Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂ

Abstract

Let S γ , A , B ∗ ( D ) {S}_{\gamma ,A,B}^{\ast }\left({\mathbb{D}}) be the usual class of g g -starlike functions of complex order γ \gamma in the unit disk D = { ζ ∈ C : ∣ ζ ∣ < 1 } {\mathbb{D}}=\left\{\zeta \in {\mathbb{C}}:| \zeta | \lt 1\right\} , where g ( ζ ) = ( 1 + A ζ ) ∕ ( 1 + B ζ ) g\left(\zeta )=\left(1+A\zeta )/\left(1+B\zeta ) , with γ ∈ C \ { 0 } , − 1 ≤ A < B ≤ 1 , ζ ∈ D \gamma \left\in {\mathbb{C}}\backslash \left\{0\right\}\right,-1\le A\lt B\le 1,\zeta \in {\mathbb{D}} . First, we obtain the bounds of all the coefficients of homogeneous expansions for the functions f ∈ S γ , A , B ∗ ( D ) f\in {S}_{\gamma ,A,B}^{\ast }\left({\mathbb{D}}) when ζ = 0 \zeta =0 is a zero of order k + 1 k+1 of f ( ζ ) − ζ f\left(\zeta )-\zeta . Second, we generalize this result to several complex variables by considering the corresponding biholomorphic mappings defined in a bounded complete Reinhardt domain. These main theorems unify and extend many known results.

Full text

Research Article
Xiaoying Sima, Zhenhan Tu, and Liangpeng Xiong*
Some results of homogeneous expansions
for a class of biholomorphic mappings
defined on a Reinhardt domain in ℂn
https://doi.org/10.1515/dema-2022-0242
received February 2, 2023; accepted April 25, 2023
Abstract: Let
∗
S
γAB,,
()
be the usual class of
g
-starlike functions of complex order
γ
in the unit disk
=∈ <ζζ:1

{∣∣}
, where
=+ ∕+
g
ζAζBζ11() ( )( )
, with ∈−≤<≤∈γABζ\0, 1 1,{} . First, we obtain
the bounds of all the coefficients of homogeneous expansions for the functions ∈∗
fS
γAB,, ()
when =
ζ0
is a
zero of order
+
k1
of −fζ
ζ
() . Second, we generalize this result to several complex variables by considering
the corresponding biholomorphic mappings defined in a bounded complete Reinhardt domain. These main
theorems unify and extend many known results.
Keywords: biholomorphic mappings, coefficient problems, Reinhardt domain, g-starlike mappings of complex
order gamma
MSC 2020: 32H02, 30C45, 26B10, 32A30
1 Introduction
Let
n

be the
n
-dimensional complex variable space. Let

be the unit open disk in
1

. Suppose that
⊂
Ω
,Ω
n
12

are two domains. Let
H
Ω,Ω
12
(
)
be the family of all holomorphic mappings from
Ω
1
into
Ω
2
. Let
ϕ
1
,∈
ϕ
H,
21
()
. We say that
ϕ
1
is subordinate to
ϕ
2
, and write
≺
ϕ
ϕ
12
, if there exists a Schwarz function
ω
on

such that
=
ϕ
ϕωz
12
(())
on

(see, Amini et al. [1]). Let
Ω
be a domain (connected open set) in
n

, which
contains 0. It is said that =z0is a zero of order
k
of
fz()
if
==
−
fDf00,…, 00
k1
() ()
, but
≠
D
f00
k
()
, where
∈
k
(see, Lin and Hong [2]). In one complex variable, the following Theorem A concerning starlike functions
of order
α
is classical and well known.
Theorem A. (Boyd [3]) Let ∈
α
0, 1(
)
and
∈
+
k

.If =+
∑=+
∞
fz z az
mk mm
1
() is a starlike function of order
α
on the
unit disk

,then
≤∏−+− +≤ ≤ + =
=
aμk α
sk sk m s k s
122
!,1 1,1,2,….
mμ
s
s
1
∣∣ (( ) ) ()
These estimates are sharp for
=+ =msk s1, 1, 2,….
Especially, when
=
k1
,
Xiaoying Sima: School of Mathematics and Computer Science, Jiangxi Science and Technology Normal University, Nanchang 330038,
Jiangxi, People’s Republic of China, e-mail: xiaoying_math@163.com
Zhenhan Tu: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, People’s Republic of China,
e-mail: zhhtu.math@whu.edu.cn

* Corresponding author: Liangpeng Xiong, School of Mathematics and Computer Science, Jiangxi Science and Technology Normal
University, Nanchang 330038, Jiangxi, People’s Republic of China, e-mail: lpxiong2016@whu.edu.cn
Demonstratio Mathematica 2023; 56: 20220242
Open Access. © 2023 the author(s), published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0
International License.

∏
≤−−=
=
amμα m
11! 2, 2,3,….
mμ
m
2
∣∣()
()
It is known that the coefficient inequalities are related to the Bieberbach conjectures [4], which was settled
by de Branges [5]. However, Cartan [6] stated that the Bieberbach conjecture does not hold in several complex
variables. In fact, only a few complete results are known for the inequalities of homogeneous expansions for
subclasses of biholomorphic mappings in
n

(see, e.g., Długosz and Liczberski [7], Hamada and Honda [8], Liu
and Wu [9], Liu and Liu [10], Liu et al. [11], and Xu et al. [12]). Many works are concentrating on the bounds of
second- and third-order terms of homogeneous expansions for starlike mappings and the sharp bounds of all
homogeneous expansions for the special subclasses of starlike mappings with some restricted conditions (see,
e.g., Hamada et al. [13], Liu and Liu [14], Tu and Xiong [15], Xiong [16], Xu et al. [17], and Xu et al. [18]). In
Graham et al. [19], the estimate of the second-order coefficients of the first elements of g-Loewner chains in
several complex variables was first obtained. In Bracci [20], a sharp estimate for the second-order coefficients
for the first elements of g-Loewner chains on the Euclidean unit ball of
2

, where =+∕−
g
ζζζ11()()()
, which
gives a support point for the family, was obtained. Generalizations of this result to the unit polydisk in
2

and
to bounded symmetric domains were considered in Graham et al. [21], Hamada and Kohr [22], respectively. In
Xu and Liu [23], the Fekete-Szegö inequality for starlike mappings in several complex variables was first
obtained. Very recent important results related to the Fekete-Szegö inequality in several complex variables
were obtained in other articles (see, e.g., Długosz and Liczberski [24], Elin and Jacobzon [25], Hamada [26], and
Lai and Xu [27]). In particular, the Fekete-Szegö inequality for univalent mappings in several complex vari-
ables was first obtained in Hamada et al. [28]. Also, the other related results may consult in Długosz and
Liczberski [29], Graham and Kohr [30], and Nunokawa and Sokol [31]. Liu et al. [32] considered only the main
coefficients that are analogous with the diagonal elements of a square matrix and generalized Theorem A to
the case on a Reinhardt domain in
n

from a new viewpoint. Let
∑
=⎧
⎨
⎩∈<
⎫
⎬
⎭>=
=
D
zz pln:1,1,1,2,…,
pp p n
l
n
lpl
,,…, 1
nl
12 ∣∣ (
)
be a bounded complete Reinhardt domain in
n

, its Minkowski function
ρz(
)
is
C
1
except for some lower-
dimensional manifolds in
n

, and →+∞ρ:0,
n
[)
is defined by:
=⎧
⎨
⎩>∈ ⎫
⎬
⎭∈ρz t z
tDzinf 0 : ,
.
pp p n
,,…,
n12 () (1)
Let ∈=
∗
γ\0{}. Now, we introduce the following classes ()
∗D
gγ pp p
,,,…,
n12

, which extend the usual class ∗γ

(
)
of starlike functions of complex order
γ
on

in

to the classes of
g
-starlike mapping of complex order
γ
on
the bounded Reinhardt domain
D
pp p,,…,
n
12
in
n

. The function class ∗γ

(
)
was considered earlier by Nasr and
Aouf [33] (also, see Srivastava et al. [34]).
Definition 1.1. Suppose that ∈
∗
γand
→
g
:
be a biholomorphic function such that =>
g
gζ01, 0R() ()
on

. A normalized locally biholomorphic mapping →fD:pp p
n
,,…,
n12 is called a
g
-starlike mappings of
complex order
γ
on
D
pp p,,…,
n
12
if
+⎛
⎝
⎜
⎜−⎞
⎠
⎟
⎟∈∀∈
∂
∂−
γρz
Df z f z gzD
1
1
21, \0
,
ρz
z
pp p
1,,…,
n12

()
[()]() () {}
()
() (2)
where
ρ
is the Minkowski function of
D
pp p,,…,
n
12
. We denote by ()
∗D
gγ pp p
,,,…,
n12

the set of all
g
-starlike mappings
of complex order
γ
on
D
pp p,,…,
n
12
in
n

.
Remark 1.1. (i) If
=
+
+
g
ζ
Aζ
Bζ
1
1
()
(
−
≤<≤AB1
1
,
∈
ζ

)inDefinition 1.1, then we write ()
∗+
+D
γpp p
,,,…,
Aζ
Bζ n
1
112

by ()
∗D
γAB pp p
,, ,,…,
n12

.
2Xiaoying Sima et al.

(ii) If
=
n1
and
=
+
+
g
ζ
Aζ
Bζ
1
1
()
in Definition 1.1, then it is obvious that Condition (2) is equivalent to
⎜⎟
+⎛
⎝
′−⎞
⎠≺+
+−≤ < ≤ ∈ ∈
∗
γζf ζ
fζ Aζ
Bζ AB γ ζ
1
111
1,1 1, ,
.

()
()
We denote by ∗
γAB,, 

()the set of all
g
-starlike mappings of complex order
γ
on

in

, where
=+ ∕+
g
ζAζBζ11() ( )( )
,
∈
ζ

. In particular,
∗AB1, ,


(
)
is identical with the well-known class of Janowski
starlike functions (see, Janowski [35]) and −
∗
γ,1,1


(
)
is the set of all starlike functions of complex order
γ
in

.
(iii) If
=γ
1
and
=− ∈Aα α21 0,1([)
)
,
=
B1
in Definition 1.1 (the case
g
is defined as (ii)), then we obtain
the starlike mappings of order
α
on
D
pp p,,…,
n
12
(see, e.g., Liu et al. [32]).
(iv) By choosing the suitable functions
g
and parameters
γ
in Definition 1.1, we can obtain kinds of
subclasses of starlike mappings defined on the Reinhardt domain
D
pp p,,…,
n
12
in
n

.
In this article, we first extend the definition of
g
-starlike mappings of complex order
γ
from the case of one-
dimensional space to the case of higher-dimensional space (see Definition 1.1). Next, we obtain the bound of all
coefficients of homogeneous expansions for the functions
∈
∗
f
γAB,,
(
)
when =
ζ0
is a zero of order
+
k1
of
−fζ
ζ
() (see Lemma 2.2). Finally, by applying the results in Section 2, we consider the bound of main coefficients of
the homogeneous expansions for the functions
()
∈
∗
fD
γAB pp p
,, ,,…,
n12

in several complex variables (Theorems 3.1
and 3.2). Also, our results extend some theorems given in the previous literature (see Remarks 1.1–3.1).
2 Preliminaries
The following lemmas are needed in order to prove our estimates. Actually, we may use the similar way to those in
the proof of Liu and Liu [36] (Lemmas 2.1 and 2.3). Here, we give the proof for the sake of completeness.
Lemma 2.1. Let
∈
+
k

,
≥C0
,∈
∗
γ.Then, for =
q
2, 3,…, we have
∑∏ ∏
++
⎛
⎝
⎜⎛
⎝+⎞
⎠
⎞
⎠
⎟=⎛
⎝
⎜−⎛
⎝+⎞
⎠
⎞
⎠
⎟
=
−
=
−
=
−
C γ Cγ mk Cγ mμCγ
kk
qμCγ
k
21!1!
.
m
q
μ
m
μ
q
22
1
1
0
12
0
12
∣∣ ∣∣ ( ∣∣) ∣∣ () ∣∣ (3)
Proof. We try to prove this lemma by mathematical induction. First, if
=
q
2
, it is easy to see that (3) is true.
Next, for all
=
q
2
,
l
3,…
, assume that (3) holds true. Then, we need to show that (3) holds true when
=+
q
l
1
.By
a simple computation, we have
∑∏
∏∏
∏
++
⎛
⎝
⎜⎛
⎝+⎞
⎠
⎞
⎠
⎟
=⎛
⎝
⎜−⎛
⎝+⎞
⎠
⎞
⎠
⎟++
⎛
⎝
⎜⎛
⎝+⎞
⎠
⎞
⎠
⎟
=⎛
⎝
⎜⎛
⎝+⎞
⎠
⎞
⎠
⎟
==
−
=
−
=
−
=
C γ Cγ mk Cγ mμCγ
k
k
lμCγ
kCγ lk Cγ lμCγ
k
k
lμCγ
k
21!
1! 21
!
!.
m
l
μ
m
μ
l
μ
l
μ
l
22
10
12
0
12
0
12
0
2
∣∣ ∣∣ ( ∣∣) ∣∣
() ∣∣ ∣∣( ∣∣) ∣∣
∣∣
This completes the proof. □
Lemma 2.2. Let =+
∑∈
=+
∞∗
fz z az S
mk mmγAB1,,
() ()with
∈
+
k

,∈
∗
γ,
−
≤<≤AB1
1
.Then, for =
s
1, 2,…,
we have
≤∏−+− +≤ ≤ +
=
aμkABγ
sk sk m s k
1
!,1 1
.
mμ
s
s
1
∣∣ (( ) ∣ ∣∣∣) ()
In particular, if
=
k1
,then
≤∏+− =
−=
−
aμABγm,2,3,….
mmμ
m
11! 0
2
∣ ∣ ( ∣ ∣∣∣)
()
Some results of homogeneous expansions 3

Proof. Since ∈∗
fS
γAB,, ()
, so we can write that
⎜⎟
+⎛
⎝
′−⎞
⎠≺+
+∈
γzf z
fz Az
Bz z
1
111
1,
.

()
()
Thus, there is a function
∈
φ
,(
)
with
<φz
1
∣()∣
, such that
⎜⎟
+
+=+⎛
⎝
′−⎞
⎠∈
Aφz
Bφz γ zf z
fz z
1
1111,
.

(())
(()) ()
() (4)
Using (4), a simple computation shows that
=′−
−+ −′
=+ +⋯∈
++
φ
zzf z f z
ABγBfz Bzfz bz b z z,
.
kkkk
11() () ()
(( ) )() () (5)
(5) is equal to
′− = − + − ′zfzfzφzABγBfzBzfz() () (){(( ) )() ()} (6)
and
−=− =++
−
maABγb mk k k1,1,2,…,2
.
mm1
()()
(7)
In view of the relations
<φz
1
∣()∣
and ∑≤
=
∞b
1
mk m2
∣∣ , then we have
∑
≤
=
−
b1
.
mk
k
m
21 2
∣∣
(8)
Using (7) and (8), it can be easily shown that
∑−≤−
=+ maABγ1
.
m
k
k
m
1
222 22
()∣∣∣∣∣∣ (9)
By applying (6), then it can also be written as:
∑∑
∑∑
−=
⎧
⎨
⎩−+ −+− ⎫
⎬
⎭
=⎧
⎨
⎩−+ −+− ⎫
⎬
⎭+
=+
∞
=+
∞
=+
−
=+
∞
m az φz A Bγz A Bγ B mBa z
φzABγz ABγBmBaz cz
1
.
mk mm
mk mm
mk
pk
mm
mp mm
11
11
() ()() (() )
()( ) (( ) ) (10)
Using the equality in (10), we have
∑∑ ∑
−+ =
⎧
⎨
⎩−+ −+− ⎫
⎬
⎭
=+ =+
∞
=+
−
m az dz φz A Bγz A Bγ B mBa z1
.
m
k
p
mm
mp mm
mk
pk
mm
11 1
() ()() (() ) (11)
Taking
→r
1
in (11) and applying the similar argument being used in Theorem 1 by Boyd [3], we obtain
∑∑
−≤−+ −+−
=+ =+
−
maABγ ABγBmBa1
.
m
k
p
mmk
pk
m
1
22 22
1
22
( )∣∣∣ ∣∣∣ ∣( ) ∣∣∣
(12)
In fact, with (12), a simple computation shows that
∑∑
−≤−+ −+−−−
=−+ =+
−
maABγ ABγBmBm a11.
m
pk
p
mmk
pk
m
1
22 22
1
222
( )∣∣∣ ∣∣∣ (∣( ) ∣( ))∣∣ (13)
Furthermore, in view of (13), we have
∑∑
⎜⎟
−≤−
⎛
⎝
−+⎛
⎝−+ −⎞
⎠⎞
⎠
=−+ =+
−
ma ABγ
ABγ mABγa12 212
.
m
pk
p
mmk
pk
m
1
22
1
2
()∣∣∣∣∣∣
∣∣∣∣ ∣∣∣∣
∣∣
(14)
4Xiaoying Sima et al.

Here, we will prove that the following inequalities
∑∏
−≤
⎛
⎝
⎜−⎛
⎝+−⎞
⎠
⎞
⎠
⎟
=+
+
=
−
ma k
sμABγ
k
11!
m
sk
sk
mμ
s
1
122
0
12
()∣∣() ∣∣∣∣
() (15)
and
∑∏
⎛
⎝−+ −⎞
⎠≤⎛
⎝+−⎞
⎠
⎛
⎝
⎜⎛
⎝+−⎞
⎠
⎞
⎠
⎟
=+
+
=
−
mABγask
ABγsμABγ
k
122
1!
m
sk
sk
mμ
s
1
12
0
12
∣∣∣∣
∣∣ ∣∣∣∣ ∣∣∣∣
() (16)
hold true for =
s
1, 2,….
If
=
s1
, then (15) holds from (9). Moreover, by using Lemma 2.2 in Liu and Liu [36] and (9), we obtain
∑
∑
∑
⎛
⎝−+ −⎞
⎠
=+
+
⎛
⎝−+ −⎞
⎠
≤+−≤
+−
=⎛
⎝+−⎞
⎠⎛
⎝−⎞
⎠
=+
−
=+ −
−
=+
−
mABγa
k
kk
kmABγa
k
kmak
kABγ
kABγ ABγ
k
12
12
1
2.
mk
k
m
ABγ
mk
k
ABγ m
ABγ
mk
k
m
ABγ
1
22
2
21
22
2
2
2
21
222 2
222
2
∣∣∣∣
∣∣ ∣∣∣∣
∣∣
()∣∣ ∣∣∣∣
∣∣∣∣∣∣∣∣
∣∣∣∣ ∣∣∣∣
∣∣∣∣ ∣∣∣∣ (17)
Thus, (17) implies that (16) is true for
=
s1
.
At last, assume that (15) and (16) are valid for
=−
s
q1, 2,…,
1
. Taking
=+
p
qk1()
in (14) and using
Lemma 2.1, it gives that
∑∑
∑∑
∑∏
∏
⎟
⎜⎟
⎜
−≤−
⎛
⎝
−+⎛
⎝−+ −⎞
⎠⎞
⎠
=− ⎛
⎝
−+⎛
⎝−+ −⎞
⎠⎞
⎠
≤− ⎛
⎝
⎜−+⎛
⎝+−⎞
⎠
⎛
⎝
⎜⎛
⎝+−⎞
⎠
⎞
⎠
⎟⎞
⎠
⎟
=⎛
⎝
⎜−⎛
⎝+−⎞
⎠
⎞
⎠
⎟
=+
+
=+
=
−
=+
+
=
−
=
−
=
−
ma ABγ
ABγ mABγa
ABγ
ABγ mABγa
ABγ
ABγ sk ABγsμABγ
k
k
qμABγ
k
12 212
2212
222
1!
1! .
mqk
qk
mmk
qk
m
s
q
msk
sk
m
s
q
μ
s
μ
q
1
122
1
2
1
1
1
12
1
1
0
12
0
12
()∣∣∣∣∣∣
∣∣∣∣ ∣∣∣∣
∣∣
∣∣∣∣
∣∣∣∣ ∣∣∣∣
∣∣
∣∣∣∣
∣∣∣∣ ∣∣∣∣ ∣∣∣∣
() ∣∣∣∣
()
()
The aforementioned inequality shows that (15) holds for
=
s
q
. On the other hand, when
=
s
q
,wefind that
∑
∑
∑
∏
∏
⎛
⎝−+ −⎞
⎠
=+
+
⎛
⎝−+ −⎞
⎠
≤+−
≤+⎛
⎝
⎜−⎛
⎝+−⎞
⎠
⎞
⎠
⎟
=⎛
⎝+−⎞
⎠
⎛
⎝
⎜⎛
⎝+−⎞
⎠
⎞
⎠
⎟
=+
+
−
=+
+
−
−
=+
+
−
=
−
=
−
mABγa
qk
qk qk
qk mABγa
qk
qk ma
qk
qk k
qμABγ
k
qk ABγqμABγ
k
12
12
1
1!
21!.
mqk
qk
m
ABγ
mqk
qk
ABγ m
ABγ
nqk
qk
m
ABγ
μ
q
μ
q
1
12
2
21
12
2
2
2
21
122
2
20
12
0
12
∣∣∣∣
∣∣
() () ∣ ∣∣∣∣∣
() ()∣∣
() ( ) ∣∣∣∣
∣∣∣∣ ∣∣∣∣
()
∣∣∣∣
() ∣∣∣∣
∣∣∣∣
()
∣∣∣∣
Some results of homogeneous expansions 5

The aforementioned inequality shows that (16) holds for
=
s
q
.
In view of (15), for +≤ ≤ +
s
kms
k
11()
, we have
∏
∏
∏
∏
≤−− ⎛
⎝+−⎞
⎠
≤⎛
⎝+−⎞
⎠
=⎛
⎝−+ −⎞
⎠
=−+−
=
−
=
−
=
=
ak
ms μABγ
k
sμABγ
k
sμABγ
k
μkABγ
sk
11!
1!
1!1
1
!.
mμ
s
μ
s
μ
s
μ
s
s
0
1
0
1
1
1
∣∣()() ∣∣∣∣
∣∣∣∣
∣∣∣∣
(( ) ∣ ∣∣∣)
This completes the proof. □
Remark 2.1. If
=− ∈Aα α21 0,1([)
)
,
=
B1
, and
=γ
1
in Lemma 2.2, then it reduces to Theorem A.
3 Main results
The following theorems give the estimates of main coefficients of homogeneous expansions for the class of
g
-starlike mappings of complex order
γ
defined on the bounded complete Reinhardt domain
D
pp p,,…,
n
12
in
n

,
where
=+ ∕+
g
ζAζBζ11() ( )( )
,
−
≤<≤ ∈AB ζ11,
. Theorems 3.1 and 3.2 will give generalizations of the
results in Boyd [3] and Liu et al. [32].
Theorem 3.1. Suppose that
()
=′∈
∗
fz fz f z f z S D,,…,
nγAB pp p
12 ,, ,,…,
n12
() (() () ())
and =z0is a zero of order
+
k1
of
−fz z() .Define
∑
==
Df z
mazzz
0
!…,
mqm
ll l
n
ql l l l l l
,,…, 1 ,,…,
m
mm
12
12 12
()( )
where
=
qn
1, 2,…,
,+≤ ≤ + =
s
kmsks11,1,2,…() ,and
∈
+
k

.Let
a
qt
m
be the =
⋯
a
tn1, 2, …,
qt t t,,
m



({ }
)
.If
=
a
0
qj
m
(
≠
qj
), then we have
≤∏−+− +≤ ≤ + = ∈
=
aμkABγ
sk sk m s k s j n
1
!, 1 1 , 1, 2,…, 1, 2, …,
.
jj
mμ
s
s
1
∣∣ (( ) ∣ ∣∣∣) () { }
The aforementioned estimates are sharp for <≤γ0
1
,
=−A
1
,
=
B1
,=+msk
1
,=
s
1, 2,…,and <≤γ0
1
,
=− ∈Aα α21 0,1([)
)
,
=
B1
,=+msk
1
,=
s
1, 2,…. In particular, if
=
k1
,then
≤∏+−
−∈=
=
−
aμABγ
mjnm
1! ,1,2,…,;2,3,….
jj
mμ
m0
2
∣∣ ( ∣ ∣∣∣)
() {}
Proof. Since =′fz fz f z f z,,…,
n12
() (() () ()), then
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
∂
∂⋯∂
∂⋯∂
∂
⋮⋱⋮⋱⋮
∂
∂⋯∂
∂⋯∂
∂
⋮⋱⋮⋱⋮
∂
∂⋯∂
∂⋯∂
∂
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
D
fz
fz
zfz
zfz
z
fz
z
fz
z
fz
z
fz
zfz
zfz
z
.
jn
jj
j
j
n
nn
j
n
n
1
1
11
1
1
()
() () ()
() () ()
() () ()
6Xiaoying Sima et al.

For
′∈zD0, …, , …,0
,
jppp,,…,
n12
()
it is easy to see that ′
D
fz0, …, , …,0
j
(( ))
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
∂′
∂⋯∂′
∂⋯∂′
∂
⋮⋱⋮⋱⋮
∂′
∂⋯∂′
∂⋯∂′
∂
⋮⋱⋮⋱⋮
∂′
∂⋯∂′
∂⋯∂′
∂
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
fz
z
fz
z
fz
z
fz
z
fz
z
fz
z
fz
z
fz
z
fz
z
0, …, , …,0 0, …, , …,0 0, …, , …,0
0, …, , …,0 0, …, , …,0 0, …, , …,0
0, …, , …,0 0, …, , …,0 0, …, , …,0
.
jj
j
j
n
jjjj
j
jj
n
njnj
j
nj
n
1
1
11
1
1
(( )) (( )) (( ))
(( )) (( )) (( ))
(( )) (( )) (( ))
(18)
Since =≠
a
qj0
qj
m
(
)
, it follows that
∂′
∂== ≠
fz
zqnqj
0, …, , …,0 0, 1, 2,…, ,
.
qj
j
(( )) (19)
From (18) and (19), then ′
D
fz0, …, , …,0
j
(( ))
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
∂′
∂⋯⋯
∂′
∂
⋮⋱⋮⋱⋮
∂′
∂⋯∂′
∂⋯∂′
∂
⋮⋱⋮⋱⋮
∂′
∂⋯⋯
∂′
∂
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
fz
z
fz
z
fz
z
fz
z
fz
z
fz
z
fz
z
0, …, , …,0 00, …, , …,0
0, …, , …,0 0, …, , …,0 0, …, , …,0
0, …, , …,0 00, …, , …,0
.
jj
n
jjjj
j
jj
n
njnj
n
1
1
1
1
1
(( )) (( ))
(( )) (( )) (( ))
(( )) (( ))
(20)
In view that ′
D
fz0, …, , …,0
j
(( ))is invertible, thus, it implies that
∂′
∂≠∈
fz
zjn
0, …, , …,0 0, 1, 2, …,
.
jj
j
(( )) {}
(21)
Using (20) and (21), then we obtain
′=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
∗⋯ ⋯ ∗
⋮⋱ ⋮ ⋱⋮
∗⋯ ⋯ ∗
⋮⋱ ⋮ ⋱⋮
∗⋯ ⋯ ∗
⎞
⎠
⎟
⎟
⎟
⎟
⎟
−∂′
∂
Df z0, …, , …,0
0
1
0
,
jfz
z
10,…, ,…,0
jj
j
((( ))) (( )) (22)
where the symbol
∗
means the unknown term. Furthermore, a simple computation in (22) shows that
′′
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
∗⋯ ⋯ ∗
⋮⋱ ⋮ ⋱⋮
∗⋯ ⋯ ∗
⋮⋱ ⋮ ⋱⋮
∗⋯ ⋯ ∗
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎜
⋮′
⋮
⎞
⎠
⎟
⎟
⎟
⎟
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⋮′
⋮
⎞
⎠
⎟
⎟
⎟
⎟
⎟
−
∂′
∂
∂′
∂
Df z f z
fz fz
0, …, , …,0 0, …, , …,0
0
1
0
0
0, …, , …,0
0
0
0, …, , …,0
0
.
jj
fz
zjjjj
fz
z
1
0,…, ,…,0 0,…, ,…,0
jj
j
jj
j
( (( ))) (( ))
(( )) (( ))
(( )) (( ))
Let == ′
−
DfzfzWzWzWz Wz,…, ,…,
jn
11
(()()) ()(() () ())and
=′∈
h
zf z H0, …, , …,0
jj jj
() (( )) (
)
,∈z
.
j
Since
()
∈
∗
fz S D
γAB pp p
,, ,,…,
n12
()
and =z0is a zero of order
+
k1
of −fz z() , it follows that
Some results of homogeneous expansions 7

⎟
⎜
+⎛
⎝
′−⎞
⎠=+
⎛
⎝
⎜
⎜′−
⎞
⎠
⎟
⎟
=+
⎛
⎝
⎜
⎜
′
′−
⎞
⎠
⎟
⎟∈≠
∂′
∂
∂′
∂
γ
zh z
hz γ
z
fz
γ
ρz
Wz gz
111110, …, , …,0 1
110, …, , …,0
20,…,,…,0
1,0.
jjj
jj
jfz
z
jj
j
ρz
zjj
0,…, ,…,0
0,…, ,…,0
jj
j
j
j

()
() (( ))
(( ))
(( )) ()
(( ))
(( ))
(23)
From (23), we find that ∈
∗
h
S
jγAB,,
(
)
, and =z0
jis a zero of order
+
k1

of
−
h
zz
jj
j
()
, where
≥
k
k

.We
note that
==
a
h
mm
0
!,2,3,….
jj
mjm
()
()
(24)
Thus, in view of Lemma 2.2 and (24), we obtain the desired results.
Furthermore, let <≤γ0
1
and
=⎛
⎝
⎜−− −
⎞
⎠
⎟′= ∈
−− −
fz z
z
z
z
z
zzzz z D
1,1,…, 1,,,… \0
.
kk
n
n
knppp
1
1
2
2
12 ,,…,
αγ
kαγ
kαγ
kn
21 21 21 12
() ()() () () {}
() () () (25)
It is easy to check that
()
∈
−
∗
fS D
γα pp p
,2 1,1 ,,…,
n12
. Thus, we have
∑
′= + ∈
=
∞++
fz zazj n0, 0, …, , …,0 , 1,2, …,
jjj
sjj
sk j
sk
1
11
(( )) {
}
and
=∏−+ − =∈
+=
aμkγα
sk sjn
121
!, 1,2,…, 1,2,…,
.
jj
sk μ
s
s
11
∣∣ (( ) ( )) {}
This completes the proof. □
Theorem 3.2. Suppose that
()
=′∈
∗
fz fz f z f z S D,,…,
nγAB pp p
12 ,, ,,…,
n12
() (() () ())
and =z0is a zero of order
+
k1
of
−fz z() .Define
∑
=⋯
=
Df z
mazzz
0
!
,
mqm
ll l
n
ql l l l l l
,,…, 1 ,,…,
m
mm
12
12 12
()( )
where
=
qn
1, 2,…,
,+≤ ≤ + =
s
kmsks11,1,2,…() ,and
∈
+
k

.Let
a
tq
m
be the =
⋯
−
a
tn1, 2, …,
tqt t t,,
m1



({ })
.If
=
a
0
jq
m(
≠
qj
), then we have
≤∏−+− +≤ ≤ + = ∈
=
aμkABγ
sk sk m s k s j n
1
!, 1 1 , 1, 2,…, 1, 2, …,
.
jj
mμ
s
s
1
∣∣ (( ) ∣ ∣∣ ∣) () { }
The aforementioned estimates are sharp for <≤γ0
1
,
=−A
1
,
=
B1
,=+msk
1
,=
s
1, 2,… and <≤γ0
1
,
=− ∈Aα α21 0,1([)
)
,
=
B1
,=+msk
1
,=
s
1, 2,…. In particular, if
=
k1
,then
≤∏+−
−∈=
=
−
aμABγ
mjnm
1! ,1,2,…,;2,3,….
jj
mμ
m0
2
∣∣ ( ∣ ∣∣ ∣)
() {}
Proof. Since =
a
0
jq
m(
≠
qj
), it follows that
∂′
∂=
fz
z
0, …, , …,0 0
.
jj
q
(( ))
(26)
Using (18) and (26), we have ′
D
fz0, …, , …,0
j
(( ))
8Xiaoying Sima et al.

=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
∂′
∂⋯∂′
∂⋯∂′
∂
⋮⋱⋮⋱⋮
⋯∂′
∂⋯
⋮⋱⋮⋱⋮
∂′
∂⋯∂′
∂⋯∂′
∂
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
fz
z
fz
z
fz
z
fz
z
fz
z
fz
z
fz
z
0, …, , …,0 0, …, , …,0 0, …, , …,0
00, …, , …,0 0
0, …, , …,0 0, …, , …,0 0, …, , …,0
.
jj
j
j
n
jj
j
njnj
j
nj
n
1
1
11
1
(( )) (( )) (( ))
(( ))
(( )) (( )) (( ))
(27)
In view that ′
D
fz0, …, , …,0
j
(( ))is invertible, thus, it implies that
∂′
∂≠
fz
z
0, …, , …,0 0
.
jj
j
(( ))
(28)
Thus, (27) and (28) show that
′=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
∗⋯ ∗ ⋯ ∗
⋮⋱ ⋮ ⋱⋮
⋯⋯
⋮⋱ ⋮ ⋱⋮
∗⋯ ∗ ⋯ ∗
⎞
⎠
⎟
⎟
⎟
⎟
⎟
−∂′
∂
Df z0, …, , …,0 010,
jfz
z
10,…, ,…,0
jj
j
((( ))) (( )) (29)
where the symbol
∗
means the unknown term. Furthermore, a simple computation in (29) shows that
′′
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
∗⋯ ∗ ⋯ ∗
⋮⋱ ⋮ ⋱⋮
⋯⋯
⋮⋱ ⋮ ⋱⋮
∗⋯ ∗ ⋯ ∗
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎜
′
⋮′
⋮′
⎞
⎠
⎟
⎟
⎟
⎟
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
∗
⋮′
⋮
∗
⎞
⎠
⎟
⎟
⎟
⎟
⎟
−
∂′
∂
∂′
∂
Df z f z
fz
fz
fz
fz
0, …, , …,0 0, …, , …,0
010
0, …, , …,0
0, …, , …,0
0, …, , …,0
0, …, , …,0 .
jj
fz
z
j
jj
nj
jj
fz
z
1
0,…, ,…,0
1
0,…, ,…,0
jj
j
jj
j
( (( ))) (( ))(( ))
(( ))
(( )) (( ))
(( )) (( ))
Define == ′
−
DfzfzWzWzWz Wz,…, ,…,
jn
11
(()()) ()(() () ())
and =′∈
h
zf z H0, …, , …,0
.
jj jj() (( )) ()Since
∈
fz()
()
∗
SD
γAB pp p
,, ,,…,
n12 and =z0is a zero of order
+
k1
of −fz z() , it follows that
⎟
⎜
+⎛
⎝
′−⎞
⎠=+
⎛
⎝
⎜
⎜′−
⎞
⎠
⎟
⎟
=+
⎛
⎝
⎜
⎜
′
′−
⎞
⎠
⎟
⎟∈≠
∂′
∂
∂′
∂
γ
zh z
hz γ
z
fz
γ
ρz
Wz gz
111110, …, , …,0 1
110, …, , …,0
20,…,,…,0
1,0.
jjj
jj
jfz
z
jj
j
ρz
zjj
0,…, ,…,0
0,…, ,…,0
jj
j
j
j

()
() (( ))
(( ))
(( )) ()
(( ))
(( ))
(30)
From (30), we find that ∈
∗
h
S
jγAB,,
(
)
, and =z0
jis a zero of order
+
k1

of
−
h
zz
jj
j
()
, where
≥
k
k

.We
note that
==
a
h
mm
0
!,2,3,….
jj
mjm
()
()
(31)
Thus, in view of Lemma 2.2 and (31), we obtain the desired results. We note that the sharpness of the estimates
of Theorem 3.2 is similar to that of Theorem 3.1. This completes the proof. □
Some results of homogeneous expansions 9

Remark 3.1.
(i) If
=
n1
, then Theorem 3.1 (resp. Theorem 3.2) reduces to Lemma 2.2.
(ii) If
=γ
1
,
=− ∈Aα α21 0,1([)
)
, and
=
B1
in Theorems 3.1 and 3.2, then we obtain Theorems 3.1 and 3.2 in
Liu et al. [32], respectively.
(iii) If we take different functions
g
in Theorem 3.1 (resp. Theorem 3.2), then the bounds of homogeneous
expansions for kinds of subclasses of
g
-starlike mappings of complex order
γ
defined on
D
pp p,,…,
n
12
can be
obtained immediately.
Acknowledgement: The authors would like to thank the referees for their helpful comments.
Funding information: The project is supported by the National Natural Science Foundation of China (12071354,
12061035), Jiangxi Provincial Natural Science Foundation (20212BAB201012), Research Foundation of Jiangxi
Provincial Department of Education (GJJ201104), and Research Foundation of Jiangxi Science and Technology
Normal University (2021QNBJRC003).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and
approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or
analyzed during this current study.
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Some results of homogeneous expansions 11