rephrased as follows:

###### Corollary 2.1

Let $\mu$ be a monotone measure on $(X,F)$. Then we have

$\displaystyle Per(X,F)\subset St(\mu).$

In other words, the $\mu$-stability is a necessary condition for the
periodicity.

Under the existence of an automorphism of a dynamical system $(X,F)$, we can
relate the periodic points of $(X,F)$ with the periodic points of the quotient
system $(\overline{X},\overline{F})$.

###### Proposition 2.4

Let $\varphi$ be an automorphism of a finite dynamical system $(X,F)$. For an
element $x\in X$ to belong $Per(X,F)$, it is necessary and sufficient that
$\pi(x)\in Per(\overline{X},\overline{F})$.

###### Proof.

(Necessity) Let $x\in Per(X,F)$ so that there exists a positive integer $m$
such that

$\displaystyle F^{m}(x)=x.$ (2.8)

The left hand side is mapped through $\pi$ to

$\displaystyle\pi(F^{m}(x))=\overline{F}^{m}(\pi(x))$ (2.9)

by the very definition of $\overline{F}$, namely by the equality $\pi\circ
F=\overline{F}\circ\pi$. Since $\pi$ maps the right hand side to $\pi(x)$,
this shows the equality $\overline{F}^{m}(\pi(x))=\pi(x)$. Hence $\pi(x)\in
Per(\overline{X},\overline{F})$.
(Sufficiency). Let $\pi(x)\in Per(\overline{X},\overline{F})$ so that there
exists a positive integer $m$ such that

$\displaystyle\overline{F}^{m}(\pi(x))=\pi(x).$ (2.10)

The left hand side is equal to $\overline{F}^{m}(\pi(x))=\pi(F^{m}(x))$ by
(2.8). Hence, together with (2.9), it implies that

$\displaystyle F^{m}(x)\sim_{\varphi}x.$

It follows from the definition of the relation $\sim_{\varphi}$ that there
exists a positive integer $k$ such that

$\displaystyle F^{m}(x)=\varphi^{k}(x).$ (2.11)

Since $\varphi$ commutes with $F$, applying $F^{m}$ on both sides, we see that

$\displaystyle
F^{2m}(x)=F^{m}(\varphi^{k}(x))=\varphi^{k}(F^{m}(x))=\varphi^{2k}(x),$

the last equality being a consequence of (2.10). Repeating the process, we see
that, for any positive integer $p$, we have

$\displaystyle F^{pm}(x)=\varphi^{pk}(x).$

Therefore, when $p$ is the order of the automorphism $\varphi$, we have

$\displaystyle F^{pm}(x)=\varphi^{pk}(x)=x,$

which shows that $x\in Per(X,F)$. This completes the proof. ∎

##  3 Monotone invariant measure $\mu_{m\mathbf{R}}$ on
$m\mathbf{Ref}_{N}^{n}$

We introduce a $\mathbf{Z}$-valued function $\mu_{m\mathbf{R}}$ on the
dynamical system $\mathbf{Ref}_{N}^{n}=(m\mathbf{R}_{N}^{n},Ref)$. Recall that
the self-map $mtr$ on $m\mathbf{R}_{N}^{n}$ is an automorphism of
$\mathbf{Ref}_{N}^{n}$ in the sense of Definition 2.1 (1), since we have the
equality (1.7). We will show that $\mu_{m\mathbf{R}}$ is a monotone invariant
measure with respect to the automorphism $mtr$.

###### Definition 3.1

For any $\mbox{\boldmath$a$}=(a_{0},\cdots,a_{n-1})\in\mathbf{R}_{N}^{n}$, let
$\mu_{\mathbf{R}}:\mathbf{R}_{N}^{n}\rightarrow\mathbf{Z}$ be defined by

$\displaystyle\mu_{\mathbf{R}}(\mbox{\boldmath$a$})=\prod_{i=0}^{n-1}(a_{i+_{n}1}-_{N}a_{i}),$
(3.1)

where the product on the right hand side is that of integers. For any marked
rhythm $(k,\mbox{\boldmath$a$})\in m\mathbf{R}_{N}^{n}$, we put

$\displaystyle\mu_{m\mathbf{R}}(k,\mbox{\boldmath$a$})=\mu_{\mathbf{R}}(\mbox{\boldmath$a$}).$
(3.2)

By the very definition, the function $\mu_{\mathbf{R}}$ satisfies the
invariance property

$\displaystyle\mu_{\mathbf{R}}\circ tr=\mu_{\mathbf{R}},$ (3.3)

since each factor on the right hand side of (3.1) depends only on the
difference of the neighboring entries. Furthermore, it follows from (3.2) and
(3.3) that

$\displaystyle\mu_{m\mathbf{R}}\circ mtr=\mu_{m\mathbf{R}}.$ (3.4)

Hence we obtain the following.

###### Proposition 3.1

The function $\mu_{m\mathbf{R}}$ on $m\mathbf{R}_{N}^{n}$ defines an invariant
measure on the dynamical system $\mathbf{Ref}_{N}^{n}$ with respect to the
automorphism $mtr$ of $\mathbf{Ref}_{N}^{n}$.

By the invariance (3.3), $\mu_{\mathbf{R}}$ descends to the quotient
$\mathbf{D}_{N}^{n}$. To be more specific, let
$\mu_{\mathbf{D}}:\mathbf{D}_{N}^{n}\rightarrow\mathbf{Z}$ be defined by

$\displaystyle\mu_{\mathbf{D}}(\mbox{\boldmath$d$})=\prod_{i=0}^{n-1}d_{i}$

for any
$\mbox{\boldmath$d$}=(d_{0},d_{1},\cdots,d_{n-1})\in\mathbf{D}_{N}^{n}$. Then
we have the commutative diagram:

$\displaystyle\begin{CD}\mathbf{R}_{N}^{n}@>{\mu_{\mathbf{R}}}>{}>\mathbf{Z}\\\
@V{d}V{}V@V{id_{\mathbf{Z}}}V{}V\\\
\mathbf{D}_{N}^{n}@>{\mu_{\mathbf{D}}}>{}>\mathbf{Z}\\\ \end{CD}$ (3.5)

We extend the domain $\mathbf{D}_{N}^{n}$ of $\mu_{\mathbf{D}}$ to
$m\mathbf{D}_{N}^{n}$ in the same way as above, and denote the extended
measure by $\mu_{m\mathbf{D}}$. Namely, we put

$\displaystyle\mu_{m\mathbf{D}}(k,\mbox{\boldmath$d$})=\mu_{\mathbf{D}}(\mbox{\boldmath$d$}).$

It follows that we have the commutative diagram.

$\displaystyle\begin{CD}m\mathbf{R}_{N}^{n}@>{\mu_{m\mathbf{R}}}>{}>\mathbf{Z}\\\
@V{\Delta}V{}V@V{id_{\mathbf{Z}}}V{}V\\\
m\mathbf{D}_{N}^{n}@>{\mu_{m\mathbf{D}}}>{}>\mathbf{Z}\\\ \end{CD}$ (3.6)

Thus $\mu_{m\mathbf{D}}$ is identified with $\overline{\mu_{m\mathbf{R}}}$,
the map on the quotient $m\mathbf{D}_{N}^{n}=m\mathbf{R}_{N}^{n}/\sim_{mtr}$
induce from $\mu_{m\mathbf{R}}$. Therefore the general result Proposition 2.2
gives us the following specific result.

###### Proposition 3.2

For any $\mbox{\boldmath$A$}\in m\mathbf{R}_{N}^{n}$, it belongs to
$Per(\mathbf{Ref})$ if and only if $\Delta(\mbox{\boldmath$A$})\in
Per(\mathbf{Def})$.

Furthermore we can show the monotonicity of the measures $\mu_{m\mathbf{R}}$
and $\mu_{m\mathbf{D}}$.

###### Proposition 3.3

${\rm(1)}$ For any $(k,\mbox{\boldmath$a$})\in m\mathbf{R}_{N}^{n}$, we have

$\displaystyle\mu_{m\mathbf{R}}(Ref(k,\mbox{\boldmath$a$}))\geq\mu_{m\mathbf{R}}(k,\mbox{\boldmath$a$}).$

${\rm(2)}$ For any $(k,\mbox{\boldmath$d$})\in m\mathbf{D}_{N}^{n}$, we have

$\displaystyle\mu_{m\mathbf{D}}(Def(k,\mbox{\boldmath$d$}))\geq\mu_{m\mathbf{D}}(k,\mbox{\boldmath$d$}).$
(3.7)

${\rm(3)}$ When $\mbox{\boldmath$d$}=(d_{0},\cdots,d_{n-1})$, the equality in
$(3.7)$ holds if and only if $|d_{k}-d_{k+_{n}1}|\leq 1$.

###### Proof.

By Proposition 2.2, the item (1) follows from the item (2). Hence we have only
to prove the items (2) and (3). Furthermore the deformation map changes only
an adjacent pair of the entries, we are reduced to showing the following:

###### Lemma 3.1

For any pair $(a,b)$ of positive integers, we have

$\displaystyle a\cdot
b\leq\left\lfloor\frac{a+b}{2}\right\rfloor\cdot\left\lceil\frac{a+b}{2}\right\rceil.$
(3.8)

The equality holds if and only if $|a-b|\leq 1$.

We may assume that $a\leq b$. When $a=b$, both sides of (3.8) are equal to
$a^{2}$, and hence the equality holds. When $b=a+1$, the right hand side of
(3.8) is equal to $a\cdot(a+1)$, and hence the equality holds too. Suppose
that $b>a+1$. In case $a\equiv b\pmod{2}$, the right hand side of (3.8) is
equal to $\displaystyle{\left(\frac{a+b}{2}\right)^{2}}$, and hence the
difference (RHS)-(LHS) becomes $((a+b)^{2}-4ab)/4=(a-b)^{2}/4\geq 0$, which
implies the validity of (3.8). In case $a\not\equiv b\pmod{2}$, let $a+b=2k+1$
for an integer $k$. Then the right hand side is equal to $k(k+1)$. On the
other hand, the left hand side becomes $a((2k+1)-a)$. Note that the quadratic
function $x((2k+1)-x)$ on $\mathbf{Z}$ takes the maximum value at $x=k,k+1$.
Since we are in the case that $b>a+1$, $a$ cannot equal to $k$ and hence the
product $ab=a((2k+1)-a)$ is smaller than $k(k+1)$. This completes the proof. ∎

Combining Proposition 3.1 and 3.3, we arrive at the following objective of
this section.

###### Proposition 3.4

(1) $\mu_{m\mathbf{R}}$ is a monotone invariant measure on the dynamical
system $\mathbf{Ref}_{N}^{n}$ with respect to the automorphism $mtr$ on
$\mathbf{Ref}_{N}^{n}$.
(2) $\mu_{m\mathbf{D}}$ is a monotone measure on the dynamical system
$\mathbf{Def}_{N}^{n}$.

##  4 Main theorem and its proof

In this section we prove our main theorem which characterizes quasi-smooth
rhythms in $m\mathbf{R}_{N}^{n}$. For our proof it is indispensable to focus
on the structure of the set $St(\mu_{m\mathbf{D}})$ of
$\mu_{m\mathbf{D}}$-stable points in the dynamical system
$\mathbf{Def}_{N}^{n}=(m\mathbf{D}_{N}^{n},Def)$. The following property is
fundamental.

###### Proposition 4.1

Assume that $\mbox{\boldmath$D$}=(k,\mbox{\boldmath$d$})$ is
$(\mu_{m\mathbf{D}},Def)$-invariant, where
$\mbox{\boldmath$d$}=(d_{0},\cdots,d_{n-1})$. Then we have
$(1)$ $|d_{k}-d_{k+_{n}1}|\leq 1$.
$(2.1)$ When $d_{k+_{n}1}=d_{k}+1$, we have

$\displaystyle Def(k,\mbox{\boldmath$d$})=(k+_{n}1,\mbox{\boldmath$d$}).$
(4.1)

$(2.2)$ When $d_{k+_{n}1}=d_{k}-1$ or $d_{k+_{n}1}=d_{k}$, we have

$\displaystyle
Def(k,\mbox{\boldmath$d$})=(k+_{n}1,ad_{k}(\mbox{\boldmath$d$})),$ (4.2)

where $ad_{k}:\mathbf{D}_{N}^{n}\rightarrow\mathbf{D}_{N}^{n}$ denotes the map
which transposes the $k$-th and the $(k+_{n}1)$-th coordinates.

###### Proof.

Let $Def(k,\mbox{\boldmath$d$})=(k+_{n}1,\mbox{\boldmath$e$})$ with
$\mbox{\boldmath$e$}=(e_{0},\cdots,e_{n-1})$. The item (1) is already proved
in Proposition 3.3 (3). When $d_{k+_{n}1}=d_{k}+1$, we have
$(e_{k},e_{k+_{n}1})=(d_{k},d_{k+_{n}1})$ by the definition of the map $Def$,
and hence (4.1) follows. When $d_{k+_{n}1}=d_{k}-1$, we have
$e_{k}=\lfloor\frac{2d_{k}-1}{2}\rfloor=d_{k}-1=d_{k+_{n}1}$, and
$e_{k+_{n}1}=\lceil\frac{2d_{k}-1}{2}\rceil=d_{k}$, and hence the equality
(4.2) holds. Furthermore when $d_{k+_{n}1}=d_{k}$, the equality (4.2) holds
trivially. This completes the proof. ∎

For ease of description, we introduce the following notation.

###### Definition 4.1

Let $\mbox{\boldmath$D$}=(k,\mbox{\boldmath$d$})\in\mathbf{D}_{N}^{n}$ with
$\mbox{\boldmath$d$}=(d_{0},\cdots,d_{n-1})$.
(1) For any positive integer $\ell$, we express
$Def^{\ell}(\mbox{\boldmath$D$})$ as

$\displaystyle\mbox{\boldmath$D$}^{(\ell)}=(k+_{n}\ell,\mbox{\boldmath$d$}^{(\ell)})=(k+_{n}\ell,(d_{0}^{(\ell)},\cdots,d_{n-1}^{(\ell)})).$

(2) The multiset of all the entries of $d$ is called the content of $D$, and
is denoted by $Cont(\mbox{\boldmath$D$})$.

As a direct consequence of Proposition 4.1, we obtain the following.

###### Corollary 4.1

If $\mbox{\boldmath$D$}\in St(\mu_{m\mathbf{D}})$, then for any positive
integer $\ell$ we have

$\displaystyle Cont(\mbox{\boldmath$D$}^{(\ell)})=Cont(\mbox{\boldmath$D$}).$

Now we can prove the following result which plays a crucial role for our
characterization of the quasi-smoothness. Recall that the difference of the
maximum and the minimum of the entries of
$\mbox{\boldmath$d$}\in\mathbf{D}_{N}^{n}$ is called the width of $d$, and is
denoted by $w(\mbox{\boldmath$d$})$. For any
$\mbox{\boldmath$D$}=(k,\mbox{\boldmath$d$})\in m\mathbf{D}_{N}^{n}$ we set
$w(\mbox{\boldmath$D$})=w(\mbox{\boldmath$d$})$, and call it the width of $D$
too.

###### Proposition 4.2

If $\mbox{\boldmath$D$}\in St(\mu_{m\mathbf{D}})$, then we have

$\displaystyle w(\mbox{\boldmath$D$})\leq 1.$

###### Proof.

We may assume that the marker of $D$ is equal to 0, and hence we set
$\mbox{\boldmath$D$}=(0,\mbox{\boldmath$d$})$ with
$\mbox{\boldmath$d$}=(d_{0},\cdots,d_{n-1})$. Let
$M=\max\\{d_{i}|i\in\mathbf{Z}_{n}\\}$ and let $i_{m}=\min\\{i|d_{i}=M\\}$,
the first index of the entry which attains the maximum $M$. Then it follows
from Proposition 4.1 (2.2) that, for any $\ell\geq i_{m}$, the marked entry of
$\mbox{\boldmath$D$}^{(\ell)}$ is equal to $M$, and the subsequent entry
belongs to $\\{M,M-1\\}$. Applying $Def$ repeatedly, the same argument shows
that $Cont(\mbox{\boldmath$D$}^{(i_{m}+n-1)})\subset\\{M,M-1\\}$. Therefore it
follows from Corollary 4.1 that $Cont(\mbox{\boldmath$D$})\subset\\{M,M-1\\}$,
and hence we have $w(\mbox{\boldmath$D$})\leq 1$. This completes the proof. ∎

In order to characterize the set $Per(\mathbf{Def}_{N}^{n})$ in
$m\mathbf{D}_{N}^{n}$, we need to specify the value which the marked entry
takes.

###### Definition 4.2

For any $\mbox{\boldmath$D$}\in m\mathbf{D}_{N}^{n}$, we denote the maximum of
its difference part by $\max(\mbox{\boldmath$D$})$. We say that $D$ is max-
marked, if the value of the marked entry is equal to
$\max(\mbox{\boldmath$D$})$.

The following lemma enables us to characterize the subset
$Per(\mathbf{Def}_{N}^{n})$ in $m\mathbf{D}_{N}^{n}$.

###### Lemma 4.1

Suppose that $\mbox{\boldmath$D$}\in m\mathbf{D}_{N}^{n}$ has width $\leq 1$.
$(1)$ If it is max-marked, then for any positive integer $\ell$, the $\ell$-th
deformation $\mbox{\boldmath$D$}^{(\ell)}$ is max-marked.
$(2)$ If it is not max-marked, it is not periodic under the deformation map
$Def$.

###### Proof.

(1) By the assumption that $D$ is max-marked, the alternative (2.2) in
Proposition 4.1 holds, and as a result $\mbox{\boldmath$D$}^{(1)}$ is max-
marked. Repeating this argument we see that every
$\mbox{\boldmath$D$}^{(\ell)}$ is max-marked for any positive integer $\ell$.
(2) Let $\mbox{\boldmath$D$}=(k,\mbox{\boldmath$d$})$ with
$\mbox{\boldmath$d$}=(d_{0},\cdots,d_{n-1})$, and
$M=max(\mbox{\boldmath$D$})$. It follows from the assumption that $d_{k}=M-1$.
Hence there exists an index $\ell\in\mathbf{Z}_{n}$ such that $d_{\ell}=M$ and
that $d_{m}=M-1$ for every $m\in[k,\ell-_{n}1]$. Then it follows from
Proposition 4.1 that $\mbox{\boldmath$D$}^{(\ell-_{n}k)}$ is max-marked. This
implies by (1) that $\mbox{\boldmath$D$}^{m)}$ is max-marked for any
$m\geq\ell-_{n}k$, and hence the original $D$, which is not max-marked, cannot
belong to a cycle. This completes the proof. ∎

###### Theorem 4.1

${\rm(A)}$ A marked difference $\mbox{\boldmath$D$}\in m\mathbf{D}_{N}^{n}$
belongs to $Per(\mathbf{Def}_{N}^{n})$ if and only if it satisfies the
following two conditions.
$(A.1)$ $w(\mbox{\boldmath$D$})\leq 1$.
$(A.2)$ It is max-marked.
${\rm(B)}$ A marked rhythm $\mbox{\boldmath$A$}\in m\mathbf{R}_{N}^{n}$ is
quasi-smooth if and only if its difference $\Delta(\mbox{\boldmath$A$})$
satisfies the above two conditions $(A.1)$ and $(A.2)$.

###### Proof.

(A) Only-if-part: Suppose that $\mbox{\boldmath$D$}\in
Per(\mathbf{Def}_{N}^{n})$. Since $\mu_{m\mathbf{D}}$ is a monotone measure by
Proposition 3.3 (2), it follows from Corollary 2.1 that $D$ is
$\mu_{m\mathbf{D}}$-stable, which implies by Proposition 4.2 that
$w(\mbox{\boldmath$D$})\leq 1$. Hence the icondition (A.1) holds. The validity
of the condition (A.2) is a direct consequence of Lemma 4.1 (2).
If-part: Let $\mbox{\boldmath$D$}=(k,\mbox{\boldmath$d$})$ with
$\mbox{\boldmath$d$}=(d_{0},\cdots,d_{n-1})$. Under the conditions (A.1) and
(A.2), we have

$\displaystyle\mbox{\boldmath$D$}^{(1)}=(k+_{n}1,ad_{k}(\mbox{\boldmath$d$}))$

by Proposition 4.1 (2.2). Since $\mbox{\boldmath$D$}^{(1)}$ is max-marked
itself, we have

$\displaystyle\mbox{\boldmath$D$}^{(2)}=(k+_{n}2,ad_{k+_{n}1}ad_{k}(\mbox{\boldmath$d$})).$

By repeating these processes, we see that

$\displaystyle\mbox{\boldmath$D$}^{(n-1)}=(k+_{n}(n-1),ad_{k+_{n}(n-2)}\cdots
ad_{k+_{n}1}ad_{k}(\mbox{\boldmath$d$})).$

Notice here that, as the product of adjacent transpositions on
$\mathbf{Z}_{n}$, we have the equality

$\displaystyle(n-2\hskip 5.69054ptn-1)\cdots(1\hskip 5.69054pt2)(0\hskip
5.69054pt1)=(n-1\hskip 5.69054ptn-2\hskip 5.69054pt\cdots\hskip
5.69054pt2\hskip 5.69054pt1\hskip 5.69054pt0),$

and a slightly general one

$\displaystyle(k+_{n}(n-2)\hskip 5.69054ptk+_{n}(n-1))\cdots(k+_{n}1\hskip
5.69054ptk+_{n}2)(k\hskip 5.69054ptk+_{n}1)$ $\displaystyle=(k+_{n}(n-1)\hskip
5.69054ptk+_{n}(n-2)\hskip 5.69054pt\cdots\hskip 5.69054ptk+_{n}2\hskip
5.69054ptk+_{n}1\hskip 5.69054ptk)$ $\displaystyle=(n-1\hskip
5.69054ptn-2\hskip 5.69054pt\cdots\hskip 5.69054pt2\hskip 5.69054pt1\hskip
5.69054pt0),$

the last equality coming from the fact that both of the last two cyclic
permutations map $m$ to $m-_{n}1$ for any $m\in\mathbf{Z}_{n}$. Since the
cycle $(n-1\hskip 5.69054ptn-2\hskip 5.69054pt\cdots\hskip 5.69054pt2\hskip
5.69054pt1\hskip 5.69054pt0)$ is of order $n$, we see that

$\displaystyle\mbox{\boldmath$D$}^{(n(n-1))}=\mbox{\boldmath$D$},$

and hence $\mbox{\boldmath$D$}\in Per(\mathbf{Def}_{N}^{n})$.
(B) The assertion is a direct consequence of Proposition 3.2 which establishes
the equivalence of the periodicity of $A$ and that of
$\Delta(\mbox{\boldmath$A$})$. This completes the proof. ∎

Thus we arrive at the main theorem of this paper.

###### Theorem 4.2

A rhythm is smooth if and only if it is quasi-smooth. In particular, for any
rhythm $\mbox{\boldmath$a$}\in\mathbf{R}_{N}^{n}$, there exists a nonnegative
integer $\ell$ such that $p_{2}(Ref^{\ell}(\iota_{0}(\mbox{\boldmath$a$})))$
is smooth, where $p_{2}:m\mathbf{R}_{N}^{n}\rightarrow\mathbf{R}_{N}^{n}$
denotes the projection onto the second factor.

###### Proof.

Recall that, by the main theorem of (Hazama 2022), a rhythm $a$ is smooth if
and only if

$\displaystyle w(d(\mbox{\boldmath$a$}))\leq 1.$ (4.3)

Therefore the if-part follows from Theorem 4.1 (B). As for the only-if part,
assume that $a$ is smooth. Let
$\mbox{\boldmath$d$}=d(\mbox{\boldmath$a$})=(d_{0},\cdots,d_{n-1})$, and
choose an index $k\in\mathbf{Z}_{n}$ such that
$d_{k}=\max(\mbox{\boldmath$d$})$. Then
$\mbox{\boldmath$D$}=\iota_{k}(\mbox{\boldmath$d$})\in m\mathbf{D}_{N}^{n}$ is
max-marked and satisfies the condition $w(\mbox{\boldmath$D$})\leq 1$ by
(4.3). Therefore Theorem 4.1 (B) implies that $\iota_{k}(\mbox{\boldmath$a$})$
is quasi-smooth, and hence $a$ is quasi-smooth.
∎

References

Clough J, Douthett J, Krantz R (2000), Maximally Even Sets: A Discovery in
Mathematical Music Theory is Found to Apply in Physics. In: Reza S(ed)
Bridges: Mathematical Connections in Art, Music, and Science, Conference
Proceedings 2000. Central Plain Book Manufacturing, Kansas, pp 193-200.

Demaine ED, Gomez-Martin F, Meijer H, Rappaport D, Taslakian P, Toussaint GT,
Winograd T, Wood DR (2009), The distance geometry of music. Computational
Geometry 42: 429-454.

Hazama F (2022) Iterative method of construction for smooth rhythms. Journal
of Mathematics and Music 16:216-235.

Rota GC (2001) The adventures of measure theory. In Crapo H, Senato D(eds)
Algebraic Combinatorics and Computer Science. Springer-Verlag Italia, pp
27-39.

Toussaint. G (2013) The Geometry of Musical Rhythm: What Makes a “Good” Rhythm
Good? CRC Press, New York.


# Differential Subordinations for functions with positive real part using
Admissibility conditions

Meghna Sharma  Department of Mathematics, University of Delhi, Delhi–110 007,
India<EMAIL_ADDRESS>,  Sushil Kumar  Bharati Vidyapeeth’s college
of Engineering, Delhi-110063, India<EMAIL_ADDRESS>and  Naveen
Kumar Jain  Department of Mathematics, Aryabhatta College, Delhi-110021,India
<EMAIL_ADDRESS>
###### Abstract.

Some sufficient conditions on certain constants which are involved in some
first, second and third order differential subordinations associated with
certain functions with positive real part like modified Sigmoid function,
exponential function and Janowski function are obtained so that the analytic
function $p$ normalized by the condition $p(0)=1$, is subordinate to Janowski
function. The admissibility conditions for Janowski function are used as a
tool in the proof of the results. As application, several sufficient
conditions are also computed for Janowski starlikeness.

###### Key words and phrases:

Differential subordination; admissibilty condition; modified sigmoid function;
Janowski function; Exponential function

###### 2010 Mathematics Subject Classification:

30C45, 30C50

The first author is supported by Junior Research Fellowship from Council of
Scientific and Industrial Research, New Delhi, Ref. No.:1753/(CSIR-UGC NET
JUNE, 2018).

##  1\. Introduction

Let $\mathcal{A}$ denote the class of all analytic functions $f$ on the open
unit disc $\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$ normalized by the
conditions $f(0)=0$ and $f^{\prime}(0)=1$. An analytic function defined on
$\mathbb{D}$ is univalent if $f$ is one-to-one in $\mathcal{A}$. Let
$\mathcal{S}\subset\mathcal{A}$ be the class of all univalent functions.
Denote the class of all analytic functions $f$ having Taylor series expansion
$f(z)=a+a_{n}z^{n}+a_{n+1}z^{n+1}+...$, for some $a\in\mathbb{C}$ and fixed
integer $n$ by $\mathcal{H}[a,n]$. Let $f$ and $g$ be analytic in
$\mathbb{D}$. The function $f$ is subordinate to $g$, and write $f\prec g$, if
there exists an analytic function $w:\mathbb{D}\rightarrow\mathbb{D}$ with
$|w(z)|\leq|z|$ such that $f(z)=g(w(z))$ for all $z\in\mathbb{D}$. In
particular, if $g\in\mathscr{U}$ then, $f\prec g$ if and only if $f(0)=g(0)$
and $f(\mathbb{D})\subseteq g(\mathbb{D})$.

Let $\mathcal{P}$ be the class of functions with positive real part of the
form $p(z)=1+c_{1}z+c_{2}z+\cdots$ over $\mathbb{D}$. Let $A$ and $B$ be
arbitrary fixed numbers which are satisfying the inequality $-1\leq B<A\leq
1$, then the analytic function $p\in\mathcal{P}$ is known as the Janowski
functions associated with right half plane if it satisfies the subordination
relation $p(z)\prec(1+Az)/(1+Bz)$ for all $z\in\mathbb{D}$. The class of such
functions is denoted by $\mathcal{P}[A,B]$. Let $\mathcal{S}^{*}[A,\,B]$ be
the class of Janowski starlike functions $f\in\mathcal{A}$ such that
${zf^{\prime}(z)}/{f(z)}\in\mathcal{P}[A,\,B]$ for $z\in\mathbb{D},$
introduced by Janowski [17]. Let $\mathcal{S}^{*}[A,B]$ be the class of the
functions $f\in\mathcal{A}$ such that the quantity $zf^{\prime}(z)/f(z)$ lies
in the region $\Delta=\\{w\in\mathbb{C}:|(w-1)/(A-Bw)|<1\\}$. As a special
case, we note that $S^{*}[1-2\alpha,-1]=\mathcal{S}^{*}(\alpha)$ that contains
starlike functions of order $\alpha$ [16, 31]. In 2015, authors [24]
introduced the class $\mathcal{S}^{*}_{e}$ which contains the functions
$f\in\mathcal{A}$ satisfying the subordination relation
${zf^{\prime}(z)}/{f(z)}\prec e^{z}$ for all $z\in\mathbb{D}$. In addition, if
$f\in\mathcal{S}^{*}_{e}$, then the quantity $zf^{\prime}(z)/f(z)$ lies in the
domain $\\{w\in\mathbb{C}\colon|\log w|<1\\}$. Recently, Goel and Kumar [14]
introduced and studied the class $\mathcal{S}^{*}_{SG}$ which contains
starlike functions associated with modified sigmoid function
$\phi_{SG}(z):=2/(1+e^{-z})$ and satisfy the subordination relation
$zf^{\prime}(z)/f(z)\prec\phi_{SG}$ for all $z\in\mathbb{D}$. In similar way,
if the function $f\in\mathcal{S}^{*}_{SG}$, then the quantity
$zf^{\prime}(z)/f(z)$ lies in the domain $\\{w\in\mathbb{C}\colon|\log
w/(2-w)|<1\\}$. For details, see[11, 2, 3].

Goluzin [15] studied initially the first order differential subordination
$zp^{\prime}(z)\prec zq^{\prime}(z)$, whenever $zq^{\prime}(z)$ is convex, the
subordination $p\prec q$ holds and the function $q$ is the best dominant.
After this basic result, many authors established several generalizations of
differential subordination implications. In 1981, an article titled
”Differential subordination and univalent functions” by Miller and Mocanu [25]
commenced the study of differential subordination as a generalized version of
differential inequalities. For more details, see [8, 22, 29, 26]. In 1989,
Nunokawa _et al._ [28] studied the first order differential subordination and
proved that $1+zp^{\prime}(z)\prec 1+z$ implies $p(z)\prec 1+z$. They used
this result to provide a criterion so that a normalized analytic function is
univalent in $\mathbb{D}$. Then, Ali _et al._ [4] generalized this result and
proved that $p(z)$ is subordinate to the Janowski function whenever $1+\beta
zp^{\prime}(z)/p^{j}(z)\prec(1+Dz)/(1+Ez)$ for $j=0,1,2$. Here,
$A,B,D,E\in[-1,1]$. Further, Ali _et al._ [5] determined the estimate on
$\beta$ so that the subordination $1+\beta zp^{\prime}(z)/p^{j}(z)$ is
subordinate to the function $\sqrt{1+z}$, $(j=0,1,2)$ which implies that
$p(z)$ is subordinate to $\sqrt{1+z}$. Later, Kumar _et al._ [21] computed a
bound on $\beta$ so that $p(z)\prec\sqrt{1+z}$, whenever $1+\beta
zp^{\prime}(z)/p^{j}(z)\prec(1+Dz)/(1+Ez)$, $(j=0,1,2)$ with $|D|\leq 1$ and
$-1<E<1$. Some of these results were not sharp. Also, it was difficult to
establish analogous results for certain functions with positive real parts
such as $\phi_{0}(z):=1+(z/k)((k+z)/(k-z))$, $(k=1+\sqrt{2})$,
$\phi_{\sin}(z):=1+\sin z$, $\mathcal{Q}(z):=e^{e^{z}-1}$ by the approach used
in above discussed research work. Later in 2018, Kumar and Ravichandran [19]
used some different approach and were able to established best possible bounds
on $\beta$ so that $1+\beta zp^{\prime}(z)/p^{j}(z)$ is subordinate to
$\sqrt{1+z},(1+Az)/(1+Bz)$ which implies that $p(z)\prec e^{z},(1+Az)/(1+Bz)$.
In 2018, Ahuja _et al._ [1] also obtained sharp subordination implications
results for the functions associated with lemniscate of Bernoulli. For recent
work related to first order differential subordinations, reader may refer [9,
10, 12, 13, 32, 33].

In 2018, Madaan _et al._ [23] established first and second order differential
subordinations associated with the lemniscate of Bernoulli using admissibility
technique. Further, Anand _et al._ [7] also studied the generalized first
order differential subordination for the Janowski functions. In 2019, Dorina
Răducanu [30] established second order differential subordination implications
associated with generalized Mittag-Leffler function. For related work, readers
may see [20, 27, 18].

Motivated by the aforesaid work, using admissibility conditions for Janowski
functions, we determine certain conditions on $\beta$, $\gamma$, $A$ and $B$
where $-1\leq B<A\leq 1$ so that $p$ belongs to the class $\mathcal{P}[A,B]$
whenever $1+\beta{zp^{\prime}(z)}/{p^{k}(z)}$,
$1+\beta{(zp^{\prime}(z))^{2}}/{p^{k}(z)}$, $(1-\alpha)p(z)+\alpha
p^{2}(z)+\beta z{p^{\prime}(z)}/{p^{k}(z)}$ $(\alpha\in[0,\,1])$,
$\left({1}/{p(z)}\right)-\beta z{p^{\prime}(z)}/{p^{k}(z)}$,
$p(z)+{zp^{\prime}(z)}/{(\beta p(z)+\gamma)^{k}}$ $(\gamma>0)$, $1+\gamma
zp^{\prime}(z)+\beta z^{2}p^{\prime\prime}(z)$ and $p(z)+\gamma
zp^{\prime}(z)+\beta z^{2}p^{\prime\prime}(z)$ are subordinate to some
functions with positive real part like $e^{z}$, $2/(1+e^{-z})$ and
$(1+Az)/(1+Bz)$, where $k$ is a positive integer. Certain implications of
these results are also discussed which gives sufficient conditions for an
analytic function $f$ to be in the class $\mathcal{S}^{*}[A,B]$.

##  2\. The Admissibility Condition

This section provides some basic facts related to admissibility conditions
associated with Janowski function that will be needed in the proving our main
results.

######  Definition 2.1.

Let $\psi(r,s,t;z):\mathbb{C}^{3}\times\mathbb{D}\to\mathbb{D}$ be analytic
and $h\in\mathcal{S}$. Then a function $p\in\mathcal{A}$, satisfying following
differential subordination relation

$\psi(p(z),zp^{\prime}(z),z^{2}p^{\prime\prime}(z);z)\prec h(z)$

is called its _solution_.

Let $\mathfrak{Q}$ denote the class of all functions $q\in\mathcal{S}$ defined
on $\overline{\mathbb{D}}\setminus\mathbf{E}(q)$, where
$\mathbf{E}=\\{\zeta\in\partial\mathbb{D}:\lim\limits_{z\rightarrow\zeta}q(z)=\infty\\}$
such that $q^{\prime}(\zeta)\neq 0$ for
$\zeta\in\partial\mathbb{D}\setminus\mathbf{E}(q)$.

######  Definition 2.2.

Let $\Omega\subset\mathbb{C},q\in\mathfrak{Q}$ and $n\geq 1$. Consider the
class of admissible functions $\Psi_{n}[\Omega,q]$, consists of those
functions $\psi:\mathbb{C}^{3}\times\mathbb{D}\to\mathbb{C}$ which satisfy the
admissibility condition:

$\psi(r,s,t;z)\notin\Omega$

whenever

$r=q(\zeta),s=m\zeta
q^{\prime}(\zeta)\,\,\text{and}\,\,\operatorname{Re}\left(\frac{t}{s}+1\right)\geq
m\operatorname{Re}\left(\frac{\zeta
q^{\prime\prime}(\zeta)}{q^{\prime}(\zeta)}+1\right)$

for $z\in\mathbb{D}$, $\zeta\in\partial\mathbb{D}\setminus\mathbf{E}(q)$ and
$m\geq n\geq 1$.

We write the class $\Psi_{1}[\Omega,q]$ as $\Psi[\Omega,q]$.

######  Theorem 2.3.

[25, Theorem 2.3b, p.28] Let the function $\psi\in\Psi_{n}[\Omega,q]$ and
$q(0)=a$. If $p\in\mathcal{H}[a,n]$, then

$\psi(p(z),zp^{\prime}(z),z^{2}p^{\prime\prime}(z);z)\in\Omega\implies
p(z)\prec q(z).$ (2.1)

Let $\Omega$ be a simply connected domain which is not the entire complex
plane, then there exists a conformal mapping $h$ from $\mathbb{D}$ onto
$\Omega$ with $h(0)=\psi(a,0,0;0)$. Therefore, if $p\in\mathcal{H}[a,n]$,
equation (2.1) can be written as

$\psi(p(z),zp^{\prime}(z),z^{2}p^{\prime\prime}(z);z)\prec h(z)\implies
p(z)\prec q(z).$ (2.2)

The univalent function $q$ is called _dominant of the solutions_ of the
differential subordination (2.2). The function $\tilde{q}$ is called the _best
dominant_ of (2.2) if $\tilde{q}\prec q$ for all dominants of (2.2).

Consider the function $q(z)=(1+Az)/(1+Bz)$ for $-1\leq A<B\leq 1$. Denote the
class $\Psi_{n}[\Omega,(1+Az)/(1+Bz)]$ by $\Psi_{n}[\Omega;A,B]$. Therefore,
the admissibility conditions for the function $q$ are given as follows:

######  Theorem 2.4.

[7] Let the function $p\in\mathcal{H}[1,n]$ such that $p(z)\not\equiv 1$ and
$n\geq 1$ and $\Omega$ be a subset of $\mathbb{C}$. The class
$\Psi_{n}[\Omega;A,B]$ is defined as the class of all those functions
$\psi:\mathbb{C}^{3}\times\mathbb{D}\rightarrow\mathbb{C}$ such that

$\psi(r,s,t;z)\not\in\Omega\quad\text{whenever}\quad(r,s,t;z)\in\operatorname{Dom}\psi\quad\text{and}$
$r=q(\zeta)=\frac{1+Ae^{i\theta}}{1+Be^{i\theta}},s=m\zeta
q^{\prime}(\zeta)=\frac{m(A-B)e^{i\theta}}{(1+Be^{i\theta})^{2}}\text{ and
}\operatorname{Re}\left(\frac{t}{s}+1\right)\geq\frac{m(1-B^{2})}{1+B^{2}+2B\cos\theta}$

for $z\in\mathbb{D},\theta\in(0,2\pi)$ and $m\geq 1$.

On taking $\psi\in\Psi_{n}[\Omega;A,B]$ in Theorem (2.4), we have

######  Corollary 2.5.

If $(p(z),zp^{\prime}(z),z^{2}p^{\prime\prime}(z);z)\in\operatorname{Dom}\psi$
and $\psi(p(z),zp^{\prime}(z),z^{2}p^{\prime\prime}(z);z)\in\Omega$ for
$z\in\mathbb{D}$, then $p$ belongs to $\mathcal{P}[A,B]$.

For $\zeta=e^{i\theta}$, where $\theta\in[0,2\pi)$, let us consider

$|q(\zeta)|=\sqrt{\frac{1+A^{2}+2A\cos\theta}{1+B^{2}+2B\cos\theta}}:=k(\theta)$
(2.3)

whose minimum value is $\frac{1-A}{1-B}$, attained at $\theta=\pi$. Also,
observe that

$|q^{\prime}(\zeta)|=\frac{A-B}{1+B^{2}+2B\cos\theta}:=d(\theta)$ (2.4)

and the minimum value of $d(\theta)$ is $d(0)=\frac{A-B}{1+B^{2}+2B}$ for
$B>0$ and $d(\pi)=\frac{A-B}{1+B^{2}-2B}$ for $B<0$. Note that

$\operatorname{Re}\left(\frac{\zeta
q^{\prime\prime}(\zeta)}{q^{\prime}(\zeta)}\right)=\frac{-2B(B+\cos\theta)}{1+B^{2}+2B\cos\theta}:=g(\theta)$
(2.5)

and the minimum value of $g(\theta)$ is $g(0)=\frac{-2B(B+1)}{1+B^{2}+2B}$ for
$B>0$ and $g(\pi)=\frac{-2B(B-1)}{1+B^{2}-2B}$ for $B<0$.

Using above values, we get the admissibility condition for third order
differential subordination as follows:

In order to the prove our main results, we will use the following lemmas
extensively.

######  Lemma 2.6.

[20] Let $z$ be a complex number. Then

$|\log(1+z)|\geq 1\quad\text{if and only if}\quad|z|\geq e-1.$

######  Lemma 2.7.

Consider the disc $\Delta_{\beta}=\\{w\in\mathbb{C}:|w|<\beta,0<\beta\leq
1\\}$. Then the inequality

$\left|\log\left(\frac{w}{2-w}\right)\right|\geq 1$

holds if and only if $|w|\geq\beta_{0}\approx 0.473519$, where $\beta_{0}$ is
the positive real root of the equation
$(e^{2}-1)\beta^{4}-2(e^{2}-4)\beta^{3}+4(e^{2}-6)\beta^{2}+32\beta-16=0$.

###### Proof.

For $\theta\in[0,2\pi]$, let $w=\beta e^{i\theta}$ be a boundary point of the
disc $\Delta_{\beta}$. Consider

$\displaystyle\left|\log\left(\frac{\beta e^{i\theta}}{2-\beta
e^{i\theta}}\right)\right|^{2}$
$\displaystyle=\left|\frac{1}{2}\log\left(\frac{4\beta^{2}+\beta^{4}-4\beta^{3}\cos\theta}{(4+\beta^{2}-4\beta\cos\theta)^{2}}\right)+i\arctan\left(\frac{2\beta\sin{\theta}}{2\beta\cos{\theta}-\beta^{2}}\right)\right|^{2}$
$\displaystyle=\left(\log\left(\frac{\sqrt{4\beta^{2}+\beta^{4}-2\beta^{3}\cos\theta}}{4+\beta^{2}-4\beta\cos\theta}\right)\right)^{2}+\left(\arctan\left(\frac{2\sin{\theta}}{2\cos{\theta}-\beta}\right)\right)^{2}$
$\displaystyle:=f(\beta,\theta)$

In the interval $[0,2\pi]$, the function $f(\beta,\theta)$ attains its
absolute minimum at $\theta=0$ and therefore, $f(\beta,\theta)\geq f(\beta,0)$
for all $\theta\in[0,2\pi]$. Thus, the inequality

$\left|\log\left(\frac{\beta e^{i\theta}}{2-\beta
e^{i\theta}}\right)\right|\geq 1$

holds if and only if

$f(\beta,0)\geq 1$

or equivalently,

$(e^{2}-1)\beta^{4}-2(e^{2}-4)\beta^{3}+4(e^{2}-6)\beta^{2}+32\beta-16\geq 0.$
(2.6)

Therefore, by Intermediate Value Theorem, the inequality(2.6) holds for
$\beta\geq\beta_{0}\approx 0.473519$ which is the positive real root of the
equation in (2.6). ∎

##  3\. First Order Differential Subordination

In order to prove first order differential subordination relations, we need
the following result due to Swati _et al._ [19], which is a specific case of
Theorem (2.4).

######  Theorem 3.1.

Let $p\in\mathcal{H}[1,n]$ such that $p(z)\not\equiv 1$ and $n\geq 1$. Let
$\Omega$ be a set in $\mathbb{C}$. The class $\Psi_{n}[\Omega;A,B]$ is defined
as the class of all those functions
$\psi:\mathbb{C}^{2}\times\mathbb{D}\to\mathbb{C}$ such that

$\psi(r,s;z)\not\in\Omega\quad\text{whenever}\quad(r,s;z)\in\operatorname{Dom}\psi\quad\text{and}$
$r=\frac{1+Ae^{i\theta}}{1+Be^{i\theta}}\quad\text{ and }\quad
s=\frac{m(A-B)e^{i\theta}}{(1+Be^{i\theta})^{2}}$ (3.1)

for $z\in\mathbb{D},\theta\in(0,2\pi)$ and $m\geq 1$.

Consequently, when $\psi\in\Psi_{n}[\Omega;A,B]$, the above theorem becomes:
Let $z\in\mathbb{D}$. If $(p(z),zp^{\prime}(z);z)\in\operatorname{Dom}\psi$
and $\psi(p(z),zp^{\prime}(z);z)\in\Omega$, then $p\in\mathcal{P}[A,B]$.

Using above theorem, we determine the conditions on $A$, $B$ and $\beta$ so
that the function $\psi(p(z),zp^{\prime}(z);z)$ is subordinate to Modified
Sigmoid function $\phi_{SG}$ and exponential function $e^{z}$ implies $p(z)$
is subordinate to $(1+Az)/(1+Bz)$.

######  Theorem 3.2.

Let $-1\leq B<A\leq 1$ and $k$ be a non-negative integer. Let $p$ be an
analytic function defined on $\mathbb{D}$, which satisfies $p(0)=1$ and
$\beta_{0}\approx 0.475319$. Then the following are sufficient for
$p\in\mathcal{P}[A,B]$.

  1. (a)

$1+\beta\frac{zp^{\prime}(z)}{p^{k}(z)}\prec\phi_{SG}$, where
$|\beta|\geq\begin{cases}\frac{\beta_{0}(1+|A|)^{k}(1+|B|)^{2-k}}{(A-B)},&\text{
when }0\leq k\leq 2\\\ \frac{\beta_{0}(1+|A|)^{k}}{(A-B)(1-|B|)^{k-2}},&\text{
when }k>2\end{cases}$.

  2. (b)

$1+\beta\frac{zp^{\prime}(z)}{p^{k}(z)}\prec e^{z}$, where
$|\beta|\geq\begin{cases}\frac{(e-1)(1+|A|)^{k}(1+|B|)^{2-k}}{(A-B)},&\text{
when }0\leq k\leq 2\\\ \frac{(e-1)(1+|A|)^{k}}{(A-B)(1-|B|)^{k-2}},&\text{
when }k>2\end{cases}$.

###### Proof.

  1. (a)

Let
$\Omega=\phi_{SG}(\mathbb{D})=\\{w\in\mathbb{C}:\left|\log\left(\frac{w}{2-w}\right)\right|<1\\}$.
Consider the analytic function
$\psi:\mathbb{C}\setminus\\{0\\}\times\mathbb{C}\times\mathbb{D}\rightarrow\mathbb{C}$
defined as

$\psi(r,s;z)=1+\beta\frac{s}{r^{k}}.$

In accordance with Theorem (3.1), $\psi\in\Psi[\Omega;A,B]$, if
$\psi(r,s;z)\notin\Omega$ where $r$ and $s$ are given in the equation (3.1).
Therefore, it is enough to show that the required subordination holds if

$\left|\log\left(\frac{\psi(r,s;z)}{2-\psi(r,s;z)}\right)\right|\geq 1.$ (3.2)

    1. (i)

When $0\leq k\leq 2$, let us consider

$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|1+\beta\frac{m(A-B)e^{i\theta}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2-k}}\right|$
$\displaystyle=\left|\frac{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2-k}+\beta
m(A-B)e^{i\theta}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2-k}}\right|$
$\displaystyle\geq\frac{|\beta|m(A-B)-|(1+Ae^{i\theta})^{k}||(1+Be^{i\theta})^{2-k}|}{|(1+Ae^{i\theta})^{k}||(1+Be^{i\theta})^{2-k}|}$
$\displaystyle\geq\frac{|\beta|m(A-B)-(1+|A|)^{k}(1+|B|)^{2-k}}{(1+|A|)^{k}(1+|B|)^{2-k}}$
$\displaystyle=:\phi(m).$

By First Derivative Test, $\phi(m)$ is an increasing function for $m\geq 1$.
This implies $\phi(m)\geq\phi(1)$ for all $m\geq 1$. Hence, the last
inequality reduces to

$|\psi(r,s;z)|\geq\phi(1)$

where

$|\phi(1)|=\frac{|\beta|(A-B)-(1+|A|)^{k}(1+|B|)^{2-k}}{(1+|A|)^{k}(1+|B|)^{2-k}}.$

Using Lemma (2.7), the inequality (3.2) is true if

$|\phi(1)|\geq\beta_{0}$

or

$\frac{|\beta|(A-B)-(1+|A|)^{k}(1+|B|)^{2-k}}{(1+|A|)^{k}(1+|B|)^{2-k}}\geq\beta_{0}.$

or

$|\beta|(A-B)-(1+|A|)^{k}(1+|B|)^{2-k}\geq\beta_{0}(1+|A|)^{k}(1+|B|)^{2-k},$

which shows that $\psi(r,s;z)\notin\Omega$ for
$|\beta|\geq\beta_{0}\frac{(1+|A|)^{k}(1+|B|)^{2-k}}{(A-B)}.$

    2. (ii)

When $k>2$, observe that

$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|1+\beta\frac{m(A-B)(1+Be^{i\theta})^{k-2}e^{i\theta}}{(1+Ae^{i\theta})^{k}}\right|$
$\displaystyle=\left|\frac{(1+Ae^{i\theta})^{k}+\beta
m(A-B)(1+Be^{i\theta})^{k-2}e^{i\theta}}{(1+Ae^{i\theta})^{k}}\right|$
$\displaystyle\geq\frac{|\beta|m(A-B)|(1+Be^{i\theta})^{k-2}|-|(1+Ae^{i\theta})^{k}|}{|(1+Ae^{i\theta})^{k}|}$
$\displaystyle\geq\frac{|\beta|m(A-B)(1-|B|)^{k-2}-(1+|A|)^{k}}{(1+|A|)^{k}}$
$\displaystyle=:\phi(m)$

As in the earlier case, note that $\phi(m)$ is an increasing function of $m$.
Hence, $\phi(m)\geq\phi(1)$ for all $m\geq 1$. By Lemma (2.7), (3.2) holds if

$|\psi(r,s;z)|\geq|\phi(1)|\geq\beta_{0}$

where

$\phi(1)=\frac{|\beta|(A-B)(1-|B|)^{k-2}-(1+|A|)^{k}}{(1+|A|)^{k}}.$

Therefore, for $|\beta|\geq\beta_{0}\frac{(1+|A|)^{k}}{(A-B)(1-|B|)^{k-2}}$,
$\psi\in\Psi[\Omega;A,B]$ and hence, $p\in\mathcal{P}[A,B]$.

  2. (b)

Consider the domain $\Omega=\\{w\in\mathbb{C}:|\log w|<1\\}$. Let
$\psi:\mathbb{C}\backslash\\{0\\}\times\mathbb{C}\times\mathbb{D}\rightarrow\mathbb{C}$
be defined by

$\psi(r,s;z)=1+\beta\frac{s}{r^{k}}.$

By Theorem (3.1), $\psi$ belongs to $\Psi[\Omega;A,B]$ if
$\psi(r,s;z)\notin\Omega$ for $z\in\mathbb{D}$. This implication holds if

$|\log(\psi(r,s;z))|\geq 1$ (3.3)

Since

$\displaystyle|\log(\psi(r,s;z))|$
$\displaystyle=\left|\log\left(1+\beta\frac{s}{r^{k}}\right)\right|$
$\displaystyle=\left|\log\left(1+\beta\frac{m(A-B)e^{i\theta}(1+Be^{i\theta})^{k}}{(1+Be^{i\theta})^{2}(1+Ae^{i\theta})^{k}}\right)\right|,$

by Lemma(2.6), inequality (3.3) holds if and only if

$\left|\frac{\beta
m(A-B)e^{i\theta}(1+Be^{i\theta})^{k}}{(1+Be^{i\theta})^{2}(1+Ae^{i\theta})^{k}}\right|\geq
e-1.$ (3.4)

    1. (i)

When $0\leq k\leq 2$ , consider

$\displaystyle\left|\frac{\beta
m(A-B)e^{i\theta}}{(1+Be^{i\theta})^{2-k}(1+Ae^{i\theta})^{k}}\right|$
$\displaystyle\geq\frac{|\beta|m(A-B)}{|(1+Be^{i\theta})^{2-k}||(1+Ae^{i\theta})^{k}|}$
$\displaystyle\geq\frac{|\beta|(A-B)}{(1+|B|)^{2-k}(1+|A|)^{k}}\quad(\because
m\geq 1)$ $\displaystyle\geq e-1$

if

$|\beta|\geq\frac{(e-1)(1+|A|)^{k}(1+|B|)^{2-k}}{(A-B)}.$

which shows that $\psi(r,s;z)\notin\Omega$ and hence the required result
holds.

    2. (ii)

When $k>2$, observe that

$\displaystyle\left|\frac{\beta
m(A-B)e^{i\theta}(1+Be^{i\theta})^{k-2}}{(1+Ae^{i\theta})^{k}}\right|$
$\displaystyle\geq\frac{|\beta|m(A-B)|(1+Be^{i\theta})^{k-2}|}{|(1+Ae^{i\theta})^{k}|}$
$\displaystyle\geq\frac{|\beta|(A-B)(1-|B|)^{k-2}}{(1+|A|)^{k}}\quad(\because
m\geq 1)$

Therefore, (3.4) holds if

$\frac{|\beta|(A-B)(1-|B|)^{k-2}}{(1+|A|)^{k}}\geq e-1$

or equivalently,

$|\beta|\geq\frac{(e-1)(1+|A|)^{k}}{(A-B)(1-|B|)^{k-2}}.$

which proves that $\psi(r,s;z)\notin\Omega$ and hence, $p\in\mathcal{P}[A,B]$.

∎

######  Corollary 3.3.

Let $f\in\mathcal{A}$ and $\beta_{0}\approx 0.473519$. Set
$\mathcal{G}(z):=\frac{f^{\prime}(z)}{f(z)}-z\left(\frac{f^{\prime}(z)}{f(z)}\right)^{2}+\frac{zf^{\prime\prime}(z)}{f(z)}$.
If one of the following subordination holds, then $f\in S^{*}[A,B].$

  1. (a)

$1+\beta z\mathcal{G}(z)\prec\phi_{SG}$ for
$|\beta|(A-B)\geq\beta_{0}(1+|B|)^{2}$,

  2. (b)

$1+\beta\left(\frac{f(z)}{f^{\prime}(z)}\right)\mathcal{G}(z)\prec\phi_{SG}$
for $|\beta|(A-B)\geq\beta_{0}(1+|A|)(1+|B|)$,

  3. (c)

$1+\frac{\beta}{z}\mathcal{G}(z)\prec\phi_{SG}$ for
$|\beta|(A-B)\geq\beta_{0}(1+|A|)^{2}$,

  4. (d)

$1+\beta z\mathcal{G}(z)\prec e^{z}$ for $|\beta|(A-B)\geq(e-1)(1+|B|)^{2}$,

  5. (e)

$1+\beta\left(\frac{f(z)}{f^{\prime}(z)}\right)\mathcal{G}(z)\prec e^{z}$ for
$|\beta|(A-B)\geq(e-1)(1+|A|)(1+|B|)$,

  6. (f)

$1+\frac{\beta}{z}\mathcal{G}(z)\prec e^{z}$ for
$|\beta|(A-B)\geq(e-1)(1+|A|)^{2}$.

######  Theorem 3.4.

Let $-1\leq B<A\leq 1$, $\beta_{0}\approx 0.473519$, $k$ be a non-negative
integer and $p\in\mathcal{A}$ such that $p(0)=1$. If any of the following
subordinations holds true, then $p(z)\in\mathcal{P}[A,B]$.

  1. (a)

$1+\beta\frac{(zp^{\prime}(z))^{2}}{p^{k}(z)}\prec\phi_{SG}$, where
$|\beta|\geq\begin{cases}\frac{\beta_{0}(1+|A|)^{k}(1+|B|)^{4-k}}{(A-B)^{2}},&\text{
when }0\leq k\leq 4\\\
\frac{\beta_{0}(1+|A|)^{k}}{(A-B)^{2}(1-|B|)^{k-4}},&\text{ when
}k>4\end{cases}$

  2. (b)

$1+\beta\frac{(zp^{\prime}(z))^{2}}{p^{k}(z)}\prec e^{z}$, where
$|\beta|\geq\begin{cases}\frac{(e-1)(1+|A|)^{k}(1+|B|)^{4-k}}{(A-B)^{2}},&\text{
when }0\leq k\leq 4\\\ \frac{(e-1)(1+|A|)^{k}}{(A-B)^{2}(1-|B|)^{k-4}},&\text{
when }k>4\end{cases}$

###### Proof.

  1. (a)

Consider $\Omega$ as in Theorem (3.2)(a). Define the analytic function
$\psi:\mathbb{C}\setminus\\{0\\}\times\mathbb{C}\times\mathbb{D}\to\mathbb{C}$
as

$\psi(r,s;z)=1+\beta\frac{s^{2}}{r^{k}}.$

Therefore, we have

$\psi(r,s;z)=1+\beta\frac{m^{2}(A-B)^{2}e^{2i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{4}}.$

Proceeding in the similar manner as in Theorem (3.2), we have the following
two cases.

    1. (i)

When $0\leq k\leq 4$, let us consider

$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|1+\beta\frac{m^{2}(A-B)^{2}e^{2i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{4}}\right|$
$\displaystyle=\left|\frac{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{4-k}+\beta
m^{2}(A-B)^{2}e^{2i\theta}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{4-k}}\right|$
$\displaystyle\geq\frac{|\beta|m^{2}(A-B)^{2}-|(1+Ae^{i\theta})^{k}||(1+Be^{i\theta})^{4-k}|}{|(1+Ae^{i\theta})^{k}||(1+Be^{i\theta})^{4-k}|}$
$\displaystyle\geq\frac{|\beta|m^{2}(A-B)^{2}-(1+|A|)^{k}(1+|B|)^{4-k}}{(1+|A|)^{k}(1+|B|)^{4-k}}$
$\displaystyle=:\phi(m)$

Simple observation shows that $\phi(m)$ in an increasing function for $m\geq
1$. The required subordination result holds if $\psi(r,s;z)\notin\Omega$. So,
using Lemma (2.7), it is concluded that $p\in\mathcal{P}[A,B]$ if
$|\beta|\geq\frac{\beta_{0}(1+|A|)^{k}(1+|B|)^{4-k}}{(A-B)^{2}}$.

    2. (ii)

When $k>4$, observe that

$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|1+\beta\frac{m^{2}(A-B)^{2}e^{2i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{4}}\right|$
$\displaystyle=\left|\frac{(1+Ae^{i\theta})^{k}+\beta
m^{2}(A-B)^{2}e^{2i\theta}(1+Be^{i\theta})^{k-4}}{(1+Ae^{i\theta})^{k}}\right|$
$\displaystyle\geq\frac{|\beta|m^{2}(A-B)^{2}|(1+Be^{i\theta})^{k-4}|-|(1+Ae^{i\theta})^{k}|}{|(1+Ae^{i\theta})^{k}|}$
$\displaystyle\geq\frac{|\beta|m^{2}(A-B)^{2}(1-|B|)^{k-4}-(1+|A|)^{k}}{(1+|A|)^{k}}$
$\displaystyle=:\phi(m)$

Noting that $\phi^{\prime}(m)>0$ for $m\geq 1$ and proceeding as in the part
(i), we get the desired subordination result.

  2. (b)

Let $\Omega=\\{w\in\mathbb{C}:|\log w|<1\\}$ be the domain. Let
$\psi:\mathbb{C}\backslash\\{0\\}\times\mathbb{C}\times\mathbb{D}\rightarrow\mathbb{C}$
be defined as

$\psi(r,s;z)=1+\beta\frac{s^{2}}{r^{k}}.$

On the similar lines of the proof of Theorem (3.2) and using Lemma (2.6), we
get the desired result if

$\left|\frac{\beta
m^{2}(A-B)^{2}e^{2i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{4}}\right|\geq
e-1.$ (3.5)

    1. (i)

When $0\leq k\leq 4$ , consider

$\displaystyle\left|\frac{\beta
m^{2}(A-B)^{2}e^{2i\theta}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{4-k}}\right|$
$\displaystyle\geq\frac{|\beta|m^{2}(A-B)^{2}}{|(1+Ae^{i\theta})^{k}||(1+Be^{i\theta})^{4-k}|}$
$\displaystyle\geq\frac{|\beta|(A-B)^{2}}{(1+|A|)^{k}(1+|B|)^{4-k}}\quad(\because
m\geq 1)$

Now

$\frac{|\beta|(A-B)^{2}}{(1+|A|)^{k}(1+|B|)^{4-k}}\geq e-1$

if and only if

$|\beta|\geq\frac{(e-1)(1+|A|)^{k}(1+|B|)^{4-k}}{(A-B)^{2}}.$

    2. (ii)

When $k>4$, observe that

$\displaystyle\left|\frac{\beta
m^{2}(A-B)^{2}e^{2i\theta}(1+Be^{i\theta})^{k-4}}{(1+Ae^{i\theta})^{k}}\right|$
$\displaystyle\geq\frac{|\beta|m^{2}(A-B)^{2}|(1-Be^{i\theta})^{k-4}|}{|(1+Ae^{i\theta})^{k}|}$
$\displaystyle\geq\frac{|\beta|(A-B)^{2}(1-|B|)^{k-4}}{(1+|A|)^{k}}\quad(\because
m\geq 1)$ $\displaystyle\geq e-1.$

which yields the desired estimate on $\beta$.∎

######  Corollary 3.5.

Let $f\in\mathcal{A}$ and $\mathcal{G}(z)$ be same as defined in Corollary
(3.3). Then each of the following subordination imply $f\in S^{*}[A,B]$.

  1. (a)

$1+\beta(z\mathcal{G}(z))^{2}\prec\phi_{SG}$ for
$|\beta|(A-B)^{2}\geq\beta_{0}(1+|B|)^{4}$,

  2. (b)

$1+\beta
z\left(\frac{f(z)}{f^{\prime}(z)}\right)(\mathcal{G}(z))^{2}\prec\phi_{SG}$
for $|\beta|(A-B)^{2}\geq\beta_{0}(1+|A|)(1+|B|)^{3}$,

  3. (c)

$1+\beta\left(\frac{f(z)}{f^{\prime}(z)}\right)^{2}(\mathcal{G}(z))^{2}\prec\phi_{SG}$
for $|\beta|(A-B)^{2}\geq\beta_{0}(1+|A|)^{2}(1+|B|)^{2}$,

  4. (d)

$1+\beta(z\mathcal{G}(z))^{2}\prec e^{z}$ for
$|\beta|(A-B)^{2}\geq(e-1)(1+|B|)^{4}$,

  5. (e)

$1+\beta z\left(\frac{f(z)}{f^{\prime}(z)}\right)(\mathcal{G}(z))^{2}\prec
e^{z}$ for $|\beta|(A-B)^{2}\geq(e-1)(1+|A|)(1+|B|)^{3}$,

  6. (f)

$1+\beta\left(\frac{f(z)}{f^{\prime}(z)}\right)^{2}(\mathcal{G}(z))^{2}\prec
e^{z}$ for $|\beta|(A-B)^{2}\geq(e-1)(1+|A|)^{2}(1+|B|)^{2}$.

######  Theorem 3.6.

Let $-1\leq B<A\leq 1$, $\beta_{0}\approx 0.475319$ and $\alpha\in[0,1]$. If
$p\in\mathcal{P}$ satisfy the differential subordination

$(1-\alpha)p(z)+\alpha p^{2}(z)+\beta
z\frac{p^{\prime}(z)}{p^{k}(z)}\prec\phi_{SG}(z),\text{ where }$
$|\beta|\geq\frac{\beta_{0}(1+|A|)^{k}(1+|B|)^{2}+(1-\alpha){(1+|A|)^{k+1}(1+|B|)+\alpha(1+|A|)^{k+2}}}{(A-B)(1-|B|)^{k}}.$

then $p(z)\prec(1+Az)/(1+Bz)$.

###### Proof.

Let $\Omega$ be same as in Theorem (3.2)(a). Let the function $\psi$ be
defined as

$\psi(r,s;z)=(1-\alpha)r+\alpha r^{2}+\beta\frac{s}{r^{k}}.$

Substituting the values of $r$ and $s$ from equation(3.1), we get

$\psi(r,s;z)=(1-\alpha)\frac{1+Ae^{i\theta}}{1+Be^{i\theta}}+\alpha\frac{(1+Ae^{i\theta})^{2}}{(1+Be^{i\theta})^{2}}+\beta\frac{m(A-B)e^{i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2}}$

Then

$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|(1-\alpha)\frac{1+Ae^{i\theta}}{1+Be^{i\theta}}+\alpha\frac{(1+Ae^{i\theta})^{2}}{(1+Be^{i\theta})^{2}}+\beta\frac{m(A-B)e^{i\theta}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2-k}}\right|$
$\displaystyle=\left|\frac{(1-\alpha)(1+Ae^{i\theta})^{k+1}(1+Be^{i\theta})+\alpha(1+Ae^{i\theta})^{k+2}+\beta
m(A-B)e^{i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2}}\right|$
$\displaystyle\geq\frac{|\beta|m(A-B)(1-|B|)^{k}-(1-\alpha)|(1+Ae^{i\theta})^{k+1}||(1+Be^{i\theta})|-\alpha|(1+Ae^{i\theta})^{k+2}|}{|(1+Ae^{i\theta})^{k}||(1+Be^{i\theta})^{2}|}$
$\displaystyle\geq\frac{|\beta|m(A-B)(1-|B|)^{k}-(1-\alpha)(1+|A|)^{k+1}(1+|B|)-\alpha(1+|A|)^{k+2}}{(1+|A|)^{k}(1+|B|)^{2}}$
$\displaystyle=:\phi(m)$

Verify that the function $\phi(m)$ is increasing $\forall$ $m\geq 1$ and
hence, attains its minimum value at $m=1$. Since

$\frac{|\beta|(A-B)(1-|B|)^{k}-(1-\alpha)(1+|A|)^{k+1}(1+|B|)-\alpha(1+|A|)^{k+2}}{(1+|A|)^{k}(1+|B|)^{2}}\geq\beta_{0}$

by Theorem (3.1) and Lemma (2.7), we get the desired result.

∎

######  Remark 3.7.

For $\alpha=0$, the above theorem reduces to the following result.

######  Corollary 3.8.

Let $p$ be an analytic function satisfying $p(0)=1$ and
$\beta>\beta_{0}\approx 0.475319$. Then each of the following subordinations
is sufficient to imply $p\in\mathcal{P}[A,B]$.

$p(z)+\beta z\frac{p^{\prime}(z)}{p^{k}(z)}\prec\phi_{SG},\text{ where
}|\beta|\geq\frac{\beta_{0}(1+|A|)^{k}(1+|B|)^{2}+{(1+|A|)^{k+1}(1+|B|)}}{(A-B)(1-|B|)^{k}}.$

######  Theorem 3.9.

Let $\beta_{0}\approx 0.475319$ and $k$ be a non-negative integer. If
$p\in\mathcal{P}$ and satisfies the differential subordination

$\left(\frac{1}{p(z)}\right)-\beta
z\frac{p^{\prime}(z)}{p^{k}(z)}\prec\phi_{SG}(z),\text{ where
}|\beta|\geq\begin{cases}\frac{\beta_{0}(1+|A|)^{k+1}(1+|B|)^{2-k}+(1+|A|)^{k}(1+|B|)^{3-k}}{(A-B)(1-|A|)},&\text{
when }0\leq k\leq 2\\\
\frac{\beta_{0}(1+|A|)^{k}+(1+|A|)^{k}(1+|B|)}{(A-B)(1-|A|)(1-|B|)^{k-2}},&\text{
when }k>2\end{cases},$

then $p(z)\prec(1+Az)/(1+Bz)$.

###### Proof.

Let $\Omega$ be same as in Theorem(3.2)(a). Consider the analytic function
$\psi$ defined as

$\psi(r,s;z)=\frac{1}{r}-\beta\frac{s}{r^{k}}.$

Substituting the values of $r$ and $s$ as given in equation (3.1), we get

$\psi(r,s;z)=\frac{1+Be^{i\theta}}{1+Ae^{i\theta}}-\beta\frac{m(A-B)e^{i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2}}$

Proceeding as in Theorem (3.2)(a), the following two cases arises.

  1. (i)

When $0\leq k\leq 2$, consider

$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|\frac{1+Be^{i\theta}}{1+Ae^{i\theta}}-\beta\frac{m(A-B)e^{i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2}}\right|$
$\displaystyle=\left|\frac{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{3-k}-\beta
m(A-B)e^{i\theta}(1+Ae^{i\theta})}{(1+Ae^{i\theta})^{k+1}(1+Be^{i\theta})^{2-k}}\right|$
$\displaystyle\geq\frac{|\beta|m(A-B)(1-|A|)-(1+|A|)^{k}(1+|B|)^{3-k}}{(1+|A|)^{k+1}(1+|B|)^{2-k}}$
$\displaystyle=:\phi(m)$

Observe that $\phi^{\prime}(m)>0$ for $m\geq 1$. In view of above and Lemma
(2.7), simple computations gives the desired bound on $\beta$ in terms of $A$
and $B$.

  2. (ii)

When $k>2$, consider

$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|\frac{1+Be^{i\theta}}{1+Ae^{i\theta}}-\beta\frac{m(A-B)e^{i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2}}\right|$
$\displaystyle=\left|\frac{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})-\beta
m(A-B)e^{i\theta}(1+Ae^{i\theta})(1+Be^{i\theta})^{k-2}}{(1+Ae^{i\theta})^{k+1}}\right|$
$\displaystyle\geq\frac{|\beta|m(A-B)(1-|A|)(1-|B|)^{k-2}-(1+|A|)^{k}(1+|B|)}{(1+|A|)^{k+1}}$
$\displaystyle=:\phi(m)$

Since, the function $\phi(m)$ is increasing for $m\geq 1$, similar
computations as done in case (i) gives the required result.

∎

######  Corollary 3.10.

Let $\beta_{0}\approx 0.475319$, $f$ be an analytic function and
$\mathcal{G}(z)$ be same as in Corollary (3.3). Then each of the following
subordinations imply that $f\in\mathcal{S}^{*}[A,B]$.

  1. (a)

$\frac{f(z)}{zf^{\prime}(z)}-\beta z\mathcal{G}(z)\prec\phi_{SG}$ for
$|\beta|(A-B)(1-|A|)\geq\beta_{0}(1+|A|)(1+|B|)^{2}+(1+|B|)^{3}$,

  2. (b)

$\frac{f(z)}{zf^{\prime}(z)}-\beta\frac{f(z)}{f^{\prime}(z)}\mathcal{G}(z)\prec\phi_{SG}$
for $|\beta|(A-B)(1-|A|)\geq\beta_{0}(1+|A|)^{2}(1+|B|)+(1+|A|)(1+|B|)^{2}$,

  3. (c)

$\frac{f(z)}{zf^{\prime}(z)}-\beta\left(\frac{f(z)}{f^{\prime}(z)}\right)^{2}\mathcal{G}(z)\prec\phi_{SG}$
for $|\beta|(A-B)(1-|A|)\geq\beta_{0}(1+|A|)^{3}+(1+|A|)^{2}(1+|B|)$.

######  Theorem 3.11.

Suppose $-1\leq B<A\leq 1$, $\gamma>0$, $\beta_{0}\approx 0.475319$ and $k$ be
a non-negative integer. Let $p$ be an analytic function satisfying the
differential subordination

$p(z)+\frac{zp^{\prime}(z)}{(\beta p(z)+\gamma)^{k}}\prec\phi_{SG}(z),\text{
where }$
$(A-B)(1-|B|)\geq\beta_{0}(1+|B|)^{2-k}(\beta(1+|A|)+\gamma(1+|B|))^{k}(2+|A|+|B|),\text{
when }0\leq k\leq 2,$
$(A-B)(1-|B|)^{k-1}\geq\beta_{0}(2+|A|+|B|)(\beta(1+|A|)+\gamma(1+|B|))^{k},\text{
when }k>2.$

Then $p\in\mathcal{P}[A,B]$.

###### Proof.

Let $\Omega$ be the domain as defined in Theorem(3.2)(a). Define the function
$\psi(r,s;z):\mathbb{C}\backslash\\{0\\}\times\mathbb{C}\times\mathbb{D}\rightarrow\mathbb{C}$
as

$\psi(r,s;z)=r+\frac{s}{(\beta r+\gamma)^{k}}.$

Then using equation(3.1), the function $\psi$ becomes

$\psi(r,s;z)=\frac{1+Ae^{i\theta}}{1+Be^{i\theta}}+\frac{m(A-B)(1+Be^{i\theta})^{k}e^{i\theta}}{(1+Be^{i\theta})^{2}(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}}$

In view of Theorem (3.1), the desired subordination $p\prec(1+Az)/(1+Bz)$ will
follow if we show that $\psi\in\Psi[\Omega;A,B]$. For this, it suffices to
show that

$\left|\log\left(\frac{\psi(r,s;z)}{2-\psi(r,s;z)}\right)\right|\geq 1.$

  1. (i)

When $0\leq k\leq 2$, observe that

$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|\frac{1+Ae^{i\theta}}{1+Be^{i\theta}}+\frac{m(A-B)e^{i\theta}}{(1+Be^{i\theta})^{2-k}(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}}\right|$
$\displaystyle=\left|\frac{\begin{multlined}(1+Ae^{i\theta})(1+Be^{i\theta})^{2-k}(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}\\\
+m(A-B)e^{i\theta}(1+Be^{i\theta})\end{multlined}(1+Ae^{i\theta})(1+Be^{i\theta})^{2-k}(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}\\\
+m(A-B)e^{i\theta}(1+Be^{i\theta})}{(1+Be^{i\theta})^{3-k}(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}}\right|$
$\displaystyle\geq\frac{\begin{multlined}m(A-B)|(1+Be^{i\theta})|-|(1+Ae^{i\theta})||(1+Be^{i\theta})^{2-k}|\\\
|(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}|\end{multlined}m(A-B)|(1+Be^{i\theta})|-|(1+Ae^{i\theta})||(1+Be^{i\theta})^{2-k}|\\\
|(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}|}{|(1+Be^{i\theta})^{3-k}||(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}|}$
$\displaystyle\geq\frac{m(A-B)(1-|B|)-(1+|A|)(1+|B|)^{2-k}|(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}|}{(1+|B|)^{3-k}(\beta(1+|A|)+\gamma(1+|B|))^{k}}$

Similar analysis as done in Theorem (3.2)(a) gives that
$\psi(r,s;z)\notin\Omega$ for

$(A-B)(1-|B|)\geq\beta_{0}(1+|B|)^{2-k}(\beta(1+|A|)+\gamma(1+|B|))^{k}(2+|A|+|B|).$

  2. (ii)

When $k>2$, consider

$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|\frac{1+Ae^{i\theta}}{1+Be^{i\theta}}+\frac{m(A-B)e^{i\theta}(1+Be^{i\theta})^{k-2}}{(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}}\right|$
$\displaystyle=\left|\frac{(1+Ae^{i\theta})(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}+m(A-B)e^{i\theta}(1+Be^{i\theta})^{k-1}}{(1+Be^{i\theta})(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}}\right|$
$\displaystyle\geq\frac{m(A-B)|(1+Be^{i\theta})^{k-1}|-|(1+Ae^{i\theta})||(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}|}{|(1+Be^{i\theta})||(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}|}$
$\displaystyle\geq\frac{m(A-B)(1-|B|)^{k-1}-(1+|A|)|(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}|}{(1+|B|)|(\beta(1+|A|)+\gamma(1+|B|))^{k}|}$

On the similar lines as in proof of part (i), we get the desired result.

∎

##  4\. Second order differential subordination

In this section, sufficient conditions are obtained so that the subordination
implication

$p(z)\prec\frac{1+Az}{1+Bz}$

holds whenever $\psi(p(z),zp^{\prime}(z),z^{2}p^{\prime\prime}(z);z)$ is
subordinate to Modified Sigmoid function, exponential function and Janowski
function.

######  Theorem 4.1.

Let $-1<B<A<1$, $\gamma>0$, $\beta>0$ and $\beta_{0}\approx 0.475319$. Let $p$
be an analytic function satisfying $p(0)=1$. Then, each of the following is
sufficient for $p\in\mathcal{P}[A,B]$.

$\displaystyle(a)$ $\displaystyle 1+\gamma zp^{\prime}(z)+\beta
z^{2}p^{\prime\prime}(z)\prec\phi_{SG}(z),\text{where}$
$\displaystyle(A-B)[\gamma(1+B^{2}+2B)-2B\beta(B+1)]\geq(\beta_{0}+1)(1+B^{2}+2B)^{2}\text{
for }B>0\text{ and }$
$\displaystyle(A-B)[\gamma(1+B^{2}-2B)-2B\beta(B-1)]\geq(\beta_{0}+1)(1+B^{2}-2B)^{2}\text{
for }B<0.$ $\displaystyle(b)$ $\displaystyle 1+\gamma zp^{\prime}(z)+\beta
z^{2}p^{\prime\prime}(z)\prec e^{z},\text{where}$
$\displaystyle(A-B)[\gamma(1+B^{2}+2B)-2B\beta(B+1)]\geq(e-1)(1+B^{2}+2B)^{2}\text{
for }B>0\text{ and }$
$\displaystyle(A-B)[\gamma(1+B^{2}-2B)-2B\beta(B-1)]\geq(e-1)(1+B^{2}-2B)^{2}\text{
for }B<0.$ $\displaystyle(c)$ $\displaystyle 1+\gamma zp^{\prime}(z)+\beta
z^{2}p^{\prime\prime}(z)\prec(1+Az)/(1+Bz),\text{where}$
$\displaystyle(A-B)(1-B^{2})[\gamma(1+B^{2}+2B)-2B\beta(B+1)]-|B|(A-B)(1+B^{2}+2B)^{2}$
$\displaystyle\geq(A-B)(1+B^{2}+2B)^{2},\text{ for }B>0\text{ and }$
$\displaystyle(A-B)(1-B^{2})[\gamma(1+B^{2}-2B)-2B\beta(B-1)]-|B|(A-B)(1+B^{2}-2B)^{2}$
$\displaystyle\geq(A-B)(1+B^{2}-2B)^{2},\text{ for }B<0.$

###### Proof.

  1. (a)

Let
$\Omega=\phi_{SG}(\mathbb{D})=\\{w\in\mathbb{C}:\left|\log\left({w}/{(2-w)}\right)\right|<1\\}$.
Consider the analytic function
$\psi:\mathbb{C}^{3}\times\mathbb{D}\rightarrow\mathbb{C}$ defined as

$\psi(r,s,t;z)=1+\gamma s+\beta t$

For $\psi\in\Psi[\Omega;A,B]$, we must have $\psi(r,s,t;z)\notin\Omega$. By
Theorem (2.4), this implication is true if

$\left|\log\left(\frac{\psi(r,s,t;z)}{2-\psi(r,s,t;z)}\right)\right|\geq 1.$
(4.1)

By Lemma (2.7), the inequality (4.1) holds if and only if
$|\psi(r,s,t;z)|\geq\beta_{0}$. A calculation shows that

$\displaystyle|\psi(r,s,t;z)|$ $\displaystyle=|1+\gamma s+\beta t|$
$\displaystyle\geq
1+\gamma|s|\left|1+\left(\frac{\beta}{\gamma}\right)\frac{t}{s}\right|-1$
$\displaystyle\geq\gamma|s|\operatorname{Re}\left(1+\left(\frac{\beta}{\gamma}\right)\frac{t}{s}\right)-1$
$\displaystyle\geq m\gamma
d(\theta)\left(1+\left(\frac{\beta}{\gamma}\right)(mg(\theta)+m-1)\right)-1$
$\displaystyle\geq\begin{cases}\frac{m(A-B)}{1+B^{2}+2B}\left(\gamma+\frac{-2B(B+1)\beta
m}{1+B^{2}+2B}\right)-1,B>0\\\
\frac{m(A-B)}{1+B^{2}-2B}\left(\gamma+\frac{-2B(B-1)\beta
m}{1+B^{2}-2B}\right)-1,B<0\end{cases}$ $\displaystyle:=\phi(m),$

where $d(\theta)$ and $g(\theta)$ are given by (2.4) and (2.5) respectively.
Observe that $\phi(m)$ is increasing function for $m\geq 1$. Therefore, we
have $|\psi(r,s,t;z)|\geq\phi(1)\geq\beta_{0}$ and hence,
$\psi\in\Psi[\Omega;A,B]$. By Theorem 2.4, $p(z)\prec(1+Az)/(1+Bz)$.

  2. (b)

Consider the domain $\Omega=\\{w\in\mathbb{C}:|\log w|<1\\}$. Let the function
$\psi:\mathbb{C}^{3}\times\mathbb{D}\rightarrow\mathbb{C}$ be defined as

$\psi(r,s,t;z)=1+\gamma s+\beta t.$

For $\psi\in\Psi[\Omega;A,B]$, we must have $\psi(r,s,t;z)\notin\Omega$. In
order to satisfy this relation, it is sufficient to show that

$|\log(\psi(r,s,t;z))|\geq 1.$

Since

$\displaystyle\left|\gamma s\left(1+\frac{\beta t}{\gamma s}\right)\right|$
$\displaystyle\geq\gamma|s|\left|1+\left(\frac{\beta}{\gamma}\right)\frac{t}{s}\right|$
$\displaystyle\geq\gamma|s|\operatorname{Re}\left(1+\left(\frac{\beta}{\gamma}\right)\frac{t}{s}\right)$
$\displaystyle\geq m\gamma
d(\theta)\left(1+\left(\frac{\beta}{\gamma}\right)(mg(\theta)+m-1)\right)$
$\displaystyle\geq\begin{cases}\frac{m(A-B)}{1+B^{2}+2B}\left(\gamma-\frac{2B(B+1)\beta
m}{1+B^{2}+2B}\right),B>0\\\
\frac{m(A-B)}{1+B^{2}-2B}\left(\gamma-\frac{2B(B-1)\beta
m}{1+B^{2}-2B}\right),B<0\end{cases}$ $\displaystyle:=\phi(m)$

and $\phi(m)$ is increasing function of $\phi$, by Lemma 2.6, $\left|\gamma
s\left(1+\frac{\beta t}{\gamma s}\right)\right|\geq e-1$. Thus, by Theorem
2.4, $p(z)\prec(1+Az)/(1+Bz)$.

  3. (c)

Consider the domain

$\Omega=\left\\{w\in\mathbb{C}:\left|w-\frac{1-AB}{1-B^{2}}\right|<\frac{A-B}{1-B^{2}}\right\\}.$

Let $\psi:\mathbb{C}^{3}\times\mathbb{D}\rightarrow\mathbb{C}$ be defined as
$\psi(r,s,t;z)=1+\gamma s+\beta t$. Now, $\psi\in\Psi[\Omega;A,B]$, if
$\psi(r,s,t;z)\notin\Omega$. On the similar lines on the proof of part(a),

$\displaystyle\left|\psi(r,s,t;z)-\frac{1-AB}{1-B^{2}}\right|$
$\displaystyle=\left|1+\gamma s+\beta t-\frac{1-AB}{1-B^{2}}\right|$
$\displaystyle\geq\gamma|s|\operatorname{Re}\left(1+\left(\frac{\beta}{\gamma}\right)\frac{t}{s}\right)-\frac{|B|(A-B)}{1-B^{2}}$
$\displaystyle\geq m\gamma
d(\theta)\left(1+\left(\frac{\beta}{\gamma}\right)(mg(\theta)+m-1)\right)-\frac{|B|(A-B)}{1-B^{2}}$
$\displaystyle:=\phi(m).$

Using the values of $d(\theta)$ and $g(\theta)$ as given in the equations
(2.4) and (2.5), and first derivative test for function $\phi$ we have,

for $B>0$,

$\displaystyle\left|\psi(r,s,t;z)-\frac{1-AB}{1-B^{2}}\right|$
$\displaystyle\geq\frac{(A-B)}{1+B^{2}+2B}\left(\gamma-\frac{2B(B+1)\beta}{1+B^{2}+2B}\right)-\frac{|B|(A-B)}{1-B^{2}}$
$\displaystyle\geq\frac{A-B}{1-B^{2}},$

and for $B<0$,

$\displaystyle\left|\psi(r,s,t;z)-\frac{1-AB}{1-B^{2}}\right|$
$\displaystyle\geq\frac{(A-B)}{1+B^{2}-2B}\left(\gamma-\frac{2B(B-1)\beta}{1+B^{2}-2B}\right)-\frac{|B|(A-B)}{1-B^{2}}$
$\displaystyle\geq\frac{A-B}{1-B^{2}}.$

Therefore, $\psi\in\Psi[\Omega;A,B]$ and hence, by Theorem 2.4,
$p(z)\prec(1+Az)/(1+Bz)$.∎

######  Corollary 4.2.

Let $\gamma$ and $\beta$ be positive integers and $f$ be an analytic function.
Set

$\displaystyle\mathcal{H}(z)=$ $\displaystyle
1+\gamma\Bigg{(}\frac{z^{2}f^{\prime\prime}(z)}{f(z)}-\Bigg{(}\frac{zf^{\prime}(z)}{f(z)}\Bigg{)}^{2}+\frac{zf^{\prime}(z)}{f(z)}\Bigg{)}+\beta\Bigg{(}\frac{z^{3}f^{\prime\prime\prime}(z)}{f(z)}$
$\displaystyle+\frac{2z^{2}f^{\prime\prime}(z)}{f(z)}+2\Bigg{(}\frac{zf^{\prime}(z)}{f(z)}\Bigg{)}^{3}-2\Bigg{(}\frac{zf^{\prime}(z)}{f(z)}\Bigg{)}^{2}-\frac{3z^{3}f^{\prime}(z)f^{\prime\prime}(z)}{f(z)^{2}}\Bigg{)}.$

Then, $f\in\mathcal{S}^{*}[A,B]$ if any one of the following condition hold.

$\displaystyle(a)$ $\displaystyle\mathcal{H}(z)\prec\phi_{SG}(z),\text{where}$
$\displaystyle(A-B)[\gamma(1+B^{2}+2B)-2B\beta(B+1)]\geq(\beta_{0}+1)(1+B^{2}+2B)^{2}\text{
for }B>0\text{ and }$
$\displaystyle(A-B)[\gamma(1+B^{2}-2B)-2B\beta(B-1)]\geq(\beta_{0}+1)(1+B^{2}-2B)^{2}\text{
for }B<0.$ $\displaystyle(b)$ $\displaystyle\mathcal{H}(z)\prec
e^{z},\text{where}$
$\displaystyle(A-B)[\gamma(1+B^{2}+2B)-2B\beta(B+1)]\geq(e-1)(1+B^{2}+2B)^{2}\text{
for }B>0\text{ and }$
$\displaystyle(A-B)[\gamma(1+B^{2}-2B)-2B\beta(B-1)]\geq(e-1)(1+B^{2}-2B)^{2}\text{
for }B<0.$ $\displaystyle(c)$
$\displaystyle\mathcal{H}(z)\prec(1+Az)/(1+Bz),\text{where}$
$\displaystyle(A-B)(1-B^{2})[\gamma(1+B^{2}+2B)-2B\beta(B+1)]-|B|(A-B)(1+B^{2}+2B)^{2}$
$\displaystyle\geq(A-B)(1+B^{2}+2B)^{2},\text{ for }B>0\text{ and }$
$\displaystyle(A-B)(1-B^{2})[\gamma(1+B^{2}-2B)-2B\beta(B-1)]-|B|(A-B)(1+B^{2}-2B)^{2}$
$\displaystyle\geq(A-B)(1+B^{2}-2B)^{2},\text{ for }B<0.$

######  Theorem 4.3.

Suppose $-1<B<A<1$, $\beta_{0}\approx 0.475319$, $\beta>0$ and $\gamma>0$. Let
$p$ be an analytic function which satisfies the condition $p(0)=1$ and the
following inequalities holds:

$(A-B)(1+B)[\gamma(1+B^{2}+2B)-2B(B+1)\beta]-(1+A)(1+B^{2}+2B)^{2}\geq\beta_{0}(1+B)(1+B^{2}+2B)^{2}$

for $B>0$, and

$(A-B)(1+B)[\gamma(1+B^{2}-2B)-2B(B-1)\beta]-(1+A)(1+B^{2}-2B)^{2}\geq\beta_{0}(1+B)(1+B^{2}-2B)^{2}$

for $B<0$. Then,

$p(z)+\gamma zp^{\prime}(z)+\beta z^{2}p^{\prime\prime}(z)\prec\phi_{SG}(z)$

implies

$p\prec\frac{1+Az}{1+Bz}.$

###### Proof.

Let $\Omega$ be the domain defined in Theorem (3.2)(a). Consider the function
$\psi:\mathbb{C}^{3}\times\mathbb{D}\rightarrow\mathbb{C}$ defined as

$\psi(r,s,t;z)=r+\gamma s+\beta t.$

For $\psi$ to be in $\Psi[\Omega;A,B]$, we must have
$\psi(r,s,t;z)\notin\Omega$. By Theorem (2.4), this result is true if

$\left|\log\left(\frac{\psi(r,s,t;z)}{2-\psi(r,s,t;z)}\right)\right|\geq 1$

Using Lemma (2.7), this inequality holds if and only if

$|\psi(r,s,t;z)|\geq\beta_{0}$ (4.2)

where $\beta_{0}$ is the positive real root of the equation in (2.6). If
$k(\theta)$, $g(\theta)$ and $d(\theta)$ are given by the equations (2.3),
(2.5) and (2.4) respectively, then

$\displaystyle|\psi(r,s,t;z)|$ $\displaystyle=|r+\gamma s+\beta t|$
$\displaystyle\geq\gamma|s|\left|1+\left(\frac{\beta}{\gamma}\right)\frac{t}{s}\right|-|r|$
$\displaystyle\geq\gamma|s|\operatorname{Re}\left(1+\left(\frac{\beta}{\gamma}\right)\frac{t}{s}\right)-|r|$
$\displaystyle\geq m\gamma
d(\theta)\left(1+\left(\frac{\beta}{\gamma}\right)(mg(\theta)+m-1)\right)-|k(\theta)|$
$\displaystyle\geq m\gamma
d(\theta)\left(1+\left(\frac{\beta}{\gamma}\right)(mg(\theta)+m-1)\right)-\frac{1+A}{1+B}$
$\displaystyle:=\phi(m).$

Since $\phi(m)$ is increasing function, we have, for $B>0$,

$|\psi(r,s,t;z)|\geq\frac{(A-B)}{1+B^{2}+2B}\left(\gamma-\frac{2B(B+1)\beta}{1+B^{2}+2B}\right)-\frac{1+A}{1+B}\text{
and }$

for $B<0$,

$|\psi(r,s,t;z)|\geq\frac{(A-B)}{1+B^{2}-2B}\left(\gamma-\frac{2B(B-1)\beta}{1+B^{2}-2B}\right)-\frac{1+A}{1+B}.$

Therefore, inequality (4.2) is satisfied and hence, we get the required
result. ∎

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