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
Let𝒜𝑝 denote the class of functions of the form
𝑓(𝑧) =𝑧𝑝+
∞
∑
𝑛=1
𝑎𝑛+𝑝𝑧𝑛+𝑝 (𝑝∈ℕ:={1,2,3, . . .}), (1.1)
which areanalyticin theopenunit disk
𝕌:={𝑧:𝑧∈ℂ and ∣𝑧∣<1}.
For simplicity, we write
𝒜1=:𝒜.
Let𝑓, 𝑔∈ 𝒜𝑝, where 𝑓 is given by (1.1) and𝑔 is defined by
𝑔(𝑧) =𝑧𝑝+
∞
∑
𝑛=1
𝑏𝑛+𝑝𝑧𝑛+𝑝.
Then the Hadamard product (or convolution) 𝑓 ∗𝑔 of the functions 𝑓 and 𝑔 is defined by
(𝑓∗𝑔)(𝑧) :=𝑧𝑝+
∞
∑
𝑛=1
𝑎𝑛+𝑝𝑏𝑛+𝑝𝑧𝑛+𝑝=: (𝑔∗𝑓)(𝑧).
For parameters
𝛼𝑗∈ℂ (𝑗= 1, . . . , 𝑙) and 𝛽𝑗∈ℂ∖ℤ−0 (ℤ
−
0 :={0,−1,−2, . . .}; 𝑗= 1, . . . , 𝑚),
2000Mathematics Subject Classification. 30C45, 30C80.
Key words and phrases. Analytic functions; multivalent functions; subordination between ana-lytic functions; Hadamard product (or convolution); Dziok-Srivastava operator; symmetric points; conjugate points; symmetric conjugate points.
c
⃝2010 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. 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.
the generalized hypergeometric function
𝑙𝐹𝑚(𝛼1, . . . , 𝛼𝑙;𝛽1, . . . , 𝛽𝑚;𝑧)
is defined by the following infinite series:
𝑙𝐹𝑚(𝛼1, . . . , 𝛼𝑙;𝛽1, . . . , 𝛽𝑚;𝑧) := ∞
∑
𝑛=0
(𝛼1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛 𝑧𝑛 𝑛!
(𝑙≦𝑚+ 1; 𝑙, 𝑚∈ℕ0:=ℕ∪ {0}; 𝑧∈𝕌),
where (𝜆)𝑛 is the Pochhammer symbol defined by
(𝜆)𝑛 :=
⎧ ⎨
⎩
1 (𝑛= 0),
𝜆(𝜆+ 1)⋅ ⋅ ⋅(𝜆+𝑛−1) (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) := [𝑧𝑝 𝑙𝐹𝑚(𝛼1, . . . , 𝛼𝑙;𝛽1, . . . , 𝛽𝑚)]∗𝑓(𝑧) (1.2)
(𝑙≦𝑚+ 1; 𝑙, 𝑚∈ℕ0; 𝑧∈𝕌).
If𝑓 ∈ 𝒜𝑝 is given by (1.1), then we have
𝐻𝑝(𝛼1, . . . , 𝛼𝑙;𝛽1, . . . , 𝛽𝑚)𝑓(𝑧) =𝑧𝑝+
∞
∑
𝑛=1
(𝛼1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛 𝑎𝑛+𝑝
𝑧𝑛+𝑝
𝑛! (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) =𝛼1𝐻𝑝𝑙,𝑚(𝛼1+ 1)𝑓(𝑧)−(𝛼1−𝑝)𝐻𝑝𝑙,𝑚(𝛼1)𝑓(𝑧). (1.3)
Let𝒫 denote the class of functions of the form
𝑝(𝑧) = 1 +
∞
∑
𝑛=1
𝑝𝑛𝑧𝑛,
which are analytic and convex in𝕌and satisfy the condition
ℜ(𝑝(𝑧))>0 (𝑧∈𝕌).
For two functions𝑓and𝑔, analytic in𝕌, we say that the function𝑓 is subordinate to𝑔 in𝕌, and write
𝑓(𝑧)≺𝑔(𝑧) (𝑧∈𝕌),
if there exists a Schwarz function𝜔, which is analytic in𝕌with
𝜔(0) = 0 and ∣𝜔(𝑧)∣<1 (𝑧∈𝕌)
such that
𝑓(𝑧) =𝑔(𝜔(𝑧)) (𝑧∈𝕌).
Indeed, it is known that
Furthermore, if the function𝑔 is univalent in𝕌, then we have the following
equiv-alence:
𝑓(𝑧)≺𝑔(𝑧) (𝑧∈𝕌)⇐⇒𝑓(0) =𝑔(0) and 𝑓(𝕌)⊂𝑔(𝕌).
Throughout this paper, we assume that
𝑝, 𝑘∈ℕ, 𝑙, 𝑚∈ℕ0, 𝜀𝑘= exp (2𝜋𝑖
𝑘
)
,
𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) = 1
𝑘 𝑘−1 ∑
𝑗=0
𝜀−𝑗𝑝𝑘 (𝐻𝑝𝑙,𝑚(𝛼1)𝑓
)
(𝜀𝑗𝑘𝑧) =𝑧𝑝+⋅ ⋅ ⋅ (𝑓 ∈ 𝒜𝑝), (1.4)
𝑔𝑙,𝑚𝑝 (𝛼1;𝑧) = 1 2 [
𝐻𝑝𝑙,𝑚(𝛼1)𝑓(𝑧) +𝐻
𝑙,𝑚
𝑝 (𝛼1)𝑓(𝑧) ]
=𝑧𝑝+⋅ ⋅ ⋅ (𝑓 ∈ 𝒜𝑝), (1.5)
and
ℎ𝑙,𝑚𝑝 (𝛼1;𝑧) = 1 2 [
𝐻𝑝𝑙,𝑚(𝛼1)𝑓(𝑧)−𝐻𝑝𝑙,𝑚(𝛼1)𝑓(−𝑧)
]
=𝑧𝑝+⋅ ⋅ ⋅ (𝑓 ∈ 𝒜𝑝). (1.6)
Clearly, for𝑘= 1, we have
𝑓𝑝,𝑙,𝑚1(𝛼1;𝑧) =𝐻𝑝𝑙,𝑚(𝛼1)𝑓(𝑧).
In recent years, several authors obtained many interesting results involving the Dziok-Srivastava operator𝐻𝑙,𝑚
𝑝 (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 operator𝐻𝑙,𝑚
𝑝 (𝛼1) and the above-mentioned principle of subordination between analytic functions, we introduce and investigate the following subclasses of the class𝒜𝑝 of𝑝-valent analytic functions.
Definition 1.1. A function 𝑓 ∈ 𝒜𝑝 is said to be in the class ℱ𝑝,𝑘𝑙,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)]
𝑝[(1−𝛼)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) +𝛼𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧)
] ≺𝜙(𝑧) (𝑧∈𝕌), (1.7)
where𝜙∈ 𝒫,𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) is defined by (1.4) and
𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧)∕= 0 (𝑧∈𝕌).
For simplicity, we write
ℱ𝑝,𝑘𝑙,𝑚(0;𝛼1;𝜙) =:ℱ𝑝,𝑘𝑙,𝑚(𝛼1;𝜙).
Remark 1.1. If we set
𝑝= 1, 𝑙= 2, 𝑚= 1, and 𝛼1=𝛼2=𝛽1= 1
in the class ℱ𝑝,𝑘𝑙,𝑚(𝛼;𝛼1;𝜙), then it reduces to the known class 𝒮(𝑘)(𝛼;𝜙) of func-tions 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, and 𝛼= 0
Definition 1.2. 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)(
𝐻𝑙,𝑚 𝑝 (𝛼1)𝑓
)′
(𝑧) +𝛼(
𝐻𝑙,𝑚
𝑝 (𝛼1+ 1)𝑓)
′
(𝑧)]
𝑝[(1−𝛼)𝑔𝑝𝑙,𝑚(𝛼1;𝑧) +𝛼𝑔𝑝𝑙,𝑚(𝛼1+ 1;𝑧)
] ≺𝜙(𝑧) (𝑧∈𝕌),
where𝜙∈ 𝒫,𝑔𝑝𝑙,𝑚(𝛼1;𝑧) is defined by (1.5) and
𝑔𝑝𝑙,𝑚(𝛼1+ 1;𝑧)∕= 0 (𝑧∈𝕌).
For simplicity, we write
𝒢𝑙,𝑚
𝑝 (0;𝛼1;𝜙) =:𝒢 𝑙,𝑚 𝑝 (𝛼1;𝜙).
Definition 1.3. 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)]
𝑝[(1−𝛼)ℎ𝑙,𝑚𝑝 (𝛼1;𝑧) +𝛼ℎ𝑙,𝑚𝑝 (𝛼1+ 1;𝑧)
] ≺𝜙(𝑧) (𝑧∈𝕌),
where𝜙∈ 𝒫,ℎ𝑙,𝑚
𝑝 (𝛼1;𝑧) is defined by (1.6) and
ℎ𝑙,𝑚𝑝 (𝛼1+ 1;𝑧)∕= 0 (𝑧∈𝕌).
For simplicity, we write
ℋ𝑙,𝑚
𝑝 (0;𝛼1;𝜙) =:ℋ 𝑙,𝑚 𝑝 (𝛼1;𝜙).
Remark 1.2. In 1996, Chen et al. [5] introduced and investigated a subclass
𝒮∗
𝑠𝑐(𝛼) of𝒜consisting of functions which are 𝛼-starlike with respect to symmetric
conjugate points and satisfy the inequality
ℜ
( 𝑧[(1−𝛼)𝑓′(𝑧) +𝛼(𝑧𝑓′(𝑧))′]
(1−𝛼)𝑇𝑠𝑐𝑓(𝑧) +𝛼𝑧(𝑇𝑠𝑐𝑓(𝑧))′
)
>0 (𝑧∈𝕌),
where
𝑇𝑠𝑐𝑓(𝑧) = 1 2 [
𝑓(𝑧)−𝑓(−𝑧)].
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 classℋ𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙), then it reduces to the class𝒮𝑠𝑐∗(𝛼).
Definition 1.4. A function 𝑓 ∈ 𝒜𝑝 is said to be in the class 𝔉𝑙,𝑚𝑝,𝑘(𝛼;𝛼1;𝜙) if it satisfies the subordination condition
𝑧[(1−𝛼)(
𝐻𝑙,𝑚 𝑝 (𝛼1)𝑓
)′
(𝑧) +𝛼(
𝐻𝑙,𝑚
𝑝 (𝛼1+ 1)𝑓 )′
(𝑧)]
𝑝[(1−𝛼)𝔣𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) +𝛼𝔣
𝑙,𝑚
𝑝,𝑘(𝛼1+ 1;𝑧)
] ≺𝜙(𝑧) (𝑧∈𝕌),
where𝜙∈ 𝒫,𝔣𝑙,𝑚𝑝,𝑘(𝛼1;𝑧) is defined as in (1.4) and
For simplicity, we write
𝔉𝑙,𝑚𝑝,𝑘(0;𝛼1;𝜙) =:𝔉
𝑙,𝑚 𝑝,𝑘(𝛼1;𝜙). Remark 1.3. If we set
𝑝= 1, 𝑙= 2, 𝑚= 1, and 𝛼1=𝛼2=𝛽1= 1
in the class𝔉𝑙,𝑚𝑝,𝑘(𝛼;𝛼1;𝜙), then it reduces to the known class𝒞(𝑘)(𝛼, 𝜙) of functions
𝛼-close-to-convex with respect to𝑘-symmetric points, which was also studied earlier by Parvatham and Radha [21]. Furthermore, we also note that the class𝔉𝑙,𝑚𝑝,𝑘(𝛼1;𝜙) was introduced and investigated recently by Wanget al. [34].
Definition 1.5. A function 𝑓 ∈ 𝒜𝑝 is said to be in the class 𝔊𝑙,𝑚𝑝 (𝛼;𝛼1;𝜙) if it
satisfies the subordination condition
𝑧[(1−𝛼)(
𝐻𝑙,𝑚 𝑝 (𝛼1)𝑓
)′
(𝑧) +𝛼(
𝐻𝑙,𝑚
𝑝 (𝛼1+ 1)𝑓)
′
(𝑧)]
𝑝[(1−𝛼)𝔤𝑝𝑙,𝑚(𝛼1;𝑧) +𝛼𝔤
𝑙,𝑚
𝑝 (𝛼1+ 1;𝑧)
] ≺𝜙(𝑧) (𝑧∈𝕌),
where𝜙∈ 𝒫,𝔤𝑙,𝑚
𝑝 (𝛼1;𝑧) is defined as in (1.5) and
𝔤𝑙,𝑚𝑝 (𝛼1+ 1;𝑧)∕= 0 (𝑧∈𝕌).
For simplicity, we write
𝔊𝑙,𝑚𝑝 (0;𝛼1;𝜙) =:𝔊𝑙,𝑚𝑝 (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)(
𝐻𝑙,𝑚 𝑝 (𝛼1)𝑓
)′
(𝑧) +𝛼(
𝐻𝑙,𝑚
𝑝 (𝛼1+ 1)𝑓)
′
(𝑧)]
𝑝[(1−𝛼)𝔥𝑙,𝑚𝑝 (𝛼1;𝑧) +𝛼𝔥𝑙,𝑚𝑝 (𝛼1+ 1;𝑧)
] ≺𝜙(𝑧) (𝑧∈𝕌),
where𝜙∈ 𝒫,𝔥𝑙,𝑚
𝑝 (𝛼1;𝑧) is defined as in (1.6) and
𝔥𝑙,𝑚𝑝 (𝛼1+ 1;𝑧)∕= 0 (𝑧∈𝕌).
For simplicity, we write
ℌ𝑙,𝑚𝑝 (0;𝛼1;𝜙) =:ℌ𝑙,𝑚𝑝 (𝛼1;𝜙).
In order to establish our main results, we need the following lemmas.
Lemma 1.1. (See [10, 16])Let𝛽, 𝛾∈ℂ. Suppose that𝜙(𝑧)is convex and univalent in𝕌with
𝜙(0) = 1 and ℜ(𝛽𝜙(𝑧) +𝛾)>0 (𝑧∈𝕌).
If 𝔭 is analytic in𝕌 with𝔭(0) = 1, then the subordination
𝔭(𝑧) + 𝑧𝔭
′(𝑧)
𝛽𝔭(𝑧) +𝛾 ≺𝜙(𝑧) (𝑧∈𝕌)
implies that
Lemma 1.2. (See [20]) Let 𝛽, 𝛾 ∈ℂ. Suppose that 𝜙(𝑧) is convex and univalent in𝕌with
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)
𝛽𝔮(𝑧) +𝛾 ≺𝜙(𝑧) (𝑧∈𝕌),
then
𝔭(𝑧)≺𝜙(𝑧) (𝑧∈𝕌).
Lemma 1.3. Let 𝑓 ∈ ℱ𝑝,𝑘𝑙,𝑚(𝛼;𝛼1;𝜙). Then
𝑧
[
(1−𝛼)(𝑓𝑝,𝑘𝑙,𝑚(𝛼1)𝑓 )′
(𝑧) +𝛼(𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1)𝑓 )′
(𝑧) ]
𝑝[(1−𝛼)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) +𝛼𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧)
] ≺𝜙(𝑧) (𝑧∈𝕌). (1.8)
Furthermore, if𝜙∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) +𝛼1
𝛼 −𝑝
)
>0 (𝛼 >0; 𝑧∈𝕌).
Then
𝑧(𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧))
′
𝑝𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) ≺𝜙(𝑧) (𝑧∈𝕌).
Proof. Making use of (1.4), we have
𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝜀
𝑗 𝑘𝑧) =
1
𝑘 𝑘−1 ∑
𝑛=0
𝜀−𝑛𝑝𝑘 (
𝐻𝑝𝑙,𝑚(𝛼1)𝑓 )
(𝜀𝑛𝑘+𝑗𝑧)
=𝜀𝑗𝑝𝑘 ⋅1 𝑘
𝑘−1 ∑
𝑛=0
𝜀−𝑘(𝑛+𝑗)𝑝(𝐻𝑝𝑙,𝑚(𝛼1)𝑓)(𝜀𝑛𝑘+𝑗𝑧)
=𝜀𝑗𝑝𝑘 𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) (𝑗∈ {0,1, . . . , 𝑘−1}),
(1.9)
and
(
𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) )′
= 1
𝑘 𝑘−1 ∑
𝑛=0
𝜀−𝑗𝑘 (𝑝−1)(
𝐻𝑝𝑙,𝑚(𝛼1)𝑓 )
(𝜀𝑗𝑘𝑧). (1.10)
Replacing𝛼1by𝛼1+ 1 in (1.9) and (1.10), respectively, we get
𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝜀
𝑗 𝑘𝑧) =𝜀
𝑗𝑝 𝑘 𝑓
𝑙,𝑚
𝑝,𝑘(𝛼1+ 1;𝑧) (𝑗∈ {0,1, . . . , 𝑘−1}), (1.11) and
(
𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧) )′
= 1
𝑘 𝑘−1 ∑
𝑛=0
𝜀−𝑗𝑘 (𝑝−1)(
𝐻𝑝𝑙,𝑚(𝛼1+ 1)𝑓 )
From (1.9) to (1.12), we get
𝑧
[
(1−𝛼)(𝑓𝑝,𝑘𝑙,𝑚(𝛼1)𝑓 )′
(𝑧) +𝛼(𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1)𝑓 )′
(𝑧) ]
𝑝[(1−𝛼)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) +𝛼𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧) ]
= 1
𝑘 𝑘−1 ∑
𝑗=0
𝜀−𝑗𝑘 (𝑝−1)𝑧[(1−𝛼)(
𝐻𝑙,𝑚 𝑝 (𝛼1)𝑓
)′
(𝜀𝑗𝑘𝑧) +𝛼(
𝐻𝑙,𝑚
𝑝 (𝛼1+ 1)𝑓)
′
(𝜀𝑗𝑘𝑧)]
𝑝[(1−𝛼)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) +𝛼𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧) ]
= 1
𝑘 𝑘−1 ∑
𝑗=0
𝜀𝑗𝑘𝑧[(1−𝛼)(
𝐻𝑙,𝑚 𝑝 (𝛼1)𝑓
)′
(𝜀𝑗𝑘𝑧) +𝛼(
𝐻𝑙,𝑚
𝑝 (𝛼1+ 1)𝑓)
′
(𝜀𝑗𝑘𝑧)]
𝑝[(1−𝛼)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝜀𝑘𝑗𝑧) +𝛼𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝜀𝑗𝑘𝑧)
] .
(1.13)
Moreover, since𝑓 ∈ ℱ𝑝,𝑘𝑞,𝑠(𝛼;𝛼1;𝜙), it follows that
𝜀𝑗𝑘𝑧[(1−𝛼)(𝐻𝑝𝑙,𝑚(𝛼1)𝑓
)′
(𝜀𝑗𝑘𝑧) +𝛼(𝐻𝑝𝑙,𝑚(𝛼1+ 1)𝑓 )′
(𝜀𝑗𝑘𝑧)]
𝑝[(1−𝛼)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝜀𝑘𝑗𝑧) +𝛼𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝜀𝑗𝑘𝑧)
] ≺𝜙(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
𝑧(𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧))
′
+ (𝛼1−𝑝)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) =
𝛼1
𝑘 𝑘−1 ∑
𝑗=0
𝜀−𝑗𝑝𝑘 (
𝐻𝑝𝑙,𝑚(𝛼1+ 1)𝑓)(𝜀𝑗𝑘𝑧)
=𝛼1𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧).
(1.15)
Let𝑓 ∈ ℱ𝑝,𝑘𝑙,𝑚(𝛼;𝛼1;𝜙) and suppose that
𝜓(𝑧) =
𝑧(𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧))
′
𝑝𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧)
(𝑧∈𝕌). (1.16)
Then𝜓is analytic in 𝕌and𝜓(0) = 1. It follows from (1.15) and (1.16) that
𝛼1−𝑝+𝑝𝜓(𝑧) =𝛼1
𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧)
𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧)
(𝑧∈𝕌). (1.17)
From (1.16) and (1.17), we have
𝑧(𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧) )′
= 𝑝
𝛼1
It now follows from (1.8), (1.16), (1.17) and (1.18) that
𝑧
[
(1−𝛼)(𝑓𝑝,𝑘𝑙,𝑚(𝛼1)𝑓 )′
(𝑧) +𝛼(𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1)𝑓 )′
(𝑧) ]
𝑝[(1−𝛼)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) +𝛼𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧) ]
= 𝑝(1−𝛼)𝜓(𝑧)𝑓
𝑙,𝑚
𝑝,𝑘(𝛼1;𝑧) + 𝛼 𝛼1𝑝[𝑧𝜓
′(𝑧) + (𝛼
1−𝑝+𝑝𝜓(𝑧))𝜓(𝑧)]𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧)
𝑝(1−𝛼)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) +𝛼𝛼1𝑝(𝛼1−𝑝+𝑝𝜓(𝑧))𝑓
𝑙,𝑚 𝑝,𝑘(𝛼1;𝑧)
= (1−𝛼)𝜓(𝑧) +
𝛼 𝛼1[𝑧𝜓
′(𝑧) + (𝛼
1−𝑝+𝑝𝜓(𝑧))𝜓(𝑧)] (1−𝛼) +𝛼𝛼
1(𝛼1−𝑝+𝑝𝜓(𝑧))
=
𝛼 𝛼1𝑧𝜓
′(𝑧) +[(1−𝛼) + 𝛼
𝛼1(𝛼1−𝑝+𝑝𝜓(𝑧))
]
𝜓(𝑧)
(1−𝛼) +𝛼𝛼
1(𝛼1−𝑝+𝑝𝜓(𝑧))
=𝜓(𝑧) + 𝑧𝜓
′(𝑧) 𝛼1
𝛼 −𝑝+𝑝𝜓(𝑧)
≺𝜙(𝑧).
(1.19)
Since
ℜ(𝑝𝜙(𝑧) +𝛼1
𝛼 −𝑝
)
>0 (𝛼 >0; 𝑧∈𝕌).
Thus, by (1.19) and Lemma 1.1, we know that
𝜓(𝑧) =
𝑧(𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) )′
𝑝𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) ≺𝜙(𝑧).
This completes the proof of Lemma 1.3. □
By similarly applying the method of proof of Lemma 1.3 for the classes𝒢𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙)
andℋ𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙), we get the following results.
Lemma 1.4. Let 𝑓 ∈ 𝒢𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙). Then
𝑧[(1−𝛼)(𝑔𝑝𝑙,𝑚(𝛼1)𝑓
)′
(𝑧) +𝛼(𝑔𝑙,𝑚𝑝 (𝛼1+ 1)𝑓 )′
(𝑧)]
𝑝[(1−𝛼)𝑔𝑙,𝑚𝑝 (𝛼1;𝑧) +𝛼𝑔𝑝𝑙,𝑚(𝛼1+ 1;𝑧)
] ≺𝜙(𝑧) (𝑧∈𝕌).
Furthermore, if𝜙∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) +𝛼1
𝛼 −𝑝
)
>0 (𝛼 >0; 𝑧∈𝕌).
Then
𝑧(
𝑔𝑙,𝑚 𝑝 (𝛼1;𝑧)
)′
𝑝𝑔𝑝𝑙,𝑚(𝛼1;𝑧)
≺𝜙(𝑧) (𝑧∈𝕌).
Lemma 1.5. Let 𝑓 ∈ ℋ𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙). Then
𝑧[(1−𝛼)(ℎ𝑙,𝑚𝑝 (𝛼1)𝑓
)′
(𝑧) +𝛼(ℎ𝑙,𝑚𝑝 (𝛼1+ 1)𝑓 )′
(𝑧)]
𝑝[(1−𝛼)ℎ𝑙,𝑚𝑝 (𝛼1;𝑧) +𝛼ℎ𝑙,𝑚𝑝 (𝛼1+ 1;𝑧)
Furthermore, if𝜙∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) +𝛼1
𝛼 −𝑝
)
>0 (𝛼 >0; 𝑧∈𝕌).
Then
𝑧(
ℎ𝑙,𝑚 𝑝 (𝛼1;𝑧)
)′
𝑝ℎ𝑙,𝑚𝑝 (𝛼1;𝑧)
≺𝜙(𝑧) (𝑧∈𝕌).
In the present paper, we aim at proving such results as inclusion relationships and convolution properties for the function classes ℱ𝑝,𝑘𝑙,𝑚(𝛼;𝛼1;𝜙), 𝒢𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙), ℋ𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙), 𝔉 𝑙,𝑚
𝑝,𝑘(𝛼;𝛼1;𝜙), 𝔊 𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙), and ℌ𝑙,𝑚𝑝 (𝛼;𝛼1;𝜙). The results
pre-sented here would provide extensions of those given in earlier works. Several other new results are also obtained.
2. A Set of Inclusion Relationships
At first, we provide some inclusion relationships for the classes ℱ𝑝,𝑘𝑙,𝑚(𝛼;𝛼1;𝜙),
𝒢𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙),ℋ 𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙),𝔉 𝑙,𝑚
𝑝,𝑘(𝛼;𝛼1;𝜙),𝔊 𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙),andℌ 𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙),which
were defined in the preceding section.
Theorem 2.1. Let 𝜙∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) +𝛼1
𝛼 −𝑝
)
>0 (𝛼 >0; 𝑧∈𝕌).
Then
ℱ𝑝,𝑘𝑙,𝑚(𝛼;𝛼1;𝜙)⊂ ℱ𝑝,𝑘𝑙,𝑚(𝛼1;𝜙).
Proof. Let𝑓 ∈ ℱ𝑝,𝑘𝑙,𝑚(𝛼;𝛼1;𝜙) and suppose that
𝑞(𝑧) =𝑧 (
𝐻𝑙,𝑚 𝑝 (𝛼1;𝑧)𝑓
)′ (𝑧)
𝑝𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) (𝑧∈𝕌). (2.1)
Then𝑞is analytic in 𝕌and𝑞(0) = 1. It follows from (1.3) and (2.1) that
𝑞(𝑧)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) = 𝛼1
𝑝 𝐻 𝑙,𝑚
𝑝 (𝛼1+ 1)𝑓(𝑧)−
𝛼1−𝑝
𝑝 𝐻 𝑙,𝑚
𝑝 (𝛼1)𝑓(𝑧). (2.2)
Differentiating both sides of (2.2) with respect to𝑧 and using (2.1), we have
𝑧𝑞′(𝑧) + ⎛
⎜
⎝𝛼1−𝑝+
𝑧(𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) )′
𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) ⎞
⎟ ⎠𝑞(𝑧) =
𝛼1
𝑝 𝑧(
𝐻𝑙,𝑚
𝑝 (𝛼1+ 1)𝑓 )′
(𝑧)
It now follows from (1.7), (1.17), (2.1) and (2.3) that
𝑧[(1−𝛼)(
𝐻𝑙,𝑚 𝑝 (𝛼1)𝑓
)′
(𝑧) +𝛼(
𝐻𝑙,𝑚
𝑝 (𝛼1+ 1)𝑓)
′
(𝑧)]
𝑝[(1−𝛼)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) +𝛼𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧) ]
=𝑝(1−𝛼)𝑞(𝑧)𝑓
𝑙,𝑚
𝑝,𝑘(𝛼1;𝑧) +𝛼𝛼1𝑝[𝑧𝑞
′(𝑧) + (𝛼
1−𝑝+𝑝𝜓(𝑧))𝑞(𝑧)]𝑓
𝑙,𝑚 𝑝,𝑘(𝛼1;𝑧)
𝑝[(1−𝛼)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) +𝛼𝛼1(𝛼1−𝑝+𝑝𝜓(𝑧))𝑓
𝑙,𝑚 𝑝,𝑘(𝛼1;𝑧)
]
=(1−𝛼)𝑞(𝑧) +
𝛼 𝛼1[𝑧𝑞
′(𝑧) + (𝛼
1−𝑝+𝑝𝜓(𝑧))𝑞(𝑧)] (1−𝛼) +𝛼𝛼
1(𝛼1−𝑝+𝑝𝜓(𝑧))
=
𝛼 𝛼1𝑧𝑞
′(𝑧) +[(1−𝛼) + 𝛼
𝛼1(𝛼1−𝑝+𝑝𝜓(𝑧))
]
𝑞(𝑧)
(1−𝛼) +𝛼𝛼
1(𝛼1−𝑝+𝑝𝜓(𝑧))
=𝑞(𝑧) + 𝑧𝑞
′(𝑧) 𝛼1
𝛼 −𝑝+𝑝𝜓(𝑧)
≺𝜙(𝑧) (𝑧∈𝕌).
(2.4)
Moreover, since
ℜ(𝑝𝜙(𝑧) +𝛼1
𝛼 −𝑝
)
>0 (𝛼 >0; 𝑧∈𝕌),
by Lemma 1.1, we know that
𝜓(𝑧) =
𝑧(𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧))
′
𝑝𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) ≺𝜙(𝑧) (𝑧∈𝕌).
Thus, an application of Lemma 1.2 to (2.4), we have
𝑞(𝑧)≺𝜙(𝑧) (𝑧∈𝕌),
that is𝑓 ∈ ℱ𝑝,𝑘𝑙,𝑚(𝛼1;𝜙). This implies that
ℱ𝑝,𝑘𝑙,𝑚(𝛼;𝛼1;𝜙)⊂ ℱ𝑝,𝑘𝑙,𝑚(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 𝒢𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙) and ℋ𝑙,𝑚𝑝 (𝛼;𝛼1;𝜙), we easily get the following inclusion relationships.
Corollary 2.1. Let 𝜙∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) +𝛼1
𝛼 −𝑝
)
>0 (𝛼 >0; 𝑧∈𝕌).
Then
𝒢𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙)⊂ 𝒢 𝑙,𝑚 𝑝 (𝛼1;𝜙).
Corollary 2.2. Let 𝜙∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) +𝛼1
𝛼 −𝑝
)
>0 (𝛼 >0; 𝑧∈𝕌).
Then
Theorem 2.2. Let 𝜙∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) +𝛼1
𝛼 −𝑝
)
>0 (𝛼 >0; 𝑧∈𝕌).
Then
𝔉𝑙,𝑚𝑝,𝑘(𝛼;𝛼1;𝜙)⊂𝔉𝑙,𝑚𝑝,𝑘(𝛼1;𝜙). (2.5)
Proof. Let𝑓 ∈𝔉𝑙,𝑚𝑝,𝑘(𝛼;𝛼1;𝜙) and suppose that
𝑝(𝑧) = 𝑧 (
𝐻𝑙,𝑚 𝑝 (𝛼1;𝑧)𝑓
)′ (𝑧)
𝑝𝔣𝑙,𝑚𝑝,𝑘(𝛼1;𝑧) (𝑧∈𝕌). (2.6)
Then𝑝is analytic in𝕌and𝑝(0) = 1. It follows from (1.3) and (2.6) that
𝑝(𝑧)𝔣𝑙,𝑚𝑝,𝑘(𝛼1;𝑧) =
𝛼1
𝑝 𝐻 𝑙,𝑚
𝑝 (𝛼1+ 1)𝑓(𝑧)−
𝛼1−𝑝
𝑝 𝐻 𝑙,𝑚
𝑝 (𝛼1)𝑓(𝑧). (2.7)
Differentiating both sides of (2.7) with respect to𝑧 and using (2.6), we have
𝑧𝑝′(𝑧) + ⎛
⎜
⎝𝛼1−𝑝+
𝑧(𝔣𝑙,𝑚𝑝,𝑘(𝛼1;𝑧))
′
𝔣𝑙,𝑚𝑝,𝑘(𝛼1;𝑧) ⎞
⎟ ⎠𝑝(𝑧) =
𝛼1
𝑝 𝑧(
𝐻𝑙,𝑚
𝑝 (𝛼1+ 1)𝑓)
′
(𝑧)
𝔣𝑙,𝑚𝑝,𝑘(𝛼1;𝑧)
.
Furthermore, we suppose that
𝜑(𝑧) =
𝑧(𝔣𝑙,𝑚𝑝,𝑘(𝛼1;𝑧))
′
𝑝𝔣𝑙,𝑚𝑝,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) (𝑧∈𝕌),
which implies that 𝑓 ∈ 𝔉𝑙,𝑚𝑝,𝑘(𝛼1;𝜙). The proof of Theorem 2.2 is evidently
com-pleted. □
By similarly applying the method of proof of Theorem 2.1 for the classes𝔊𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙) andℌ𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙), we easily get the following inclusion relationships.
Corollary 2.3. Let 𝜙∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) +𝛼1
𝛼 −𝑝
)
>0 (𝛼 >0; 𝑧∈𝕌).
Then
𝔊𝑙,𝑚𝑝 (𝛼;𝛼1;𝜙)⊂𝔊𝑙,𝑚𝑝 (𝛼1;𝜙).
Corollary 2.4. Let 𝜙∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) +𝛼1
𝛼 −𝑝
)
>0 (𝛼 >0; 𝑧∈𝕌).
Then
ℌ𝑙,𝑚𝑝 (𝛼;𝛼1;𝜙)⊂ℌ𝑙,𝑚𝑝 (𝛼1;𝜙).
By similarly applying the method of proof of Theorems 1 and 2 obtained by Wang
et al. [34] for the function classes𝒢𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙), 𝔊 𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙), ℋ 𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙) and
Corollary 2.5. Let 𝜙∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) +𝛼1−𝑝)>0 (𝑧∈𝕌).
Then
𝒢𝑙,𝑚
𝑝 (𝛼1+ 1;𝜙)⊂ 𝒢𝑝𝑙,𝑚(𝛼1;𝜙).
Corollary 2.6. Let 𝜙∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) +𝛼1−𝑝)>0 (𝑧∈𝕌).
Then
ℋ𝑙,𝑚
𝑝 (𝛼1+ 1;𝜙)⊂ ℋ𝑝𝑙,𝑚(𝛼1;𝜙).
Corollary 2.7. Let 𝜙∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) +𝛼1−𝑝)>0 (𝑧∈𝕌).
Then
𝔊𝑙,𝑚𝑝 (𝛼1+ 1;𝜙)⊂𝔊𝑙,𝑚𝑝 (𝛼1;𝜙).
Corollary 2.8. Let 𝜙∈ 𝒫 with
ℜ(𝑝𝜙(𝑧) +𝛼1−𝑝)>0 (𝑧∈𝕌).
Then
ℌ𝑙,𝑚𝑝 (𝛼1+ 1;𝜙)⊂ℌ𝑙,𝑚𝑝 (𝛼1;𝜙).
3. Convolution Properties
In this section, we provide some convolution properties for the function classes
ℱ𝑝,𝑘𝑙,𝑚(𝛼;𝛼1;𝜙), 𝒢𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙), andℋ𝑙,𝑚𝑝 (𝛼;𝛼1;𝜙).
Theorem 3.1. Let 𝑓 ∈ 𝒜𝑝 and𝜙∈ 𝒫. Then𝑓 ∈ ℱ𝑝,𝑘𝑙,𝑚(𝛼;𝛼1;𝜙)if and only if
1
𝑧
{
𝑓∗
[
(1−𝛼) (
𝑝𝑧𝑝+
∞
∑
𝑛=1
(𝛼1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛 𝑛+𝑝
𝑛! 𝑧
𝑛+𝑝
)
−𝑝(1−𝛼)𝜙(𝑒𝑖𝜃) (
𝑧𝑝+
∞
∑
𝑛=1
(𝛼1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛
1
𝑛!𝑧
𝑛+𝑝
)
∗
( 1
𝑘 𝑘−1 ∑
𝜐=0
𝑧𝑝
1−𝜀𝜐𝑧
)
+𝛼
(
𝑝𝑧𝑝+
∞
∑
𝑛=1
(𝛼1+ 1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛
𝑛+𝑝 𝑛! 𝑧
𝑛+𝑝
)
−𝑝 𝛼𝜙(𝑒𝑖𝜃) (
𝑧𝑝+
∞
∑
𝑛=1
(𝛼1+ 1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛
1
𝑛!𝑧
𝑛+𝑝
)
∗
( 1
𝑘 𝑘−1 ∑
𝜐=0
𝑧𝑝
1−𝜀𝜐𝑧
) ]}
∕
= 0
(𝑧∈𝕌; 0≦𝜃 <2𝜋).
Proof. Suppose that𝑓 ∈ ℱ𝑝,𝑘𝑙,𝑚(𝛼;𝛼1;𝜙). Since
𝑧[(1−𝛼)(𝐻𝑝𝑙,𝑚(𝛼1)𝑓
)′
(𝑧) +𝛼(𝐻𝑝𝑙,𝑚(𝛼1+ 1)𝑓 )′
(𝑧)]
𝑝[(1−𝛼)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) +𝛼𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧)
is equivalent to
𝑧[(1−𝛼)(𝐻𝑝𝑙,𝑚(𝛼1)𝑓)′(𝑧) +𝛼(𝐻𝑝𝑙,𝑚(𝛼1+ 1)𝑓 )′
(𝑧)]
𝑝[(1−𝛼)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) +𝛼𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧)
] ∕=𝜙(𝑒
𝑖𝜃) (0
≦𝜃 <2𝜋),
(3.1) it is easy to see that the condition (3.1) can be written as follows:
1
𝑧
{
𝑧[(1−𝛼)(
𝐻𝑝𝑙,𝑚(𝛼1)𝑓)′
(𝑧) +𝛼(
𝐻𝑝𝑙,𝑚(𝛼1+ 1)𝑓)
′
(𝑧)]
−𝑝[(1−𝛼)𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) +𝛼𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧) ]
𝜙(𝑒𝑖𝜃) }
∕
= 0 (0≦𝜃 <2𝜋).
(3.2)
On the other hand, we know from (1.2) that
𝑧(
𝐻𝑝𝑙,𝑚(𝛼1)𝑓 )′ (𝑧) = ( 𝑝𝑧𝑝+ ∞ ∑ 𝑛=1
(𝛼1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛 𝑛+𝑝
𝑛! 𝑧
𝑛+𝑝
)
∗𝑓(𝑧). (3.3)
Moreover, from the definition of𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧), we have
𝑓𝑝,𝑘𝑙,𝑚(𝛼1;𝑧) =𝐻𝑝𝑙,𝑚(𝛼1)𝑓(𝑧)∗
( 1 𝑘 𝑘−1 ∑ 𝜐=0 𝑧𝑝
1−𝜀𝜐𝑧
) = ( 𝑧𝑝+ ∞ ∑ 𝑛=1
(𝛼1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛
1 𝑛!𝑧 𝑛+𝑝 ) ∗ ( 1 𝑘 𝑘−1 ∑ 𝜐=0 𝑧𝑝
1−𝜀𝜐𝑧
)
∗𝑓(𝑧).
(3.4)
Replacing𝛼1 by𝛼1+ 1 in (3.3) and (3.4), we know that (3.3) and (3.4) also hold true, that is,
𝑧(
𝐻𝑝𝑙,𝑚(𝛼1+ 1)𝑓)
′ (𝑧) = ( 𝑝𝑧𝑝+ ∞ ∑ 𝑛=1
(𝛼1+ 1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛
𝑛+𝑝 𝑛! 𝑧
𝑛+𝑝
)
∗𝑓(𝑧), (3.5)
and
𝑓𝑝,𝑘𝑙,𝑚(𝛼1+ 1;𝑧) =𝐻𝑝𝑙,𝑚(𝛼1+ 1)𝑓(𝑧)∗ ( 1 𝑘 𝑘−1 ∑ 𝜐=0 𝑧𝑝
1−𝜀𝜐𝑧
) = ( 𝑧𝑝+ ∞ ∑ 𝑛=1
(𝛼1+ 1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛
1 𝑛!𝑧 𝑛+𝑝 ) ∗ ( 1 𝑘 𝑘−1 ∑ 𝜐=0 𝑧𝑝
1−𝜀𝜐𝑧
)
∗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 classes𝒢𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙)
andℋ𝑙,𝑚
Corollary 3.1. Let 𝑓 ∈ 𝒜𝑝 and𝜙∈ 𝒫.Then𝑓 ∈ 𝒢𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙)if and only if
1
𝑧
{
𝑓∗
[
(1−𝛼) (
𝑝𝑧𝑝+
∞
∑
𝑛=1
(𝛼1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛 𝑛+𝑝
𝑛! 𝑧
𝑛+𝑝
)
−𝑝(1−𝛼)𝜙(𝑒 𝑖𝜃)
2 ℎ1
+𝛼 ( 𝑝𝑧𝑝+ ∞ ∑ 𝑛=1
(𝛼1+ 1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛 𝑛+𝑝 𝑛! 𝑧 𝑛+𝑝 ) −𝑝 𝛼𝜙(𝑒 𝑖𝜃)
2 ℎ2
]
−𝑝(1−𝛼)𝜙(𝑒 𝑖𝜃)
2 (ℎ1∗𝑓)(𝑧)−
𝑝 𝛼𝜙(𝑒𝑖𝜃)
2 (ℎ2∗𝑓)(𝑧) }
∕
= 0 (0≦𝜃 <2𝜋),
where
ℎ1(𝑧) =𝑧𝑝+
∞
∑
𝑛=1
(𝛼1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛
1
𝑛!𝑧
𝑛+𝑝, (3.7)
and
ℎ2(𝑧) =𝑧𝑝+
∞
∑
𝑛=1
(𝛼1+ 1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛
1
𝑛!𝑧
𝑛+𝑝. (3.8)
Corollary 3.2. Let 𝑓 ∈ 𝒜𝑝 and𝜙∈ 𝒫.Then𝑓 ∈ ℋ𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙)if and only if
1
𝑧
{
𝑓∗
[
(1−𝛼) (
𝑝𝑧𝑝+
∞
∑
𝑛=1
(𝛼1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛 𝑛+𝑝
𝑛! 𝑧
𝑛+𝑝
)
−𝑝(1−𝛼)𝜙(𝑒 𝑖𝜃)
2 ℎ1
+𝛼 ( 𝑝𝑧𝑝+ ∞ ∑ 𝑛=1
(𝛼1+ 1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛
(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛
𝑛+𝑝 𝑛! 𝑧 𝑛+𝑝 ) −𝑝𝛼𝜙(𝑒 𝑖𝜃)
2 ℎ2 ]
+𝑝(1−𝛼)𝜙(𝑒
𝑖𝜃)
2 (ℎ1∗𝑓)(−𝑧) +
𝑝𝛼𝜙(𝑒𝑖𝜃)
2 (ℎ2∗𝑓)(−𝑧) }
∕
= 0 (0≦𝜃 <2𝜋),
whereℎ1 andℎ2 are given by (3.7)and (3.8), respectively.
Theorem 3.2. Let 𝑓 ∈ ℋ𝑙,𝑚
𝑝 (𝛼1;𝜙). Then
𝑓(𝑧) = [
𝑝
∫ 𝑧
0
𝜁𝑝−1𝜙(𝜔(𝜁))⋅exp (
𝑝
2 ∫ 𝜁
0
𝜙(𝜔(𝜉))−𝜙(𝜔(−𝜉))−2
𝜉 𝑑𝜉 ) 𝑑𝜁 ] ∗ (∞ ∑ 𝑛=0
𝑛!(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛
(𝛼1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛 𝑧𝑛+𝑝
)
,
(3.9)
where𝜔 is analytic in𝕌with
𝜔(0) = 0 and ∣𝜔(𝑧)∣<1 (𝑧∈𝕌).
Proof. From the definition ofℋ𝑙,𝑚
𝑝 (𝛼1;𝜙), we know that
𝑧(
𝐻𝑙,𝑚 𝑝 (𝛼1)𝑓
)′ (𝑧)
𝑝ℎ𝑙,𝑚𝑝 (𝛼1;𝑧)
= 2𝑧
(
𝐻𝑙,𝑚 𝑝 (𝛼1)𝑓
)′ (𝑧)
𝑝[(𝐻𝑝𝑙,𝑚(𝛼1)𝑓)(𝑧)−(𝐻
𝑙,𝑚
𝑝 (𝛼1)𝑓)(−𝑧)
] =𝜙(𝜔(𝑧)), (3.10)
where𝜔 is analytic in𝕌with
From (3.10), we get
2𝑧(𝐻𝑝𝑙,𝑚(𝛼1)𝑓
)′ (−𝑧)
𝑝[(𝐻𝑝𝑙,𝑚(𝛼1)𝑓)(𝑧)−(𝐻
𝑙,𝑚
𝑝 (𝛼1)𝑓)(−𝑧)
] =𝜙(𝜔(−𝑧)). (3.11)
It now follows from (3.10) and (3.11) that
𝑧(ℎ𝑙,𝑚𝑝 (𝛼1;𝑧))′
𝑝ℎ𝑙,𝑚𝑝 (𝛼1;𝑧)
= 1 2 [
𝜙(𝜔(𝑧))−𝜙(𝜔(−𝑧))]. (3.12)
We next find from (3.12) that
(
ℎ𝑙,𝑚𝑝 (𝛼1;𝑧)
)′
ℎ𝑙,𝑚𝑝 (𝛼1;𝑧) −𝑝
𝑧 = 𝑝
2⋅
𝜙(𝜔(𝑧))−𝜙(𝜔(−𝑧))−2
𝑧 . (3.13)
Upon integrating (3.13), we have
log (
ℎ𝑙,𝑚 𝑝 (𝛼1;𝑧)
𝑧𝑝
)
= 𝑝 2
∫ 𝑧
0
𝜙(𝜔(𝜉))−𝜙(𝜔(−𝜉))−2
𝜉 𝑑𝜉, (3.14)
which implies that
ℎ𝑙,𝑚𝑝 (𝛼1;𝑧) =𝑧𝑝⋅exp (
𝑝
2 ∫ 𝑧
0
𝜙(𝜔(𝜉))−𝜙(𝜔(−𝜉))−2
𝜉 𝑑𝜉
)
. (3.15)
It now follows from (3.10) and (3.15) that
(
𝐻𝑝𝑙,𝑚(𝛼1)𝑓)′
(𝑧) =𝑝ℎ
𝑙,𝑚 𝑝 (𝛼1;𝑧)
𝑧 ⋅𝜙(𝜔(𝑧))
=𝑝𝑧𝑝−1𝜙(𝜔(𝑧))⋅exp (
𝑝
2 ∫ 𝑧
0
𝜙(𝜔(𝜉))−𝜙(𝜔(−𝜉))−2
𝜉 𝑑𝜉
)
.
(3.16)
Upon integrating (3.16), we get
𝐻𝑝𝑙,𝑚(𝛼1)𝑓(𝑧) =𝑝 ∫ 𝑧
0
𝜁𝑝−1𝜙(𝜔(𝜁))⋅exp (
𝑝
2 ∫ 𝜁
0
𝜙(𝜔(𝜉))−𝜙(𝜔(−𝜉))−2
𝜉 𝑑𝜉
)
𝑑𝜁.
(3.17) Combining (1.2) and (3.17), we find that
𝑝
∫ 𝑧
0
𝜁𝑝−1𝜙(𝜔(𝜁))⋅exp (
𝑝
2 ∫ 𝜁
0
𝜙(𝜔(𝜉))−𝜙(𝜔(−𝜉))−2
𝜉 𝑑𝜉
)
𝑑𝜁
= [𝑧𝑝𝑙𝐹𝑚(𝛼1, . . . , 𝛼𝑙;𝛽1, . . . , 𝛽𝑚)]∗𝑓(𝑧).
(3.18)
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 class𝒢𝑙,𝑚
𝑝 (𝛼;𝛼1;𝜙),
Corollary 3.3. Let 𝑓 ∈ 𝒢𝑙,𝑚
𝑝 (𝛼1;𝜙). Then
𝑓(𝑧) = [
𝑝
∫ 𝑧
0
𝜁𝑝−1𝜙(𝜔(𝜁))⋅exp (
𝑝
2 ∫ 𝜁
0
𝜙(𝜔(𝜉)) +𝜙(𝜔(𝜉))−2
𝜉 𝑑𝜉
)
𝑑𝜁
]
∗
(∞ ∑
𝑛=0
𝑛!(𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛
(𝛼1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛 𝑧𝑛+𝑝
)
,
where𝜔 is analytic in𝕌with
𝜔(0) = 0 and ∣𝜔(𝑧)∣<1 (𝑧∈𝕌).
References
[1] R. Aghalary and G. Azadi,The Dziok-Srivastava operator and𝑘-uniformly starlike functions, J. Inequal. Pure Appl. Math.6(2005), Article 52, pp. 1–7 (electronic).
[2] R. Aghalary, S. B. Joshi, R. N. Mohapatra and V. Ravichandran,Subordinations for analytic functions defined by the Dziok-Srivastava linear operator,Appl. Math. Comput.187(2007), 13–19.
[3] R. M. Ali, V. Ravichandran, M. H. Khan and K. G. Subramanian, Applications of first order differential superordination to certain linear operators,Southeast Asian Bull. Math.
30(2006), 799–810.
[4] M. K. Aouf and G. Murugusundaramoorthy, On a subclass of uniformly convex functions defined by the Dziok-Srivastava operator, Aust. J. Math. Anal. Appl.5 (2008), Article 3, (electronic).
[5] M.-P. Chen, Z.-R. Wu and Z.-Z. Zou, On functions 𝛼-starlike with respect to symmetric conjugate points,J. Math. Anal. Appl.201(1996), 25–34.
[6] J. Dziok,On the convex combination of the Dziok-Srivastava operator,Appl. Math. Comput.
188(2007), 1214–1220.
[7] J. Dziok, On some applications of the Briot-Bouquet differential subordination, J. Math. Anal. Appl.328(2007), 295–301.
[8] J. Dziok and H. M. Srivastava,Classes of analytic functions associated with the generzlized hypergeometric function,Appl. Math. Comput.103(1999), 1–13.
[9] J. Dziok and H. M. Srivastava,Certain sbuclasses of analytic functions associated with the generzlized hypergeometric function,Integral Transform. Spec. Funct.14(2003), 7–18. [10] P. Eenigenburg, S. S. Miller, P. T. Mocanu and M. O. Reade,On a Briot-Bouquet differential
subordination, inGeneral Mathematics3, International Series of Numerical Mathematics, Vol.64, Birkh¨auser Verlag, Basel, 1983, pp. 339–348; see also Rev. Roumaine Math. Pures Appl.29(1984), 567–573.
[11] A. Gangadharan, T. N. Shanmugam and H. M. Srivastava,Generalized hypergeometric func-tion associated with𝑘-uniformly convex functions,Comput. Math. Appl.44(2002), 1515– 1526.
[12] Y.-Y. Huang and M.-S. Liu,Properties of certain subclasses of multivalent analytic functions involving the Dziok-Srivastava operator,Appl. Math. Comput.204(2008), 137–149. [13] S. S. Kumar, H. C. Taneja and V. Ravichandran,Classes of multivalent functions defined by
Dziok-Srivastava linear operator and multitiplier transformation, Kyungpook Math. J.46
(2006), 97–109.
[14] J.-L. Liu,Strongly starlike functions associated with the Dziok-Srivastava operator,Tamkang J. Math.35(2004), 37–42.
[15] J.-L. Liu and H. M. Srivastava, Certain properties of the Dziok-Srivastava operator,Appl. Math. Comput.159 (2004), 485–493.
[16] S. S. Miller and P. T. Mocanu, On some classes of first order differential subordinations, Michigan Math. J.32(1985), 185–195.
[19] S. Owa and H. M. Srivastava,Univalent and starlike generalized hypergermetric functions, Canad. J. Math.39(1987), 1057–1077.
[20] K. S. Padmanabhan and R. Parvatham, Some applications of differential subordination,Bull. Austral. Math. Soc.32(1985), 321–330.
[21] R. Parvatham and S. Radha, On𝛼-starlike and𝛼-close-to-convex functions with respect to 𝑛-symmetric points,Indian J. Pure Appl. Math.17(1986), 1114–1122.
[22] J. Patel, A. K. Mishra and H. M. Srivastava, Classes of multivalent analytic functions involv-ing the Dziok-Srivastava operator,Comput. Math. Appl.54(2007), 599–616.
[23] K. Piejko and J. Sok´ol,On the Dziok-Srivastava operator under multivalent analytic func-tions,Appl. Math. Comput.177(2006), 839–843.
[24] C. Ramachandran, T. N. Shanmugam, H. M. Srivastava and A. Swaminathan,A unified class of𝑘-uniformly convex functions defined by the Dziok-Srivastava linear operator,Appl. Math. Comput.190(2007), 1627–1636.
[25] V. Ravichandran,Starlike and convex functions with respect to conjugate points,Acta Math. Acad. Paedagog. Nyh´azi. (N.S.)20(2004), 31–37 (electronic).
[26] T. N. Shanmugam and M. P. Jeyaraman,Strongly Mocanu convex functions associated with the Dziok-Srivastava operator,Far East J. Math. Sci.20(2006), 49–57.
[27] T. N. Shanmugam, M. P. Jeyarama and A. Singaravelu,On sandwich theorems for certain subclass of analytic functions involving Dziok-Srivastava operator,Aust. J. Math. Anal. Appl.
4(2007), Article 17, (electronic).
[28] T. N. Shanmugam, C. Ramachandran, A. Singaravelu and M. P. Jeyaraman,On differential Sanwich theorems for certain subclass of analytic functions using Dziok-Srivastava operator, Far East J. Math. Sci.25(2007), 621–632.
[29] T. N. Shanmugam, S. Sivasubramanian, and G. Murugusundaramoorthy,On the unified class of functions of complex order involving Dziok-Srivastava operator, Gen. Math.15(2007), 149–160.
[30] H. M. Srivastava, Generalized hypergeometric functions and associated families of 𝑘-uniformly convex and𝑘-starlike functions,Gen. Math.15(2007), 201–226.
[31] H. M. Srivastava, G. Murugusundaramoorthy and S. Sivasubramanian,Hypergeometric func-tions in the parabolic starlike and uniformly convex domains, Integral Transform. Spec. Funct.18(2007), 511–520.
[32] K. Suchithra, A. Gangadharan and S. Owa, Neighborhood properties of certain classes of analytic functions using the Dziok-Srivastava operator,Tamsui Oxf. J. Math. Sci.24(2008), 185–200.
[33] Z.-G. Wang, C.-Y. Gao and S.-M. Yuan,On certain subclasses of close-to-convex and quasi-convex functions with respect to𝑘-symmetric points,J. Math. Anal. Appl.322(2006), 97– 106.
[34] Z.-G. Wang, Y.-P. Jiang and H. M. Srivastava,Some subclasses of multivalent analytic func-tions involving the Dziok-Srivastava operator, Integral Transforms Spec. Funct.19(2008), 129–146.
[35] N. Xu and D.-G. Yang,Some classes of analytic and multivalent functions involving a linear operator,Math. Comput. Modelling49(2009), 955–965.
Zhi-Gang Wang
Department of Mathematics and Computing Science, Hengyang Normal University, Hengyang 421008, Hunan, P. R. China
E-mail address:[email protected] Hai-Ping Shi
Department of Basic Courses, Guangdong Baiyun Institute, Guangzhou 510450, Guang-dong, P. R. China
