Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 2 Issue 1 (2010), Pages 25-41.
SOME PROPERTIES OF CERTAIN SUBCLASSES OF
MULTIVALENT FUNCTIONS INVOLVING THE
DZIOK-SRIVASTAVA OPERATOR
ZHI-GANG WANG, HAI-PING SHI
Abstract. The main purpose of the present paper is to derive such results as
inclusion relationships and convolution properties for certain new subclasses
of multivalent analytic functions involving the Dziok-Srivastava operator. The
results presented here would provide extensions of those given in earlier works.
Several other new results are also obtained.
1. Introduction and Preliminaries
Let 𝒜𝑝 denote the class of functions of the form
𝑓 (𝑧) = 𝑧 𝑝 +
∞
∑
𝑎𝑛+𝑝 𝑧 𝑛+𝑝
(𝑝 ∈ ℕ := {1, 2, 3, . . .}),
(1.1)
𝑛=1
which are analytic in the open unit disk
𝕌 := {𝑧 : 𝑧 ∈ ℂ and
∣𝑧∣ < 1}.
For simplicity, we write
𝒜1 =: 𝒜.
Let 𝑓, 𝑔 ∈ 𝒜𝑝 , where 𝑓 is given by (1.1) and 𝑔 is defined by
𝑔(𝑧) = 𝑧 𝑝 +
∞
∑
𝑏𝑛+𝑝 𝑧 𝑛+𝑝 .
𝑛=1
Then the Hadamard product (or convolution) 𝑓 ∗ 𝑔 of the functions 𝑓 and 𝑔 is
defined by
∞
∑
𝑝
(𝑓 ∗ 𝑔)(𝑧) := 𝑧 +
𝑎𝑛+𝑝 𝑏𝑛+𝑝 𝑧 𝑛+𝑝 =: (𝑔 ∗ 𝑓 )(𝑧).
𝑛=1
For parameters
𝛼𝑗 ∈ ℂ (𝑗 = 1, . . . , 𝑙)
−
and 𝛽𝑗 ∈ ℂ ∖ ℤ−
0 (ℤ0 := {0, −1, −2, . . .}; 𝑗 = 1, . . . , 𝑚),
2000 Mathematics Subject Classification. 30C45, 30C80.
Key words and phrases. Analytic functions; multivalent functions; subordination between analytic functions; Hadamard product (or convolution); Dziok-Srivastava operator; symmetric points;
conjugate points; symmetric conjugate points.
c
⃝2010
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted February, 2010. Published February, 2010.
The present investigation was supported by the Scientific Research Fund of Hunan Provincial
Education Department under Grant 08C118 of the People’s Republic of China.
25
26
Z.-G. WANG, H.-P. SHI
the generalized hypergeometric function
𝑙 𝐹𝑚 (𝛼1 , . . . , 𝛼𝑙 ; 𝛽1 , . . . , 𝛽𝑚 ; 𝑧)
is defined by the following infinite series:
𝑙 𝐹𝑚 (𝛼1 , . . . , 𝛼𝑙 ; 𝛽1 , . . . , 𝛽𝑚 ; 𝑧)
:=
∞
∑
(𝛼1 )𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛 𝑧 𝑛
(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛 𝑛!
𝑛=0
(𝑙 ≦ 𝑚 + 1; 𝑙, 𝑚 ∈ ℕ0 := ℕ ∪ {0}; 𝑧 ∈ 𝕌),
where (𝜆)𝑛 is the Pochhammer symbol defined by
⎧
⎨ 1
(𝜆)𝑛 :=
⎩
𝜆(𝜆 + 1) ⋅ ⋅ ⋅ (𝜆 + 𝑛 − 1)
(𝑛 = 0),
(𝑛 ∈ ℕ).
Recently, Dziok and Srivastava [8] introduced a linear operator
𝐻𝑝 (𝛼1 , . . . , 𝛼𝑙 ; 𝛽, . . . , 𝛽𝑚 ) : 𝒜𝑝 −→ 𝒜𝑝
defined by the Hadamard product
𝐻𝑝 (𝛼1 , . . . , 𝛼𝑙 ; 𝛽1 , . . . , 𝛽𝑚 )𝑓 (𝑧) := [𝑧 𝑝 𝑙 𝐹𝑚 (𝛼1 , . . . , 𝛼𝑙 ; 𝛽1 , . . . , 𝛽𝑚 )] ∗ 𝑓 (𝑧)
(1.2)
(𝑙 ≦ 𝑚 + 1; 𝑙, 𝑚 ∈ ℕ0 ; 𝑧 ∈ 𝕌).
If 𝑓 ∈ 𝒜𝑝 is given by (1.1), then we have
𝐻𝑝 (𝛼1 , . . . , 𝛼𝑙 ; 𝛽1 , . . . , 𝛽𝑚 )𝑓 (𝑧) = 𝑧 𝑝 +
∞
∑
𝑧 𝑛+𝑝
(𝛼1 )𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛
𝑎𝑛+𝑝
(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛
𝑛!
𝑛=1
(𝑛 ∈ ℕ).
In order to make the notation simple, we write
𝐻𝑝𝑙,𝑚 (𝛼1 ) := 𝐻𝑝 (𝛼1 , . . . , 𝛼𝑙 ; 𝛽1 , . . . , 𝛽𝑚 )
(𝑙 ≦ 𝑚 + 1; 𝑙, 𝑚 ∈ ℕ0 ).
It is easily verified from the definition (1.2) that
(
)′
𝑧 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧) = 𝛼1 𝐻𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝑧) − (𝛼1 − 𝑝)𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧).
(1.3)
Let 𝒫 denote the class of functions of the form
∞
∑
𝑝𝑛 𝑧 𝑛 ,
𝑝(𝑧) = 1 +
𝑛=1
which are analytic and convex in 𝕌 and satisfy the condition
ℜ(𝑝(𝑧)) > 0
(𝑧 ∈ 𝕌).
For two functions 𝑓 and 𝑔, analytic in 𝕌, we say that the function 𝑓 is subordinate
to 𝑔 in 𝕌, and write
𝑓 (𝑧) ≺ 𝑔(𝑧)
(𝑧 ∈ 𝕌),
if there exists a Schwarz function 𝜔, which is analytic in 𝕌 with
𝜔(0) = 0
and
∣𝜔(𝑧)∣ < 1
(𝑧 ∈ 𝕌)
such that
(
)
𝑓 (𝑧) = 𝑔 𝜔(𝑧)
(𝑧 ∈ 𝕌).
Indeed, it is known that
𝑓 (𝑧) ≺ 𝑔(𝑧)
(𝑧 ∈ 𝕌) =⇒ 𝑓 (0) = 𝑔(0)
and 𝑓 (𝕌) ⊂ 𝑔(𝕌).
SOME PROPERTIES OF CERTAIN SUBCLASSES OF MULTIVALENT FUNCTIONS
27
Furthermore, if the function 𝑔 is univalent in 𝕌, then we have the following equivalence:
𝑓 (𝑧) ≺ 𝑔(𝑧) (𝑧 ∈ 𝕌) ⇐⇒ 𝑓 (0) = 𝑔(0) and 𝑓 (𝕌) ⊂ 𝑔(𝕌).
Throughout this paper, we assume that
(
)
2𝜋𝑖
𝑝, 𝑘 ∈ ℕ, 𝑙, 𝑚 ∈ ℕ0 , 𝜀𝑘 = exp
,
𝑘
𝑘−1
)
1 ∑ −𝑗𝑝 ( 𝑙,𝑚
𝐻𝑝 (𝛼1 )𝑓 (𝜀𝑗𝑘 𝑧) = 𝑧 𝑝 + ⋅ ⋅ ⋅
𝜀𝑘
(𝑓 ∈ 𝒜𝑝 ),
(1.4)
𝑘 𝑗=0
]
1 [ 𝑙,𝑚
𝐻𝑝 (𝛼1 )𝑓 (𝑧) + 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧) = 𝑧 𝑝 + ⋅ ⋅ ⋅
(𝑓 ∈ 𝒜𝑝 ), (1.5)
𝑔𝑝𝑙,𝑚 (𝛼1 ; 𝑧) =
2
and
]
1 [ 𝑙,𝑚
ℎ𝑙,𝑚
𝐻𝑝 (𝛼1 )𝑓 (𝑧) − 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (−𝑧) = 𝑧 𝑝 + ⋅ ⋅ ⋅
(𝑓 ∈ 𝒜𝑝 ). (1.6)
𝑝 (𝛼1 ; 𝑧) =
2
Clearly, for 𝑘 = 1, we have
𝑙,𝑚
𝑓𝑝,𝑘
(𝛼1 ; 𝑧) =
𝑙,𝑚
𝑓𝑝,1
(𝛼1 ; 𝑧) = 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧).
In recent years, several authors obtained many interesting results involving the
Dziok-Srivastava operator 𝐻𝑝𝑙,𝑚 (𝛼1 ) (see, for details, [1, 2, 3, 4, 6, 7, 8, 9, 11, 13,
14, 15, 17, 18, 19, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 34]). In the present paper,
by making use of the Dziok-Srivastava operator 𝐻𝑝𝑙,𝑚 (𝛼1 ) and the above-mentioned
principle of subordination between analytic functions, we introduce and investigate
the following subclasses of the class 𝒜𝑝 of 𝑝-valent analytic functions.
𝑙,𝑚
Definition 1.1. A function 𝑓 ∈ 𝒜𝑝 is said to be in the class ℱ𝑝,𝑘
(𝛼; 𝛼1 ; 𝜙) if it
satisfies the subordination condition
]
[
)′
)′
(
(
𝑧 (1 − 𝛼) 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧) + 𝛼 𝐻𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝑧)
[
]
≺ 𝜙(𝑧)
(𝑧 ∈ 𝕌), (1.7)
𝑙,𝑚
𝑙,𝑚
𝑝 (1 − 𝛼)𝑓𝑝,𝑘
(𝛼1 ; 𝑧) + 𝛼𝑓𝑝,𝑘
(𝛼1 + 1; 𝑧)
𝑙,𝑚
where 𝜙 ∈ 𝒫, 𝑓𝑝,𝑘
(𝛼1 ; 𝑧) is defined by (1.4) and
𝑙,𝑚
𝑓𝑝,𝑘
(𝛼1 + 1; 𝑧) ∕= 0
(𝑧 ∈ 𝕌).
For simplicity, we write
𝑙,𝑚
𝑙,𝑚
ℱ𝑝,𝑘
(0; 𝛼1 ; 𝜙) =: ℱ𝑝,𝑘
(𝛼1 ; 𝜙).
Remark 1.1. If we set
𝑝 = 1, 𝑙 = 2, 𝑚 = 1,
and 𝛼1 = 𝛼2 = 𝛽1 = 1
𝑙,𝑚
ℱ𝑝,𝑘
(𝛼; 𝛼1 ; 𝜙),
in the class
then it reduces to the known class 𝒮 (𝑘) (𝛼; 𝜙) of functions 𝛼-starlike with respect to 𝑘-symmetric points, which was studied earlier by
Parvatham and Radha [21]. If we set
𝑝 = 1, 𝑙 = 2, 𝑚 = 1, 𝛼1 = 𝛼2 = 𝛽1 = 1,
𝑙,𝑚
ℱ𝑝,𝑘
(𝛼; 𝛼1 ; 𝜙),
and 𝛼 = 0
(𝑘)
in the class
then it reduces to the class 𝒮 (𝜙) of functions starlike
with respect to 𝑘-symmetric points, which was considered by Wang et al. [33].
𝑙,𝑚
Furthermore, we note that the class ℱ𝑝,𝑘
(𝛼1 ; 𝜙) was introduced and investigated
recently by Wang et al. [34] (see also Huang and Liu [12] and Xu and Yang [35]).
28
Z.-G. WANG, H.-P. SHI
Definition 1.2. A function 𝑓 ∈ 𝒜𝑝 is said to be in the class 𝒢𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙) if it
satisfies the subordination condition
[
]
(
)′
(
)′
𝑧 (1 − 𝛼) 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧) + 𝛼 𝐻𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝑧)
[
]
≺ 𝜙(𝑧)
(𝑧 ∈ 𝕌),
𝑝 (1 − 𝛼)𝑔𝑝𝑙,𝑚 (𝛼1 ; 𝑧) + 𝛼𝑔𝑝𝑙,𝑚 (𝛼1 + 1; 𝑧)
where 𝜙 ∈ 𝒫, 𝑔𝑝𝑙,𝑚 (𝛼1 ; 𝑧) is defined by (1.5) and
𝑔𝑝𝑙,𝑚 (𝛼1 + 1; 𝑧) ∕= 0
(𝑧 ∈ 𝕌).
For simplicity, we write
𝒢𝑝𝑙,𝑚 (0; 𝛼1 ; 𝜙) =: 𝒢𝑝𝑙,𝑚 (𝛼1 ; 𝜙).
Definition 1.3. A function 𝑓 ∈ 𝒜𝑝 is said to be in the class ℋ𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙) if it
satisfies the subordination condition
[
]
(
)′
(
)′
𝑧 (1 − 𝛼) 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧) + 𝛼 𝐻𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝑧)
]
[
≺ 𝜙(𝑧)
(𝑧 ∈ 𝕌),
𝑙,𝑚
𝑝 (1 − 𝛼)ℎ𝑙,𝑚
𝑝 (𝛼1 ; 𝑧) + 𝛼ℎ𝑝 (𝛼1 + 1; 𝑧)
where 𝜙 ∈ 𝒫, ℎ𝑙,𝑚
𝑝 (𝛼1 ; 𝑧) is defined by (1.6) and
ℎ𝑙,𝑚
𝑝 (𝛼1 + 1; 𝑧) ∕= 0
(𝑧 ∈ 𝕌).
For simplicity, we write
ℋ𝑝𝑙,𝑚 (0; 𝛼1 ; 𝜙) =: ℋ𝑝𝑙,𝑚 (𝛼1 ; 𝜙).
Remark 1.2. In 1996, Chen et al. [5] introduced and investigated a subclass
∗
𝒮𝑠𝑐
(𝛼) of 𝒜 consisting of functions which are 𝛼-starlike with respect to symmetric
conjugate points and satisfy the inequality
(
)
𝑧 [(1 − 𝛼)𝑓 ′ (𝑧) + 𝛼(𝑧𝑓 ′ (𝑧))′ ]
ℜ
>0
( 𝑧 ∈ 𝕌),
(1 − 𝛼)𝑇𝑠𝑐 𝑓 (𝑧) + 𝛼𝑧(𝑇𝑠𝑐 𝑓 (𝑧))′
where
𝑇𝑠𝑐 𝑓 (𝑧) =
]
1[
𝑓 (𝑧) − 𝑓 (−𝑧) .
2
It is easy to see that, if we set
𝑝 = 1, 𝑙 = 2, 𝑚 = 1, 𝛼1 = 𝛼2 = 𝛽1 = 1, and 𝜙(𝑧) =
1+𝑧
1−𝑧
∗
in the class ℋ𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙), then it reduces to the class 𝒮𝑠𝑐
(𝛼).
Definition 1.4. A function 𝑓 ∈ 𝒜𝑝 is said to be in the class 𝔉𝑙,𝑚
𝑝,𝑘 (𝛼; 𝛼1 ; 𝜙) if it
satisfies the subordination condition
[
]
(
)′
(
)′
𝑧 (1 − 𝛼) 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧) + 𝛼 𝐻𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝑧)
[
]
≺ 𝜙(𝑧)
(𝑧 ∈ 𝕌),
𝑙,𝑚
𝑝 (1 − 𝛼)𝔣𝑙,𝑚
𝑝,𝑘 (𝛼1 ; 𝑧) + 𝛼𝔣𝑝,𝑘 (𝛼1 + 1; 𝑧)
where 𝜙 ∈ 𝒫, 𝔣𝑙,𝑚
𝑝,𝑘 (𝛼1 ; 𝑧) is defined as in (1.4) and
𝔣𝑙,𝑚
𝑝,𝑘 (𝛼1 + 1; 𝑧) ∕= 0
(𝑧 ∈ 𝕌).
SOME PROPERTIES OF CERTAIN SUBCLASSES OF MULTIVALENT FUNCTIONS
29
For simplicity, we write
𝑙,𝑚
𝔉𝑙,𝑚
𝑝,𝑘 (0; 𝛼1 ; 𝜙) =: 𝔉𝑝,𝑘 (𝛼1 ; 𝜙).
Remark 1.3. If we set
𝑝 = 1, 𝑙 = 2, 𝑚 = 1,
and 𝛼1 = 𝛼2 = 𝛽1 = 1
𝔉𝑙,𝑚
𝑝,𝑘 (𝛼; 𝛼1 ; 𝜙),
in the class
then it reduces to the known class 𝒞 (𝑘) (𝛼, 𝜙) of functions
𝛼-close-to-convex with respect to 𝑘-symmetric points, which was also studied earlier
by Parvatham and Radha [21]. Furthermore, we also note that the class 𝔉𝑙,𝑚
𝑝,𝑘 (𝛼1 ; 𝜙)
was introduced and investigated recently by Wang et al. [34].
Definition 1.5. A function 𝑓 ∈ 𝒜𝑝 is said to be in the class 𝔊𝑙,𝑚
𝑝 (𝛼; 𝛼1 ; 𝜙) if it
satisfies the subordination condition
]
[
)′
)′
(
(
𝑧 (1 − 𝛼) 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧) + 𝛼 𝐻𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝑧)
]
[
≺ 𝜙(𝑧)
(𝑧 ∈ 𝕌),
𝑙,𝑚
(𝛼
+
1;
𝑧)
(𝛼
;
𝑧)
+
𝛼𝔤
𝑝 (1 − 𝛼)𝔤𝑙,𝑚
𝑝
𝑝
1
1
where 𝜙 ∈ 𝒫, 𝔤𝑙,𝑚
𝑝 (𝛼1 ; 𝑧) is defined as in (1.5) and
𝔤𝑙,𝑚
𝑝 (𝛼1 + 1; 𝑧) ∕= 0
(𝑧 ∈ 𝕌).
For simplicity, we write
𝑙,𝑚
𝔊𝑙,𝑚
𝑝 (0; 𝛼1 ; 𝜙) =: 𝔊𝑝 (𝛼1 ; 𝜙).
Definition 1.6. A function 𝑓 ∈ 𝒜𝑝 is said to be in the class ℌ𝑙,𝑚
𝑝 (𝛼; 𝛼1 ; 𝜙) if it
satisfies the subordination condition
]
[
)′
)′
(
(
𝑧 (1 − 𝛼) 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧) + 𝛼 𝐻𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝑧)
]
[
≺ 𝜙(𝑧)
(𝑧 ∈ 𝕌),
𝑙,𝑚
(𝛼
+
1;
𝑧)
(𝛼
;
𝑧)
+
𝛼𝔥
𝑝 (1 − 𝛼)𝔥𝑙,𝑚
𝑝
𝑝
1
1
where 𝜙 ∈ 𝒫, 𝔥𝑙,𝑚
𝑝 (𝛼1 ; 𝑧) is defined as in (1.6) and
𝔥𝑙,𝑚
𝑝 (𝛼1 + 1; 𝑧) ∕= 0
(𝑧 ∈ 𝕌).
For simplicity, we write
𝑙,𝑚
ℌ𝑙,𝑚
𝑝 (0; 𝛼1 ; 𝜙) =: ℌ𝑝 (𝛼1 ; 𝜙).
In order to establish our main results, we need the following lemmas.
Lemma 1.1. (See [10, 16]) Let 𝛽, 𝛾 ∈ ℂ. Suppose that 𝜙(𝑧) is convex and univalent
in 𝕌 with
𝜙(0) = 1 and ℜ(𝛽𝜙(𝑧) + 𝛾) > 0
(𝑧 ∈ 𝕌).
If 𝔭 is analytic in 𝕌 with 𝔭(0) = 1, then the subordination
𝔭(𝑧) +
𝑧𝔭′ (𝑧)
≺ 𝜙(𝑧)
𝛽𝔭(𝑧) + 𝛾
(𝑧 ∈ 𝕌)
implies that
𝔭(𝑧) ≺ 𝜙(𝑧)
(𝑧 ∈ 𝕌).
30
Z.-G. WANG, H.-P. SHI
Lemma 1.2. (See [20]) Let 𝛽, 𝛾 ∈ ℂ. Suppose that 𝜙(𝑧) is convex and univalent
in 𝕌 with
𝜙(0) = 1 and ℜ(𝛽𝜙(𝑧) + 𝛾) > 0
(𝑧 ∈ 𝕌).
Also let
𝔮(𝑧) ≺ 𝜙(𝑧)
(𝑧 ∈ 𝕌).
If 𝔭 ∈ 𝒫 and satisfies the subordination
𝔭(𝑧) +
𝑧𝔭′ (𝑧)
≺ 𝜙(𝑧)
𝛽𝔮(𝑧) + 𝛾
(𝑧 ∈ 𝕌),
then
𝔭(𝑧) ≺ 𝜙(𝑧)
(𝑧 ∈ 𝕌).
𝑙,𝑚
Lemma 1.3. Let 𝑓 ∈ ℱ𝑝,𝑘
(𝛼; 𝛼1 ; 𝜙). Then
[
]
(
)′
(
)′
𝑙,𝑚
𝑙,𝑚
𝑧 (1 − 𝛼) 𝑓𝑝,𝑘
(𝛼1 )𝑓 (𝑧) + 𝛼 𝑓𝑝,𝑘
(𝛼1 + 1)𝑓 (𝑧)
[
]
≺ 𝜙(𝑧)
𝑙,𝑚
𝑙,𝑚
𝑝 (1 − 𝛼)𝑓𝑝,𝑘
(𝛼1 ; 𝑧) + 𝛼𝑓𝑝,𝑘
(𝛼1 + 1; 𝑧)
(𝑧 ∈ 𝕌). (1.8)
Furthermore, if 𝜙 ∈ 𝒫 with
)
(
𝛼1
−𝑝 >0
(𝛼 > 0; 𝑧 ∈ 𝕌).
ℜ 𝑝𝜙(𝑧) +
𝛼
Then
(
)′
𝑙,𝑚
𝑧 𝑓𝑝,𝑘
(𝛼1 ; 𝑧)
≺ 𝜙(𝑧)
(𝑧 ∈ 𝕌).
𝑙,𝑚
𝑝𝑓𝑝,𝑘
(𝛼1 ; 𝑧)
Proof. Making use of (1.4), we have
𝑙,𝑚
𝑓𝑝,𝑘
(𝛼1 ; 𝜀𝑗𝑘 𝑧) =
𝑘−1
)
1 ∑ −𝑛𝑝 ( 𝑙,𝑚
𝜀𝑘
𝐻𝑝 (𝛼1 )𝑓 (𝜀𝑛+𝑗
𝑧)
𝑘
𝑘 𝑛=0
= 𝜀𝑗𝑝
𝑘 ⋅
𝑘−1
)
1 ∑ −(𝑛+𝑗)𝑝 ( 𝑙,𝑚
𝜀𝑘
𝐻𝑝 (𝛼1 )𝑓 (𝜀𝑛+𝑗
𝑧)
𝑘
𝑘 𝑛=0
𝑙,𝑚
= 𝜀𝑗𝑝
𝑘 𝑓𝑝,𝑘 (𝛼1 ; 𝑧)
(1.9)
(𝑗 ∈ {0, 1, . . . , 𝑘 − 1}),
and
(
𝑘−1
)′
)
1 ∑ −𝑗(𝑝−1) ( 𝑙,𝑚
𝑙,𝑚
𝑓𝑝,𝑘
(𝛼1 ; 𝑧) =
𝜀𝑘
𝐻𝑝 (𝛼1 )𝑓 (𝜀𝑗𝑘 𝑧).
𝑘 𝑛=0
(1.10)
Replacing 𝛼1 by 𝛼1 + 1 in (1.9) and (1.10), respectively, we get
𝑙,𝑚
𝑙,𝑚
𝑓𝑝,𝑘
(𝛼1 + 1; 𝜀𝑗𝑘 𝑧) = 𝜀𝑗𝑝
𝑘 𝑓𝑝,𝑘 (𝛼1 + 1; 𝑧)
(𝑗 ∈ {0, 1, . . . , 𝑘 − 1}),
(1.11)
𝑘−1
)′
)
1 ∑ −𝑗(𝑝−1) ( 𝑙,𝑚
𝑙,𝑚
𝜀𝑘
𝐻𝑝 (𝛼1 + 1)𝑓 (𝜀𝑗𝑘 𝑧).
𝑓𝑝,𝑘
(𝛼1 + 1; 𝑧) =
𝑘 𝑛=0
(1.12)
and
(
SOME PROPERTIES OF CERTAIN SUBCLASSES OF MULTIVALENT FUNCTIONS
31
From (1.9) to (1.12), we get
[
]
(
)′
(
)′
𝑙,𝑚
𝑙,𝑚
𝑧 (1 − 𝛼) 𝑓𝑝,𝑘
(𝛼1 )𝑓 (𝑧) + 𝛼 𝑓𝑝,𝑘
(𝛼1 + 1)𝑓 (𝑧)
[
]
𝑙,𝑚
𝑙,𝑚
𝑝 (1 − 𝛼)𝑓𝑝,𝑘
(𝛼1 ; 𝑧) + 𝛼𝑓𝑝,𝑘
(𝛼1 + 1; 𝑧)
]
[
(
)′
(
)′
𝑘−1 −𝑗(𝑝−1) 𝑧 (1 − 𝛼) 𝐻 𝑙,𝑚 (𝛼 )𝑓 (𝜀𝑗 𝑧) + 𝛼 𝐻 𝑙,𝑚 (𝛼 + 1)𝑓 (𝜀𝑗 𝑧)
1
1
𝑝
𝑝
𝑘
𝑘
1 ∑ 𝜀𝑘
[
]
=
𝑙,𝑚
𝑙,𝑚
𝑘 𝑗=0
𝑝 (1 − 𝛼)𝑓𝑝,𝑘
(𝛼1 ; 𝑧) + 𝛼𝑓𝑝,𝑘
(𝛼1 + 1; 𝑧)
[
( 𝑙,𝑚
)′ 𝑗
( 𝑙,𝑚
)′ 𝑗 ]
𝑗
𝑘−1
𝜀
𝑧
(1
−
𝛼)
𝐻
(𝛼
)𝑓
(𝜀
𝑧)
+
𝛼
𝐻
(𝛼
+
1)𝑓
(𝜀𝑘 𝑧)
∑
1
1
𝑝
𝑝
𝑘
𝑘
1
[
]
=
.
𝑘 𝑗=0
𝑝 (1 − 𝛼)𝑓 𝑙,𝑚 (𝛼 ; 𝜀𝑗 𝑧) + 𝛼𝑓 𝑙,𝑚 (𝛼 + 1; 𝜀𝑗 𝑧)
𝑝,𝑘
1
𝑘
𝑝,𝑘
1
𝑘
(1.13)
𝑞,𝑠
Moreover, since 𝑓 ∈ ℱ𝑝,𝑘
(𝛼; 𝛼1 ; 𝜙), it follows that
[
]
)′
)′
(
(
𝜀𝑗𝑘 𝑧 (1 − 𝛼) 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝜀𝑗𝑘 𝑧) + 𝛼 𝐻𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝜀𝑗𝑘 𝑧)
[
]
≺ 𝜙(𝑧)
𝑙,𝑚
𝑙,𝑚
𝑝 (1 − 𝛼)𝑓𝑝,𝑘
(𝛼1 ; 𝜀𝑗𝑘 𝑧) + 𝛼𝑓𝑝,𝑘
(𝛼1 + 1; 𝜀𝑗𝑘 𝑧)
(1.14)
(𝑧 ∈ 𝕌; 𝑗 ∈ {0, 1, . . . , 𝑘 − 1}).
By noting that 𝜙(𝑧) is convex and univalent in 𝕌, from (1.13) and (1.14), we
conclude that the assertion (1.8) of Lemma 1.3 holds true.
Next, making use of the relationships (1.3) and (1.4), we know that
𝑘−1
(
)′
)
𝛼1 ∑ −𝑗𝑝 ( 𝑙,𝑚
𝑙,𝑚
𝑙,𝑚
𝑧 𝑓𝑝,𝑘
𝜀𝑘
𝐻𝑝 (𝛼1 + 1)𝑓 (𝜀𝑗𝑘 𝑧)
(𝛼1 ; 𝑧) + (𝛼1 − 𝑝)𝑓𝑝,𝑘
(𝛼1 ; 𝑧) =
𝑘 𝑗=0
𝑙,𝑚
= 𝛼1 𝑓𝑝,𝑘
(𝛼1 + 1; 𝑧).
(1.15)
𝑙,𝑚
Let 𝑓 ∈ ℱ𝑝,𝑘
(𝛼; 𝛼1 ; 𝜙) and suppose that
𝜓(𝑧) =
(
)′
𝑙,𝑚
𝑧 𝑓𝑝,𝑘
(𝛼1 ; 𝑧)
𝑙,𝑚
𝑝𝑓𝑝,𝑘
(𝛼1 ; 𝑧)
(𝑧 ∈ 𝕌).
(1.16)
Then 𝜓 is analytic in 𝕌 and 𝜓(0) = 1. It follows from (1.15) and (1.16) that
𝛼1 − 𝑝 + 𝑝𝜓(𝑧) = 𝛼1
𝑙,𝑚
𝑓𝑝,𝑘
(𝛼1 + 1; 𝑧)
𝑙,𝑚
𝑓𝑝,𝑘
(𝛼1 ; 𝑧)
(𝑧 ∈ 𝕌).
(1.17)
From (1.16) and (1.17), we have
(
)′
𝑝
𝑙,𝑚
𝑙,𝑚
𝑧 𝑓𝑝,𝑘
(𝛼1 + 1; 𝑧) =
[𝑧𝜓 ′ (𝑧) + (𝛼1 − 𝑝 + 𝑝𝜓(𝑧))𝜓(𝑧)]𝑓𝑝,𝑘
(𝛼1 ; 𝑧).
𝛼1
(1.18)
32
Z.-G. WANG, H.-P. SHI
It now follows from (1.8), (1.16), (1.17) and (1.18) that
[
]
(
)′
(
)′
𝑙,𝑚
𝑙,𝑚
𝑧 (1 − 𝛼) 𝑓𝑝,𝑘
(𝛼1 )𝑓 (𝑧) + 𝛼 𝑓𝑝,𝑘
(𝛼1 + 1)𝑓 (𝑧)
[
]
𝑙,𝑚
𝑙,𝑚
𝑝 (1 − 𝛼)𝑓𝑝,𝑘
(𝛼1 ; 𝑧) + 𝛼𝑓𝑝,𝑘
(𝛼1 + 1; 𝑧)
=
=
=
𝑙,𝑚
𝛼
′
𝛼1 𝑝[𝑧𝜓 (𝑧) + (𝛼1 − 𝑝 + 𝑝𝜓(𝑧))𝜓(𝑧)]𝑓𝑝,𝑘 (𝛼1 ; 𝑧)
𝑙,𝑚
𝑙,𝑚
𝑝(1 − 𝛼)𝑓𝑝,𝑘
(𝛼1 ; 𝑧) + 𝛼𝛼1 𝑝(𝛼1 − 𝑝 + 𝑝𝜓(𝑧))𝑓𝑝,𝑘
(𝛼1 ; 𝑧)
𝛼
′
𝛼)𝜓(𝑧) + 𝛼1 [𝑧𝜓 (𝑧) + (𝛼1 − 𝑝 + 𝑝𝜓(𝑧))𝜓(𝑧)]
(1 − 𝛼) + 𝛼𝛼1 (𝛼1 − 𝑝 + 𝑝𝜓(𝑧))
𝑙,𝑚
𝑝(1 − 𝛼)𝜓(𝑧)𝑓𝑝,𝑘
(𝛼1 ; 𝑧) +
(1 −
𝛼
′
𝛼1 𝑧𝜓 (𝑧)
[
+ (1 − 𝛼) +
(1 − 𝛼) +
𝛼
𝛼1 (𝛼1
𝛼
𝛼1 (𝛼1
]
− 𝑝 + 𝑝𝜓(𝑧)) 𝜓(𝑧)
− 𝑝 + 𝑝𝜓(𝑧))
′
= 𝜓(𝑧) +
𝛼1
𝛼
𝑧𝜓 (𝑧)
≺ 𝜙(𝑧).
− 𝑝 + 𝑝𝜓(𝑧)
(1.19)
Since
)
(
𝛼1
−𝑝 >0
(𝛼 > 0; 𝑧 ∈ 𝕌).
ℜ 𝑝𝜙(𝑧) +
𝛼
Thus, by (1.19) and Lemma 1.1, we know that
(
)′
𝑙,𝑚
𝑧 𝑓𝑝,𝑘
(𝛼1 ; 𝑧)
𝜓(𝑧) =
≺ 𝜙(𝑧).
𝑙,𝑚
𝑝𝑓𝑝,𝑘
(𝛼1 ; 𝑧)
This completes the proof of Lemma 1.3.
□
By similarly applying the method of proof of Lemma 1.3 for the classes 𝒢𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙)
and ℋ𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙), we get the following results.
Lemma 1.4. Let 𝑓 ∈ 𝒢𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙). Then
]
[
)′
)′
(
(
𝑧 (1 − 𝛼) 𝑔𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧) + 𝛼 𝑔𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝑧)
]
[
≺ 𝜙(𝑧)
𝑝 (1 − 𝛼)𝑔𝑝𝑙,𝑚 (𝛼1 ; 𝑧) + 𝛼𝑔𝑝𝑙,𝑚 (𝛼1 + 1; 𝑧)
(𝑧 ∈ 𝕌).
Furthermore, if 𝜙 ∈ 𝒫 with
(
)
𝛼1
ℜ 𝑝𝜙(𝑧) +
−𝑝 >0
(𝛼 > 0; 𝑧 ∈ 𝕌).
𝛼
Then
(
)′
𝑧 𝑔𝑝𝑙,𝑚 (𝛼1 ; 𝑧)
≺ 𝜙(𝑧)
(𝑧 ∈ 𝕌).
𝑝𝑔𝑝𝑙,𝑚 (𝛼1 ; 𝑧)
Lemma 1.5. Let 𝑓 ∈ ℋ𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙). Then
[
]
(
)′
( 𝑙,𝑚
)′
𝑧 (1 − 𝛼) ℎ𝑙,𝑚
(𝑧)
+
𝛼
ℎ
(𝑧)
(𝛼
)𝑓
(𝛼
+
1)𝑓
1
1
𝑝
𝑝
[
]
≺ 𝜙(𝑧)
𝑙,𝑚
𝑙,𝑚
𝑝 (1 − 𝛼)ℎ𝑝 (𝛼1 ; 𝑧) + 𝛼ℎ𝑝 (𝛼1 + 1; 𝑧)
(𝑧 ∈ 𝕌).
SOME PROPERTIES OF CERTAIN SUBCLASSES OF MULTIVALENT FUNCTIONS
33
Furthermore, if 𝜙 ∈ 𝒫 with
(
)
𝛼1
ℜ 𝑝𝜙(𝑧) +
−𝑝 >0
𝛼
(𝛼 > 0; 𝑧 ∈ 𝕌).
Then
(
)′
𝑧 ℎ𝑙,𝑚
𝑝 (𝛼1 ; 𝑧)
𝑝ℎ𝑙,𝑚
𝑝 (𝛼1 ; 𝑧)
≺ 𝜙(𝑧)
(𝑧 ∈ 𝕌).
In the present paper, we aim at proving such results as inclusion relationships
𝑙,𝑚
and convolution properties for the function classes ℱ𝑝,𝑘
(𝛼; 𝛼1 ; 𝜙), 𝒢𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙),
𝑙,𝑚
𝑙,𝑚
ℋ𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙), 𝔉𝑙,𝑚
𝑝,𝑘 (𝛼; 𝛼1 ; 𝜙), 𝔊𝑝 (𝛼; 𝛼1 ; 𝜙), and ℌ𝑝 (𝛼; 𝛼1 ; 𝜙). The results presented here would provide extensions of those given in earlier works. Several other
new results are also obtained.
2. A Set of Inclusion Relationships
𝑙,𝑚
At first, we provide some inclusion relationships for the classes ℱ𝑝,𝑘
(𝛼; 𝛼1 ; 𝜙),
𝑙,𝑚
𝑙,𝑚
𝒢𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙), ℋ𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙), 𝔉𝑙,𝑚
𝑝,𝑘 (𝛼; 𝛼1 ; 𝜙), 𝔊𝑝 (𝛼; 𝛼1 ; 𝜙), and ℌ𝑝 (𝛼; 𝛼1 ; 𝜙), which
were defined in the preceding section.
Theorem 2.1. Let 𝜙 ∈ 𝒫 with
(
)
𝛼1
ℜ 𝑝𝜙(𝑧) +
−𝑝 >0
𝛼
(𝛼 > 0; 𝑧 ∈ 𝕌).
Then
𝑙,𝑚
𝑙,𝑚
ℱ𝑝,𝑘
(𝛼; 𝛼1 ; 𝜙) ⊂ ℱ𝑝,𝑘
(𝛼1 ; 𝜙).
𝑙,𝑚
Proof. Let 𝑓 ∈ ℱ𝑝,𝑘
(𝛼; 𝛼1 ; 𝜙) and suppose that
𝑞(𝑧) =
(
)′
𝑧 𝐻𝑝𝑙,𝑚 (𝛼1 ; 𝑧)𝑓 (𝑧)
𝑙,𝑚
𝑝𝑓𝑝,𝑘
(𝛼1 ; 𝑧)
(𝑧 ∈ 𝕌).
(2.1)
Then 𝑞 is analytic in 𝕌 and 𝑞(0) = 1. It follows from (1.3) and (2.1) that
𝑙,𝑚
𝑞(𝑧)𝑓𝑝,𝑘
(𝛼1 ; 𝑧) =
𝛼1 𝑙,𝑚
𝛼1 − 𝑝 𝑙,𝑚
𝐻𝑝 (𝛼1 + 1)𝑓 (𝑧) −
𝐻𝑝 (𝛼1 )𝑓 (𝑧).
𝑝
𝑝
(2.2)
Differentiating both sides of (2.2) with respect to 𝑧 and using (2.1), we have
(
)′ ⎞
(
)′
𝑙,𝑚
𝑧
𝑓
(𝛼
;
𝑧)
1
𝑝,𝑘
𝛼1 𝑧 𝐻𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝑧)
⎜
⎟
′
𝑧𝑞 (𝑧) + ⎝𝛼1 − 𝑝 +
.
⎠ 𝑞(𝑧) =
𝑙,𝑚
𝑙,𝑚
𝑝
𝑓𝑝,𝑘
(𝛼1 ; 𝑧)
𝑓𝑝,𝑘
(𝛼1 ; 𝑧)
⎛
(2.3)
34
Z.-G. WANG, H.-P. SHI
It now follows from (1.7), (1.17), (2.1) and (2.3) that
[
]
(
)′
(
)′
𝑧 (1 − 𝛼) 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧) + 𝛼 𝐻𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝑧)
[
]
𝑙,𝑚
𝑙,𝑚
𝑝 (1 − 𝛼)𝑓𝑝,𝑘
(𝛼1 ; 𝑧) + 𝛼𝑓𝑝,𝑘
(𝛼1 + 1; 𝑧)
=
=
=
𝑙,𝑚
𝑙,𝑚
𝑝(1 − 𝛼)𝑞(𝑧)𝑓𝑝,𝑘
(𝛼1 ; 𝑧) + 𝛼𝛼1 𝑝 [𝑧𝑞 ′ (𝑧) + (𝛼1 − 𝑝 + 𝑝𝜓(𝑧))𝑞(𝑧)] 𝑓𝑝,𝑘
(𝛼1 ; 𝑧)
]
[
𝑙,𝑚
𝑙,𝑚
(𝛼1 ; 𝑧)
𝑝 (1 − 𝛼)𝑓𝑝,𝑘
(𝛼1 ; 𝑧) + 𝛼𝛼1 (𝛼1 − 𝑝 + 𝑝𝜓(𝑧))𝑓𝑝,𝑘
𝛼
𝛼1
(1 − 𝛼)𝑞(𝑧) +
[𝑧𝑞 ′ (𝑧) + (𝛼1 − 𝑝 + 𝑝𝜓(𝑧))𝑞(𝑧)]
(1 − 𝛼) + 𝛼𝛼1 (𝛼1 − 𝑝 + 𝑝𝜓(𝑧))
[
]
𝛼
𝛼
′
𝑧𝑞
(𝑧)
+
(1
−
𝛼)
+
(𝛼
−
𝑝
+
𝑝𝜓(𝑧))
𝑞(𝑧)
1
𝛼1
𝛼1
(1 − 𝛼) +
𝛼
𝛼1 (𝛼1
− 𝑝 + 𝑝𝜓(𝑧))
′
= 𝑞(𝑧) +
𝛼1
𝛼
𝑧𝑞 (𝑧)
≺ 𝜙(𝑧)
− 𝑝 + 𝑝𝜓(𝑧)
(𝑧 ∈ 𝕌).
(2.4)
Moreover, since
(
)
𝛼1
ℜ 𝑝𝜙(𝑧) +
−𝑝 >0
(𝛼 > 0; 𝑧 ∈ 𝕌),
𝛼
by Lemma 1.1, we know that
(
)′
𝑙,𝑚
𝑧 𝑓𝑝,𝑘
(𝛼1 ; 𝑧)
𝜓(𝑧) =
≺ 𝜙(𝑧)
(𝑧 ∈ 𝕌).
𝑙,𝑚
𝑝𝑓𝑝,𝑘
(𝛼1 ; 𝑧)
Thus, an application of Lemma 1.2 to (2.4), we have
𝑞(𝑧) ≺ 𝜙(𝑧)
that is 𝑓 ∈
𝑙,𝑚
ℱ𝑝,𝑘
(𝛼1 ; 𝜙).
(𝑧 ∈ 𝕌),
This implies that
𝑙,𝑚
𝑙,𝑚
ℱ𝑝,𝑘
(𝛼; 𝛼1 ; 𝜙) ⊂ ℱ𝑝,𝑘
(𝛼1 ; 𝜙).
Hence the proof of Theorem 2.1 is complete.
□
In view of Lemmas 1.4 and 1.5, by similarly applying the method of proof of
Theorem 2.1 for the classes 𝒢𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙) and ℋ𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙), we easily get the
following inclusion relationships.
Corollary 2.1. Let 𝜙 ∈ 𝒫 with
)
(
𝛼1
−𝑝 >0
(𝛼 > 0; 𝑧 ∈ 𝕌).
ℜ 𝑝𝜙(𝑧) +
𝛼
Then
𝒢𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙) ⊂ 𝒢𝑝𝑙,𝑚 (𝛼1 ; 𝜙).
Corollary 2.2. Let 𝜙 ∈ 𝒫 with
(
)
𝛼1
−𝑝 >0
(𝛼 > 0; 𝑧 ∈ 𝕌).
ℜ 𝑝𝜙(𝑧) +
𝛼
Then
ℋ𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙) ⊂ ℋ𝑝𝑙,𝑚 (𝛼1 ; 𝜙).
SOME PROPERTIES OF CERTAIN SUBCLASSES OF MULTIVALENT FUNCTIONS
Theorem 2.2. Let 𝜙 ∈ 𝒫 with
(
)
𝛼1
ℜ 𝑝𝜙(𝑧) +
−𝑝 >0
(𝛼 > 0; 𝑧 ∈ 𝕌).
𝛼
Then
𝑙,𝑚
𝔉𝑙,𝑚
𝑝,𝑘 (𝛼; 𝛼1 ; 𝜙) ⊂ 𝔉𝑝,𝑘 (𝛼1 ; 𝜙).
Proof. Let 𝑓 ∈ 𝔉𝑙,𝑚
𝑝,𝑘 (𝛼; 𝛼1 ; 𝜙) and suppose that
(
)′
𝑧 𝐻𝑝𝑙,𝑚 (𝛼1 ; 𝑧)𝑓 (𝑧)
𝑝(𝑧) =
𝑝𝔣𝑙,𝑚
𝑝,𝑘 (𝛼1 ; 𝑧)
(𝑧 ∈ 𝕌).
35
(2.5)
(2.6)
Then 𝑝 is analytic in 𝕌 and 𝑝(0) = 1. It follows from (1.3) and (2.6) that
𝛼1 𝑙,𝑚
𝛼1 − 𝑝 𝑙,𝑚
𝑝(𝑧)𝔣𝑙,𝑚
𝐻𝑝 (𝛼1 + 1)𝑓 (𝑧) −
𝐻𝑝 (𝛼1 )𝑓 (𝑧).
(2.7)
𝑝,𝑘 (𝛼1 ; 𝑧) =
𝑝
𝑝
Differentiating both sides of (2.7) with respect to 𝑧 and using (2.6), we have
⎛
(
)′ ⎞
(
)′
𝑙,𝑚
𝑧 𝔣𝑝,𝑘 (𝛼1 ; 𝑧) ⎟
𝛼1 𝑧 𝐻𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝑧)
⎜
′
.
𝑧𝑝 (𝑧) + ⎝𝛼1 − 𝑝 +
⎠ 𝑝(𝑧) =
𝑝
𝔣𝑙,𝑚
𝔣𝑙,𝑚
𝑝,𝑘 (𝛼1 ; 𝑧)
𝑝,𝑘 (𝛼1 ; 𝑧)
Furthermore, we suppose that
𝜑(𝑧) =
(
)′
𝑧 𝔣𝑙,𝑚
(𝛼
;
𝑧)
1
𝑝,𝑘
𝑝𝔣𝑙,𝑚
𝑝,𝑘 (𝛼1 ; 𝑧)
(𝑧 ∈ 𝕌).
The remainder of the proof of Theorem 2.2 is similar to that of Theorem 2.1. We,
therefore, choose to omit the analogous details involved. We thus find that
𝑝(𝑧) ≺ 𝜙(𝑧)
which implies that 𝑓 ∈
pleted.
𝔉𝑙,𝑚
𝑝,𝑘 (𝛼1 ; 𝜙).
(𝑧 ∈ 𝕌),
The proof of Theorem 2.2 is evidently com□
By similarly applying the method of proof of Theorem 2.1 for the classes 𝔊𝑙,𝑚
𝑝 (𝛼; 𝛼1 ; 𝜙)
and ℌ𝑙,𝑚
(𝛼;
𝛼
;
𝜙),
we
easily
get
the
following
inclusion
relationships.
1
𝑝
Corollary 2.3. Let 𝜙 ∈ 𝒫 with
(
)
𝛼1
ℜ 𝑝𝜙(𝑧) +
−𝑝 >0
(𝛼 > 0; 𝑧 ∈ 𝕌).
𝛼
Then
𝑙,𝑚
𝔊𝑙,𝑚
𝑝 (𝛼; 𝛼1 ; 𝜙) ⊂ 𝔊𝑝 (𝛼1 ; 𝜙).
Corollary 2.4. Let 𝜙 ∈ 𝒫 with
(
)
𝛼1
ℜ 𝑝𝜙(𝑧) +
−𝑝 >0
(𝛼 > 0; 𝑧 ∈ 𝕌).
𝛼
Then
𝑙,𝑚
ℌ𝑙,𝑚
𝑝 (𝛼; 𝛼1 ; 𝜙) ⊂ ℌ𝑝 (𝛼1 ; 𝜙).
By similarly applying the method of proof of Theorems 1 and 2 obtained by Wang
𝑙,𝑚
et al. [34] for the function classes 𝒢𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙), 𝔊𝑙,𝑚
𝑝 (𝛼; 𝛼1 ; 𝜙), ℋ𝑝 (𝛼; 𝛼1 ; 𝜙) and
𝑙,𝑚
ℌ𝑝 (𝛼; 𝛼1 ; 𝜙), we also easily get the following inclusion relationships.
36
Z.-G. WANG, H.-P. SHI
Corollary 2.5. Let 𝜙 ∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) + 𝛼1 − 𝑝) > 0
(𝑧 ∈ 𝕌).
Then
𝒢𝑝𝑙,𝑚 (𝛼1 + 1; 𝜙) ⊂ 𝒢𝑝𝑙,𝑚 (𝛼1 ; 𝜙).
Corollary 2.6. Let 𝜙 ∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) + 𝛼1 − 𝑝) > 0
(𝑧 ∈ 𝕌).
Then
ℋ𝑝𝑙,𝑚 (𝛼1 + 1; 𝜙) ⊂ ℋ𝑝𝑙,𝑚 (𝛼1 ; 𝜙).
Corollary 2.7. Let 𝜙 ∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) + 𝛼1 − 𝑝) > 0
(𝑧 ∈ 𝕌).
Then
𝑙,𝑚
𝔊𝑙,𝑚
𝑝 (𝛼1 + 1; 𝜙) ⊂ 𝔊𝑝 (𝛼1 ; 𝜙).
Corollary 2.8. Let 𝜙 ∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) + 𝛼1 − 𝑝) > 0
(𝑧 ∈ 𝕌).
Then
𝑙,𝑚
ℌ𝑙,𝑚
𝑝 (𝛼1 + 1; 𝜙) ⊂ ℌ𝑝 (𝛼1 ; 𝜙).
3. Convolution Properties
In this section, we provide some convolution properties for the function classes
𝑙,𝑚
ℱ𝑝,𝑘
(𝛼; 𝛼1 ; 𝜙), 𝒢𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙), and ℋ𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙).
𝑙,𝑚
Theorem 3.1. Let 𝑓 ∈ 𝒜𝑝 and 𝜙 ∈ 𝒫. Then 𝑓 ∈ ℱ𝑝,𝑘
(𝛼; 𝛼1 ; 𝜙) if and only if
{
[
(
)
∞
∑
1
(𝛼1 )𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛 𝑛 + 𝑝 𝑛+𝑝
𝑓 ∗ (1 − 𝛼) 𝑝𝑧 𝑝 +
𝑧
𝑧
(𝛽
1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛 𝑛!
𝑛=1
)
(
) ( 𝑘−1
∞
∑
(𝛼1 )𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛 1 𝑛+𝑝
1 ∑ 𝑧𝑝
𝑖𝜃
𝑝
− 𝑝 (1 − 𝛼)𝜙(𝑒 ) 𝑧 +
𝑧
∗
(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛 𝑛!
𝑘 𝜐=0 1 − 𝜀𝜐 𝑧
𝑛=1
(
)
∞
∑
(𝛼1 + 1)𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛 𝑛 + 𝑝 𝑛+𝑝
+ 𝛼 𝑝𝑧 𝑝 +
𝑧
(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛
𝑛!
𝑛=1
(
) ( 𝑘−1
) ]}
∞
∑
(𝛼1 + 1)𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛 1 𝑛+𝑝
1 ∑ 𝑧𝑝
𝑖𝜃
𝑝
− 𝑝 𝛼𝜙(𝑒 ) 𝑧 +
𝑧
∗
∕= 0
(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛 𝑛!
𝑘 𝜐=0 1 − 𝜀𝜐 𝑧
𝑛=1
(𝑧 ∈ 𝕌; 0 ≦ 𝜃 < 2𝜋).
𝑙,𝑚
Proof. Suppose that 𝑓 ∈ ℱ𝑝,𝑘
(𝛼; 𝛼1 ; 𝜙). Since
[
]
( 𝑙,𝑚
)′
(
)′
𝑧 (1 − 𝛼) 𝐻𝑝 (𝛼1 )𝑓 (𝑧) + 𝛼 𝐻𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝑧)
[
]
≺ 𝜙(𝑧)
𝑙,𝑚
𝑙,𝑚
𝑝 (1 − 𝛼)𝑓𝑝,𝑘
(𝛼1 ; 𝑧) + 𝛼𝑓𝑝,𝑘
(𝛼1 + 1; 𝑧)
(𝑧 ∈ 𝕌)
SOME PROPERTIES OF CERTAIN SUBCLASSES OF MULTIVALENT FUNCTIONS
is equivalent to
]
[
(
)′
(
)′
𝑧 (1 − 𝛼) 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧) + 𝛼 𝐻𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝑧)
[
]
∕= 𝜙(𝑒𝑖𝜃 )
𝑙,𝑚
𝑙,𝑚
𝑝 (1 − 𝛼)𝑓𝑝,𝑘 (𝛼1 ; 𝑧) + 𝛼𝑓𝑝,𝑘 (𝛼1 + 1; 𝑧)
37
(0 ≦ 𝜃 < 2𝜋),
(3.1)
it is easy to see that the condition (3.1) can be written as follows:
{
[
]
(
)′
(
)′
1
𝑧 (1 − 𝛼) 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧) + 𝛼 𝐻𝑝𝑙,𝑚 (𝛼1 + 1)𝑓 (𝑧)
𝑧
}
[
]
𝑙,𝑚
𝑙,𝑚
𝑖𝜃
− 𝑝 (1 − 𝛼)𝑓𝑝,𝑘 (𝛼1 ; 𝑧) + 𝛼𝑓𝑝,𝑘 (𝛼1 + 1; 𝑧) 𝜙(𝑒 ) ∕= 0
(0 ≦ 𝜃 < 2𝜋).
(3.2)
On the other hand, we know from (1.2) that
)
(
∞
∑
( 𝑙,𝑚
)′
(𝛼1 )𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛 𝑛 + 𝑝 𝑛+𝑝
𝑝
𝑧
∗ 𝑓 (𝑧).
𝑧 𝐻𝑝 (𝛼1 )𝑓 (𝑧) = 𝑝𝑧 +
(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛 𝑛!
𝑛=1
(3.3)
𝑙,𝑚
Moreover, from the definition of 𝑓𝑝,𝑘
(𝛼1 ; 𝑧), we have
( 𝑘−1
)
1 ∑ 𝑧𝑝
𝑙,𝑚
𝑙,𝑚
𝑓𝑝,𝑘 (𝛼1 ; 𝑧) = 𝐻𝑝 (𝛼1 )𝑓 (𝑧) ∗
𝑘 𝜐=0 1 − 𝜀𝜐 𝑧
(
) ( 𝑘−1
)
∞
∑
∑ 𝑧𝑝
(𝛼
)
⋅
⋅
⋅
(𝛼
)
1
1
1
𝑛
𝑙
𝑛
= 𝑧𝑝 +
𝑧 𝑛+𝑝 ∗
∗ 𝑓 (𝑧).
(𝛽
)
⋅
⋅
⋅
(𝛽
)
𝑛!
𝑘 𝜐=0 1 − 𝜀𝜐 𝑧
1
𝑛
𝑚
𝑛
𝑛=1
(3.4)
Replacing 𝛼1 by 𝛼1 + 1 in (3.3) and (3.4), we know that (3.3) and (3.4) also hold
true, that is,
)
(
∞
∑
( 𝑙,𝑚
)′
(𝛼1 + 1)𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛 𝑛 + 𝑝 𝑛+𝑝
𝑝
𝑧
∗ 𝑓 (𝑧), (3.5)
𝑧 𝐻𝑝 (𝛼1 + 1)𝑓 (𝑧) = 𝑝𝑧 +
(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛
𝑛!
𝑛=1
and
(
𝑙,𝑚
𝑓𝑝,𝑘
(𝛼1
+ 1; 𝑧)
=𝐻𝑝𝑙,𝑚 (𝛼1
+ 1)𝑓 (𝑧) ∗
𝑘−1
1 ∑ 𝑧𝑝
𝑘 𝜐=0 1 − 𝜀𝜐 𝑧
)
∞
∑
(𝛼1 + 1)𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛 1 𝑛+𝑝
= 𝑧 +
𝑧
(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛 𝑛!
𝑛=1
(
𝑝
) (
∗
𝑘−1
1 ∑ 𝑧𝑝
𝑘 𝜐=0 1 − 𝜀𝜐 𝑧
)
∗ 𝑓 (𝑧).
(3.6)
Upon substituting from (3.3) to (3.6) into (3.2), we easily deduce the convolution
property asserted by Theorem 3.1.
By similarly applying the method of proof of Theorem 3.1 for the classes 𝒢𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙)
□
and ℋ𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙), we easily get the following convolution properties.
38
Z.-G. WANG, H.-P. SHI
Corollary 3.1. Let 𝑓 ∈ 𝒜𝑝 and 𝜙 ∈ 𝒫. Then 𝑓 ∈ 𝒢𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙) if and only if
[
(
{
)
∞
∑
1
(𝛼1 )𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛 𝑛 + 𝑝 𝑛+𝑝
𝑝(1 − 𝛼)𝜙(𝑒𝑖𝜃 )
𝑝
𝑓 ∗ (1 − 𝛼) 𝑝𝑧 +
𝑧
ℎ1
−
𝑧
(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛 𝑛!
2
𝑛=1
(
)
]
∞
𝑖𝜃
∑
𝑛
+
𝑝
𝑝
𝛼𝜙(𝑒
)
(𝛼
+
1)
⋅
⋅
⋅
(𝛼
)
1
𝑛
𝑙
𝑛
+ 𝛼 𝑝𝑧 𝑝 +
𝑧 𝑛+𝑝 −
ℎ2
(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛
𝑛!
2
𝑛=1
}
𝑝 𝛼𝜙(𝑒𝑖𝜃 )
𝑝 (1 − 𝛼)𝜙(𝑒𝑖𝜃 )
(ℎ1 ∗ 𝑓 )(𝑧) −
(ℎ2 ∗ 𝑓 )(𝑧) ∕= 0
(0 ≦ 𝜃 < 2𝜋),
−
2
2
where
∞
∑
(𝛼1 )𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛 1 𝑛+𝑝
𝑧
,
(𝛽
1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛 𝑛!
𝑛=1
(3.7)
∞
∑
(𝛼1 + 1)𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛 1 𝑛+𝑝
𝑧
.
(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛 𝑛!
𝑛=1
(3.8)
ℎ1 (𝑧) = 𝑧 𝑝 +
and
ℎ2 (𝑧) = 𝑧 𝑝 +
Corollary 3.2. Let 𝑓 ∈ 𝒜𝑝 and 𝜙 ∈ 𝒫. Then 𝑓 ∈ ℋ𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙) if and only if
{
[
(
)
∞
∑
1
(𝛼1 )𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛 𝑛 + 𝑝 𝑛+𝑝
𝑝(1 − 𝛼)𝜙(𝑒𝑖𝜃 )
𝑝
𝑓 ∗ (1 − 𝛼) 𝑝𝑧 +
𝑧
−
ℎ1
𝑧
(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛 𝑛!
2
𝑛=1
)
]
(
∞
∑
(𝛼1 + 1)𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛 𝑛 + 𝑝 𝑛+𝑝
𝑝𝛼𝜙(𝑒𝑖𝜃 )
𝑝
𝑧
−
ℎ2
+ 𝛼 𝑝𝑧 +
(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛
𝑛!
2
𝑛=1
}
𝑝(1 − 𝛼)𝜙(𝑒𝑖𝜃 )
𝑝𝛼𝜙(𝑒𝑖𝜃 )
+
(ℎ1 ∗ 𝑓 )(−𝑧) +
(ℎ2 ∗ 𝑓 )(−𝑧) ∕= 0
(0 ≦ 𝜃 < 2𝜋),
2
2
where ℎ1 and ℎ2 are given by (3.7) and (3.8), respectively.
Theorem 3.2. Let 𝑓 ∈ ℋ𝑝𝑙,𝑚 (𝛼1 ; 𝜙). Then
[ ∫
( ∫
) ]
𝑧
𝜁
𝑝
𝜙(𝜔(𝜉))
−
𝜙(𝜔(−𝜉))
−
2
𝑓 (𝑧) = 𝑝
𝜁 𝑝−1 𝜙(𝜔(𝜁)) ⋅ exp
𝑑𝜉 𝑑𝜁
2 0
𝜉
0
(∞
)
∑ 𝑛!(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛
𝑛+𝑝
∗
𝑧
,
(𝛼1 )𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛
𝑛=0
(3.9)
where 𝜔 is analytic in 𝕌 with
𝜔(0) = 0
and
∣𝜔(𝑧)∣ < 1
(𝑧 ∈ 𝕌).
Proof. From the definition of ℋ𝑝𝑙,𝑚 (𝛼1 ; 𝜙), we know that
(
)′
(
)′
𝑧 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧)
2𝑧 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (𝑧)
] = 𝜙(𝜔(𝑧)), (3.10)
= [
𝑝ℎ𝑙,𝑚
𝑝 (𝛼1 ; 𝑧)
𝑝 (𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 )(𝑧) − (𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 )(−𝑧)
where 𝜔 is analytic in 𝕌 with
𝜔(0) = 0
and
∣𝜔(𝑧)∣ < 1
(𝑧 ∈ 𝕌).
SOME PROPERTIES OF CERTAIN SUBCLASSES OF MULTIVALENT FUNCTIONS
39
From (3.10), we get
(
)′
2𝑧 𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 (−𝑧)
[
] = 𝜙(𝜔(−𝑧)).
𝑝 (𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 )(𝑧) − (𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓 )(−𝑧)
It now follows from (3.10) and (3.11) that
(
)′
]
𝑧 ℎ𝑙,𝑚
1[
𝑝 (𝛼1 ; 𝑧)
𝜙(𝜔(𝑧)) − 𝜙(𝜔(−𝑧)) .
=
𝑙,𝑚
2
𝑝ℎ𝑝 (𝛼1 ; 𝑧)
We next find from (3.12) that
( 𝑙,𝑚
)′
ℎ𝑝 (𝛼1 ; 𝑧)
ℎ𝑙,𝑚
𝑝 (𝛼1 ; 𝑧)
−
𝑝
𝑝 𝜙(𝜔(𝑧)) − 𝜙(𝜔(−𝑧)) − 2
= ⋅
.
𝑧
2
𝑧
Upon integrating (3.13), we have
(
)
∫
ℎ𝑙,𝑚
𝑝 𝑧 𝜙(𝜔(𝜉)) − 𝜙(𝜔(−𝜉)) − 2
𝑝 (𝛼1 ; 𝑧)
log
=
𝑑𝜉,
𝑧𝑝
2 0
𝜉
(3.11)
(3.12)
(3.13)
(3.14)
which implies that
ℎ𝑙,𝑚
𝑝 (𝛼1 ; 𝑧)
)
( ∫
𝑝 𝑧 𝜙(𝜔(𝜉)) − 𝜙(𝜔(−𝜉)) − 2
𝑑𝜉 .
= 𝑧 ⋅ exp
2 0
𝜉
𝑝
(3.15)
It now follows from (3.10) and (3.15) that
(
𝐻𝑝𝑙,𝑚 (𝛼1 )𝑓
)′
𝑝ℎ𝑙,𝑚
𝑝 (𝛼1 ; 𝑧)
⋅ 𝜙(𝜔(𝑧))
𝑧
( ∫
)
𝑝 𝑧 𝜙(𝜔(𝜉)) − 𝜙(𝜔(−𝜉)) − 2
𝑝−1
=𝑝𝑧
𝜙(𝜔(𝑧)) ⋅ exp
𝑑𝜉 .
2 0
𝜉
(3.16)
(𝑧) =
Upon integrating (3.16), we get
( ∫
)
∫ 𝑧
𝑝 𝜁 𝜙(𝜔(𝜉)) − 𝜙(𝜔(−𝜉)) − 2
𝑙,𝑚
𝑝−1
𝐻𝑝 (𝛼1 )𝑓 (𝑧) = 𝑝
𝜁
𝜙(𝜔(𝜁)) ⋅ exp
𝑑𝜉 𝑑𝜁.
2 0
𝜉
0
(3.17)
Combining (1.2) and (3.17), we find that
)
( ∫
∫ 𝑧
𝑝 𝜁 𝜙(𝜔(𝜉)) − 𝜙(𝜔(−𝜉)) − 2
𝑝−1
𝑑𝜉 𝑑𝜁
𝑝
𝜁
𝜙(𝜔(𝜁)) ⋅ exp
2 0
𝜉
(3.18)
0
= [𝑧 𝑝 𝑙 𝐹𝑚 (𝛼1 , . . . , 𝛼𝑙 ; 𝛽1 , . . . , 𝛽𝑚 )] ∗ 𝑓 (𝑧).
Thus, from (3.18), we easily get the convolution property (3.9).
□
Remark 3.1. Putting
𝑝 = 1, 𝑙 = 2, 𝑚 = 1
and 𝛼1 = 𝛼2 = 𝛽1 = 1
in Theorem 3.2, we get the corresponding result obtained by Ravichandran [25].
By similarly applying the method of proof of Theorem 3.2 for the class 𝒢𝑝𝑙,𝑚 (𝛼; 𝛼1 ; 𝜙),
we easily get the following convolution property.
40
Z.-G. WANG, H.-P. SHI
Corollary 3.3. Let 𝑓 ∈ 𝒢𝑝𝑙,𝑚 (𝛼1 ; 𝜙). Then
( ∫
[ ∫
) ]
𝑧
𝑝 𝜁 𝜙(𝜔(𝜉)) + 𝜙(𝜔(𝜉)) − 2
𝑝−1
𝜁
𝜙(𝜔(𝜁)) ⋅ exp
𝑓 (𝑧) = 𝑝
𝑑𝜉 𝑑𝜁
2 0
𝜉
0
(∞
)
∑ 𝑛!(𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛
𝑛+𝑝
∗
,
𝑧
(𝛼1 )𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛
𝑛=0
where 𝜔 is analytic in 𝕌 with
𝜔(0) = 0
and
∣𝜔(𝑧)∣ < 1
(𝑧 ∈ 𝕌).
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Zhi-Gang Wang
Department of Mathematics and Computing Science, Hengyang Normal University, Hengyang
421008, Hunan, P. R. China
E-mail address: zhigangwang@foxmail.com
Hai-Ping Shi
Department of Basic Courses, Guangdong Baiyun Institute, Guangzhou 510450, Guangdong, P. R. China
E-mail address: shp7971@163.com