ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 1 Issue 3 (2009), Pages 49-63.
CERTAIN CLASSES OF 𝑘-UNIFORMLY CLOSE-TO-CONVEX FUNCTIONS AND OTHER RELATED FUNCTIONS DEFINED
BY USING THE DZIOK-SRIVASTAVA OPERATOR
(DEDICATED IN OCCASION OF THE 65-YEARS OF PROFESSOR R.K. RAINA)
H. M. SRIVASTAVA, SHU-HAI LI, HUO TANG
Abstract. Several interesting classes of𝑘-uniformly close-to-convex functions and𝑘-uniformly quasi-convex functions are defined here by using the Dziok- Srivastava operator. We provide necessary and sufficient coefficient conditions, extreme points, integral representations, and distortion bounds for functions belonging to each of these classes of𝑘-uniformly close-to-convex functions and 𝑘-uniformly quasi-convex functions.
1. Introduction, Definitions and Preliminaries Let𝒜denote the class of functions of the form:
𝑓(𝑧) =𝑧+
∞
∑
𝑛=2
𝑎𝑛𝑧𝑛, (1.1)
which are analytic in the open unit disk
𝕌={𝑧:𝑧∈ℂ and ∣𝑧∣<1}.
Also let𝒜− denote a subclass of𝒜consisting of functions of the form:
𝑓(𝑧) =𝑧−
∞
∑
𝑛=2
𝑎𝑛𝑧𝑛 (𝑎𝑛 ≧0), (1.2)
which are analytic in𝕌.
A function𝑓(𝑧)∈ 𝒜is said to be in the class of𝑘-uniformly convex functions of order𝛽 (0≦𝛽 <1),denoted by𝒰 𝒦(𝑘, 𝛽) (cf. [10]; see also [6] and [8]) if
ℜ (
1 +𝑧𝑓′′(𝑧) 𝑓′(𝑧)
)
> 𝑘
𝑧𝑓′′(𝑧) 𝑓′(𝑧)
+𝛽 (𝑘≧0; 0≦𝛽 <1; 𝑧∈𝕌). (1.3)
2000Mathematics Subject Classification. Primary 30C45, 33C20; Secondary 30C50.
Key words and phrases. Analytic functions;𝑘-uniformly convex functions;𝑘-uniformly starlike functions;𝑘-uniformly close-to-convex functions;𝑘-uniformly quasi-convex functions; Hadamard product (or convolution); generalized hypergeometric function; Pochhammer symbol; Gamma function; Dziok-Srivastava operator; Fox-Wright generalization of the hypergeometric function.
c
⃝2009 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.
Submitted October, 2009. Published November, 2009.
49
A corresponding class of 𝑘-uniformly starlike functions, denoted by 𝒰 𝒮(𝑘, 𝛽) consists of functions𝑓(𝑧)∈ 𝒜such that
ℜ
(𝑧𝑓′(𝑧) 𝑓(𝑧)
)
> 𝑘
𝑧𝑓′(𝑧) 𝑓(𝑧) −1
+𝛽 (𝑘≧0; 0≦𝛽 <1; 𝑧∈𝕌). (1.4) It is obvious from the inequalities in (1.3) and (1.4) that (see [10])
𝑓(𝑧)∈ 𝒰 𝒦(𝑘, 𝛽) ⇐⇒ 𝑧𝑓′(𝑧)∈ 𝒰 𝒮(𝑘, 𝛽). (1.5) Each of the function classes 𝒰 𝒦(𝑘, 𝛽) and 𝒰 𝒮(𝑘, 𝛽) provides unifications and generalizations various other (known or new) subclasses of𝒜. Several properties of some of the subclasses of the function classes𝒰 𝒦(𝑘, 𝛽) and𝒰 𝒮(𝑘, 𝛽) were studied recently in [9] (see also [6] and [8]).
Definition 1(see [1]). Define 𝒰 𝒞(𝑘, 𝛾, 𝛽) to be the family of functions 𝑓(𝑧)∈ 𝒜 such that
ℜ
(𝑧𝑓′(𝑧) 𝑔(𝑧)
)
> 𝑘
𝑧𝑓′(𝑧) 𝑔(𝑧) −1
+𝛾 (
𝑘≧0; 𝛾∈[0,1); 𝑧∈𝕌)
(1.6) for some function𝑔(𝑧)∈ 𝒰 𝒮(𝑘, 𝛽).
Definition 2 (see [1]). Define𝒰 𝒬(𝑘, 𝛾, 𝛽) to be the family of functions𝑓(𝑧)∈ 𝒜 such that
ℜ
((𝑧𝑓′(𝑧))′ 𝑔′(𝑧)
)
> 𝑘
(𝑧𝑓′(𝑧))′ 𝑔′(𝑧) −1
+𝛾 (
𝑘≧0; 𝛾∈[0,1); 𝑧∈𝕌)
(1.7) for some function𝑔(𝑧)∈ 𝒰 𝒦(𝑘, 𝛽).
It readily follows from Definitions 1 and 2 that
𝑓(𝑧)∈ 𝒰 𝒬(𝑘, 𝛾, 𝛽) ⇐⇒ 𝑧𝑓′(𝑧)∈ 𝒰 𝒞(𝑘, 𝛾, 𝛽). (1.8) We say that𝒰 𝒞(0, 𝛾, 𝛽) is the class ofclose-to-convex functions of order𝛾and type 𝛽 in 𝕌 and that 𝒰 𝒬(0, 𝛾, 𝛽) is the class of quasi-convex functions of order 𝛾 and type𝛽 in𝕌.
Definition 3. For functions 𝑓(𝑧)∈ 𝒜given by (1.1), and𝑔(𝑧)∈ 𝒜given by 𝑔(𝑧) =𝑧+
∞
∑
𝑛=2
𝑏𝑛𝑧𝑛, (1.9)
we define the Hadamard product (or convolution) of𝑓(𝑧) and𝑔(𝑧) by (𝑓 ∗𝑔)(𝑧) :=𝑧+
∞
∑
𝑛=2
𝑎𝑛𝑏𝑛𝑧𝑛 =: (𝑔∗𝑓)(𝑧) (𝑧∈𝕌). (1.10) For complex parameters
𝛼𝑗 ∈ℂ (𝑗= 1,⋅ ⋅ ⋅, 𝑙) and 𝛽𝑗∈ℂ∖ℤ−0 (𝑗= 1,⋅ ⋅ ⋅, 𝑚; ℤ−0 :={0,−1,−2,⋅ ⋅ ⋅ }), the generalized hypergeometric function 𝑙𝐹𝑚(with𝑙numerator and𝑚denominator parameters) is defined by
𝑙𝐹𝑚(𝛼1,⋅ ⋅ ⋅ , 𝛼𝑙;𝛽1,⋅ ⋅ ⋅, 𝛽𝑚) =
∞
∑
𝑛=0
(𝛼1)𝑛⋅ ⋅ ⋅(𝛼𝑙)𝑛 (𝛽1)𝑛⋅ ⋅ ⋅(𝛽𝑚)𝑛
⋅ 𝑧𝑛
𝑛! (1.11)
(𝑙≦𝑚+𝑙; 𝑙, 𝑚∈ℕ0:={0,1,2,⋅ ⋅ ⋅ }=ℕ∪ {0}),
where (𝜆)𝜈denotes the Pochhammer symbol (or theshiftedfactorial, since (1)𝑛=𝑛!
for𝑛∈ℕ) defined, in terms of the familiar Gamma functions, by (𝜆)𝜈:= Γ(𝜆+𝜈)
Γ(𝜆) =
⎧
⎨
⎩
1 (𝜈= 0; 𝜆∈ℂ∖ {0})
𝜆(𝜆+ 1)⋅ ⋅ ⋅(𝜆+𝑛−1) (𝜈=𝑛∈ℕ; 𝜆∈ℂ).
Now, corresponding to the function
ℎ(𝛼1,⋅ ⋅ ⋅ , 𝛼𝑙;𝛽1,⋅ ⋅ ⋅, 𝛽𝑚;𝑧) =𝑧𝑙𝐹𝑚(𝛼1,⋅ ⋅ ⋅, 𝛼𝑙;𝛽1,⋅ ⋅ ⋅ , 𝛽𝑚),
theDziok-Srivastava linear operator (see [3], [4], [5] and [11]; see also [7], [14] and [15])
𝐻𝑚𝑙 (𝛼1,⋅ ⋅ ⋅ , 𝛼𝑙;𝛽1,⋅ ⋅ ⋅, 𝛽𝑚)
is defined as follows by using the Hadamard product (or convolution):
𝐻𝑚𝑙 (𝛼1,⋅ ⋅ ⋅, 𝛼𝑙;𝛽1,⋅ ⋅ ⋅ , 𝛽𝑚)𝑓(𝑧)
=ℎ(𝛼1,⋅ ⋅ ⋅ , 𝛼𝑙;𝛽1,⋅ ⋅ ⋅, 𝛽𝑚;𝑧)∗𝑓(𝑧)
=𝑧+
∞
∑
𝑛=2
𝜑𝑛(𝛼1,⋅ ⋅ ⋅ , 𝛼𝑙;𝛽1,⋅ ⋅ ⋅ , 𝛽𝑚)𝑎𝑛𝑧𝑛, (1.12) where, for convenience,
𝜑𝑛(𝛼1,⋅ ⋅ ⋅ , 𝛼𝑙;𝛽1,⋅ ⋅ ⋅, 𝛽𝑚) is given by
𝜑𝑛(𝛼1,⋅ ⋅ ⋅, 𝛼𝑙;𝛽1,⋅ ⋅ ⋅, 𝛽𝑚) := (𝛼1)𝑛−1⋅ ⋅ ⋅(𝛼𝑙)𝑛−1 (𝛽1)𝑛−1⋅ ⋅ ⋅(𝛽𝑚)𝑛−1 ⋅ 1
(𝑛−1)!. (1.13) It is well known (see, for example, [5]) that
𝛼1𝐻𝑚𝑙 (𝛼1+ 1, 𝛼2,⋅ ⋅ ⋅, 𝛼𝑙;𝛽1,⋅ ⋅ ⋅, 𝛽𝑚)𝑓(𝑧)
=𝑧(
𝐻𝑚𝑙 (𝛼1+ 1, 𝛼2,⋅ ⋅ ⋅, 𝛼𝑙;𝛽1,⋅ ⋅ ⋅, 𝛽𝑚)𝑓(𝑧))′
+ (𝛼1−1)𝐻𝑚𝑙 (𝛼1, 𝛼2,⋅ ⋅ ⋅, 𝛼𝑙;𝛽1,⋅ ⋅ ⋅ , 𝛽𝑚)𝑓(𝑧). (1.14) For notational simplification in our investigation, we write
𝐻𝑚𝑙 [𝛼1]𝑓(𝑧) =𝐻𝑚𝑙 (𝛼1,⋅ ⋅ ⋅ , 𝛼𝑙;𝛽1,⋅ ⋅ ⋅, 𝛽𝑚)𝑓(𝑧). (1.15) We now define the linear operator𝐿𝜏,𝛼𝜆,𝑗,𝑚1 as follows:
𝐿0𝜆,𝛼1𝑓(𝑧) =𝑓(𝑧), (1.16)
𝐿1,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧) = (1−𝜆)𝐻𝑚𝑙 [𝛼1]𝑓(𝑧) +𝜆𝑧(
𝐻𝑚𝑙 [𝛼1]𝑓(𝑧))′
=𝐿𝛼𝜆,𝑗,𝑚1 𝑓(𝑧) (𝜆≧0), (1.17)
𝐿2,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧) =𝐿𝛼𝜆,𝑗,𝑚1 (
𝐿1,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))
(1.18) and, in general,
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧) =𝐿𝛼𝜆,𝑗,𝑚1 (
𝐿𝜏−1,𝛼𝜆,𝑗,𝑚1𝑓(𝑧))
(𝑙≦𝑚+ 1; 𝑙, 𝑚∈ℕ0; 𝜏 ∈ℕ). (1.19) If the function𝑓(𝑧) is given by (1.1), then we see from (1.12), (1.13), (1.17) and (1.19) that
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧) =𝑧+
∞
∑
𝑛=2
𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)𝑎𝑛𝑧𝑛 (𝜏 ∈ℕ0), (1.20)
where
𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚) =
((𝛼1)𝑛−1⋅ ⋅ ⋅(𝛼𝑙)𝑛−1
(𝛽1)𝑛−1⋅ ⋅ ⋅(𝛽𝑚)𝑛−1 ⋅[1 +𝜆(𝑛−1)]
(𝑛−1)!
)𝜏
(1.21) (𝑛∈ℕ∖ {1}; 𝜏∈ℕ0).
When
𝜏 = 1 and 𝜆= 0,
the linear operator 𝐿𝜏,𝛼𝜆,𝑗,𝑚1 would reduce to the familiar Dziok-Srivastava linear operator given by (1.12) above (see, for example, [3]). For a linear operator which is essentially analogous to the Dziok-Srivastava operator in (1.12), but uses instead the Fox-Wright generalization of the hypergeometric function 𝑙𝐹𝑚defined here by (1.11), the interested reader may be referred to the recent works [2] and [12] as well as to the closely-related works cited in each of these recent works.
By applying the general operator 𝐿𝜏,𝛼𝜆,𝑗,𝑚1 , we define the following subclasses of the function class𝒜.
I. Let 𝒰 𝒮𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛽) be the class of functions 𝑓(𝑧) ∈ 𝒜 satisfying the following inequality:
ℜ (𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧) )
> 𝑘
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧) −1
+𝛽 (
𝑘≧0; 𝛽∈[0,1))
. (1.22) Observe that
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧)∈ 𝒰 𝒮(𝑘, 𝛽).
II. Let𝒰 𝒦𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛽) be the class of functions 𝑓(𝑧)∈ 𝒜satisfying the following inequality:
ℜ (
1 +𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′′
(𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
)
> 𝑘
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′′
(𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
+𝛽 (
𝑘≧0; 𝛽 ∈[0,1)) . (1.23) Observe that
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧)∈ 𝒰 𝒦(𝑘, 𝛽).
III.Let𝒰 𝒞𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽) be the class of functions𝑓 ∈ 𝒜such that ℜ
(𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) )
> 𝑘
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) −1
+𝛾 (
𝑘≧0; 𝛾∈[0,1))
(1.24) for some function𝑔(𝑧)∈ 𝒰 𝒮𝑙𝑚(𝜏, 𝑘, 𝛽). Observe that
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧)∈ 𝒰 𝒞(𝑘, 𝛾, 𝛽).
IV.Let𝒰 𝒬𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽) be the class of functions𝑓 ∈ 𝒜such that
ℜ (
1 + 𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′′
(𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧))′
)
> 𝑘
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′′
(𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧))′
+𝛾 (
𝑘≧0; 𝛾∈[0,1)) (1.25) for some function𝑔(𝑧)∈ 𝒰 𝒦𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛽). Observe that
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧)∈ 𝒰 𝒦(𝑘, 𝛾, 𝛽).
It is clear from two of the above definitions that
𝑓(𝑧)∈ 𝒰 𝒦𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛽)⇐⇒𝑧𝑓′(𝑧)∈ 𝒰 𝒞𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛽). (1.26) Finally, in terms of the above-defined function classes, we write
𝒰 𝒮−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛽) =𝒜−∩ 𝒰 𝒮𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛽), 𝒰 𝒦−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛽) =𝒜−∩ 𝒰 𝒦𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛽), 𝒰 𝒞−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽) =𝒜−∩ 𝒰 𝒞𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽) and
𝒰 𝒬−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽) =𝒜−∩ 𝒰 𝒬𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽).
The various properties and characteristics of functions in the class𝒰 𝒮𝑙𝑚(1,0, 𝑘, 𝛽) were investigated by Dziok and Srivastava [3]. In this paper, we obtain several relationships and properties of the convolution operators considered here. Our paper mainly studies the functions in the class 𝒰 𝒞𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛽). We first prove a sufficient condition for a function 𝑓 ∈ 𝒜 to be in the class 𝒰 𝒞𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛽). We then provide necessary and sufficient coefficient conditions, extreme points, integral representations, distortion bounds, radii of starlikeness and convexity for functions in the class𝒰 𝒞𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛽).
2. First Set of Main Results
First of all, we obtain a sufficient condition for a function 𝑓 ∈ 𝒜to be in the class𝒰 𝒞𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽).
Theorem 1. Let 𝑓(𝑧)∈ 𝒜 be given by (1.1). Suppose also that 𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚) is given by(1.21).If
𝑘≧0, 𝛽 ∈[0,1), 𝛾∈[0,1), 𝜆≧0, 𝜏 ∈ℕ0
and ∞
∑
𝑛=2
[2𝑘∣𝑛𝑎𝑛−𝑏𝑛∣+ (1−𝛾)∣𝑏𝑛∣]
𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)<1−𝛾, then𝑓(𝑧)∈ 𝒰 𝒞𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽).
Proof. By the definition of the function class𝒰 𝒞𝑙𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽), it suffices to show for a function𝑓(𝑧)∈ 𝒜given by (1.1) that
𝑘
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) −1
− ℜ (𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) −𝛾 )
≦2𝑘
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) −1
≦2𝑘
∞
∑
𝑛=2
𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)∣𝑛𝑎𝑛−𝑏𝑛∣ ⋅ ∣𝑧∣𝑛−1 1−
∞
∑
𝑛=2
𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)∣𝑏𝑛∣ ⋅ ∣𝑧∣𝑛−1
. (2.1)
Now the last expression in (2.1) is bounded above by 1−𝛾 if and only if
∞
∑
𝑛=2
[2𝑘∣𝑛𝑎𝑛−𝑏𝑛∣+ (1−𝛾)∣𝑏𝑛∣]
𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)<1−𝛾,
which evidently completes the proof of Theorem 1. □ We next provide a necessary and sufficient coefficient bound for a given function 𝑓(𝑧)∈ 𝒜− to belong to the class𝒰 𝒞−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽).
Theorem 2. Let 𝑓(𝑧)∈ 𝒜− be given by(1.2). Also let𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)be given by (1.21). Then𝑓 ∈ 𝒰 𝒞−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽)if and only if
∞
∑
𝑛=2
[𝑛(1 +𝑘)𝑎𝑛−(𝑘+𝛾)𝑏𝑛]
𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)<1−𝛾. (2.2) Proof. Suppose that𝑓(𝑧)∈ 𝒰 𝒞−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽). Then, making use of the fact that
ℜ(𝜔)> 𝑘∣𝜔−1∣+𝛾⇐⇒ ℜ(
𝜔(1 +𝑘𝑒𝑖𝜙)−𝑘𝑒𝑖𝜙)
> 𝛾 (𝛾∈ℝ) and letting
𝜔= 𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧)
in (1.3), we obtain ℜ
(𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) (1 +𝑘𝑒𝑖𝜙)−𝑘𝑒𝑖𝜙 )
> 𝛾 or, equivalently,
ℜ
⎛
⎜
⎜
⎜
⎝
(1 +𝑘𝑒𝑖𝜙)𝑧 (
𝑧− ∑∞
𝑛=2
𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)𝑎𝑛𝑧𝑛 )′
−(𝑘𝑒𝑖𝜙+𝛾) (
𝑧− ∑∞
𝑛=2
𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)𝑏𝑛𝑧𝑛 )
𝑧−
∞
∑
𝑛=2
𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)𝑏𝑛𝑧𝑛
⎞
⎟
⎟
⎟
⎠
>0,
which holds true for all𝑧∈𝕌. By letting𝑧→1−through real values, we thus find that
ℜ
⎛
⎜
⎜
⎝
(1−𝛾)−(1 +𝑘𝑒𝑖𝜙)
∞
∑
𝑛=2
𝑛𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)𝑎𝑛+ (𝛾+𝑘𝑒𝑖𝜙)
∞
∑
𝑛=2
𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)𝑏𝑛
1−
∞
∑
𝑛=2
𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)𝑏𝑛
⎞
⎟
⎟
⎠
>0,
and so (by the mean value theorem) we have ℜ
(
(1−𝛽)−(1 +𝑘𝑒𝑖𝛾)
∞
∑
𝑛=2
𝑛𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)𝑎𝑛+ (𝛽+𝑘𝑒𝑖𝜙)
∞
∑
𝑛=2
𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)𝑏𝑛
)
>0.
Therefore, we get
∞
∑
𝑛=2
[𝑛(1 +𝑘)𝑎𝑛−(𝑘+𝛾)𝑏𝑛]
𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)<1−𝛾, which proves the first part of Theorem 2.
Conversely, we let the inequality (2.2) hold true.
Then, in light of the fact that
ℜ(𝜔)> 𝛾⇐⇒ ∣𝜔−(1 +𝛾)∣<∣𝜔+ (1−𝛾)∣ (𝛾∈ℝ),
we need only to show that
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) − (
1 +𝑘
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) −1 )
+𝛾
<
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) + (
1−𝑘
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) −1 )
−𝛾 By setting
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧)
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧)
=𝑒𝑖𝜗, we may write
𝔈=
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) + (
1−𝑘
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) −1
−𝛾 )
= ∣𝑧∣
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧)
(𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
+ (1−𝛾)𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧)
𝑧 −𝑘
(𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
−𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) 𝑧
= ∣𝑧∣
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧)
(2−𝛾)−
∞
∑
𝑛=2
[𝑛𝑎𝑛+ (1−𝛾)𝑏𝑛]𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)𝑧𝑛−1
−𝑒𝑖𝜗
−
∞
∑
𝑛=2
(𝑘𝑛𝑎𝑛−𝑘𝑏𝑛)𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)𝑧𝑛−1
> ∣𝑧∣
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧)
(
(2−𝛾)−
∞
∑
𝑛=2
(𝑛(1 +𝑘)𝑎𝑛+ (1−𝑘−𝛾)𝑏𝑛)𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚) )
and 𝔉=
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) − (
1 +𝑘
𝑧(𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′ 𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) −1
+𝛾 )
= ∣𝑧∣
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧)
(𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
−(1 +𝛾)𝐿𝜏,𝛼𝜆,𝑙,𝑚1 𝑔(𝑧)
𝑧 −𝑘
(𝐻𝑚𝑙 [𝛼1]𝑓(𝑧))′
−𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) 𝑧
= ∣𝑧∣
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧)
−𝛾−
∞
∑
𝑛=2
[𝑛𝑎𝑛−(1 +𝛾)𝑏𝑛]𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)𝑧𝑛−1
−𝑒𝑖𝜗
−
∞
∑
𝑛=2
(𝑘𝑛𝑎𝑛−𝑘𝑏𝑛)𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)𝑧𝑛−1
< ∣𝑧∣
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧)
( 𝛾+
∞
∑
𝑛=2
[𝑛(1 +𝑘)𝑎𝑛−(1 +𝑘+𝛾)𝑏𝑛]𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚) )
.
It is easy to verify that
𝔈−𝔉>0
in case the inequality (2.2) holds true. The proof of Theorem 2 is thus completed.
□
When
𝑓(𝑧) =𝑔(𝑧) (𝑧∈𝕌), Theorem 2 would yield the following corollary.
Corollary 1. Let 𝑔(𝑧)∈ 𝒜− be given by
𝑔(𝑧) =𝑧−
∞
∑
𝑛=2
𝑏𝑛𝑧𝑛 (𝑏𝑛≧0), (2.3)
Then𝑔(𝑧)∈ 𝒰 𝒮−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛽)if and only if
∞
∑
𝑛=2
[(𝑛−1)𝑘+𝑛−𝛽]𝑏𝑛𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)
1−𝛽 <1.
Corollary 2. If 𝑔(𝑧)∈ 𝒰 𝒮−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛽) is given by(2.3),then
∞
∑
𝑛=2
𝑏𝑛< 1−𝛽
(2 +𝑘−𝛽)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚).
Proof. Since𝑔(𝑧)∈ 𝒰 𝒮−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛽) is given by (2.3), we can apply Corollary 1 to obtain
(𝑘+ 2−𝛽)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚)
∞
∑
𝑛=2
𝑏𝑛
≦
∞
∑
𝑛=2
𝑏𝑛[(𝑛−1)𝑘+𝑛−𝛽]𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)
<1−𝛽.
We thus find that
∞
∑
𝑛=2
𝑏𝑛< 1−𝛽
(2 +𝑘−𝛽)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚),
which proves Corollary 2. □
Corollary 3. If 𝑔(𝑧)∈ 𝒰 𝒮−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛽) is given by(2.3),then
𝑏𝑛 < 1−𝛽
(2 +𝑘−𝛽)𝑎𝑛𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚). 3. Further Results and Consequences
In this section, several further results involving the various function classes which were introduced in Section 1.
Theorem 3. If 𝑔(𝑧)∈ 𝒰 𝒮−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛽),then 𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) = exp
(∫ 𝑧 0
𝑘−𝛽𝑄(𝑡) 𝑡[𝑘−𝑄(𝑡)]𝑑𝑡
)
(∣𝑄(𝑧)∣<1; 𝑧∈𝕌)
(3.1) and
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) = exp (∫
∣𝑥∣=1
log[
(𝑘−𝑥𝑧)−1−𝛽] 𝑑𝜇(𝑥)
)
, (3.2)
where𝜇(𝑥)is a probability measure on the set:
𝑋 ={𝑥:∣𝑥∣= 1}.
Proof. The case 𝑘= 0 of the assertion (3.1) if Theorem 3 is obvious. Let 𝑘∕= 0.
Then, for
𝑔(𝑧)∈𝑈 𝑆−𝑙,𝑚(𝑘, 𝛽) and 𝜔= 𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) , we have
ℜ(𝜔)> 𝑘∣𝜔−1∣+𝛽.
We thus find that
𝜔−1 𝜔−𝛽
< 1
𝑘 and 𝜔−1
𝜔−𝛽 =𝑄(𝑧) 𝑘
(∣𝑄(𝑧)∣<1; 𝑧∈𝕌) , which readily yields
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) = 𝑘−𝛽𝑄(𝑧) 𝑧[𝑘−𝑄(𝑧)]
and, therefore,
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) = exp (∫ 𝑧
0
𝑘−𝛽𝑄(𝑡) 𝑡[𝑘−𝑄(𝑡)]𝑑𝑡
) .
In order to derive the second representation (3.2), corresponding to the set:
𝑋 ={𝑥:∣𝑥∣= 1}, we observe that
𝜔−1 𝜔−𝛽 < 1
𝑘𝑥𝑧 or, equivalently, that
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) = 𝑘−𝛽𝑄(𝑧) 𝑧[𝑘−𝑄(𝑧)]
=⇒log
(𝐻𝑚𝑙 [𝛼1]𝑔(𝑧) 𝑧
)
=−(1 +𝛽) log(𝑘−𝑥𝑧).
Thus, if𝜇(𝑥) is the probability measure on𝑋, then 𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) = exp
(∫
∣𝑥∣=1
log[
(𝑘−𝑥𝑧)−1−𝛽] 𝑑𝜇(𝑥)
) .
□ Theorem 4. If 𝑓(𝑧)∈ 𝒰 𝒞−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽),then
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧) =
∫ 𝑧 0
[𝑘−𝛾𝑄(𝑡) 𝑘−𝑄(𝑡) exp
(∫
∣𝑥∣=1
log[
(𝑘−𝑥𝑡)−1−𝛽] 𝑑𝜇(𝑥)
)]
𝑑𝑡, (3.3) where𝜇(𝑥)is a probability measure on the following set:
𝑋 ={𝑥:∣𝑥∣= 1}.
Proof. The case𝑘 = 0 of the assertion (3.3) of Theorem 4 is obvious. Let𝑘 ∕= 0.
Then, for
𝑓 ∈ 𝒰 𝒞−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛽) and 𝜔= 𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) , we have
ℜ(𝜔)> 𝑘∣𝜔−1∣+𝛾.
We thus find that
𝜔−1 𝜔−𝛾
< 1
𝑘 and 𝜔−1
𝜔−𝛾 = 𝑄(𝑧) 𝑘
(∣𝑄(𝑧)∣<1; 𝑧∈𝕌) , which easily yields
𝑧(
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑓(𝑧))′
𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) = 𝑘−𝛾𝑄(𝑧)
𝑧[𝑘−𝑄(𝑧)]. (3.4)
Moreover, from Theorem 3, we have 𝐿𝜏,𝛼𝜆,𝑗,𝑚1 𝑔(𝑧) = exp
(∫
∣𝑥∣=1
log[
(𝑘−𝑥𝑧)−1−𝛽] 𝑑𝜇(𝑥)
)
, (3.5)
where𝜇(𝑥) is a probability measure on the set:
𝑋 ={𝑥:∣𝑥∣= 1}.
The assertion (3.3) of Theorem 4 would now follow from (3.4) and (3.5). □ Next we obtain a distortion bounds for the functions𝑓(𝑧) and𝑔(𝑧).
Theorem 5. If 𝑔(𝑧)∈ 𝒰 𝒮−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛽),then
∣𝑧∣ − 1−𝛽
(2 +𝑘−𝛽)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚)∣𝑧∣2
<∣𝑔(𝑧)∣<∣𝑧∣+ 1−𝛽
(2 +𝑘−𝛽)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚) ∣𝑧∣2 (𝑧∈𝕌) (3.6) and
1− 2(1−𝛽)
(2 +𝑘−𝛽)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚) ∣𝑧∣
<∣𝑔′(𝑧)∣<1 + 2(1−𝛽)
(2 +𝑘−𝛽)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚) ∣𝑧∣ (𝑧∈𝕌). (3.7) Proof. For𝑔(𝑧)∈ 𝒰 𝒮−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛽) given by (2.3), we find from Corollary 2 that
∞
∑
𝑛=2
𝑏𝑛< 1−𝛽
(2 +𝑘−𝛽)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚), (3.8) which implies that
∣𝑔(𝑧)∣≦∣𝑧∣+∣𝑧∣2
∞
∑
𝑛=2
𝑏𝑛<∣𝑧∣+ 1−𝛽
(2 +𝑘−𝛽)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚)∣𝑧∣2 (𝑧∈𝕌) and
∣𝑔(𝑧)∣≧∣𝑧∣ − ∣𝑧∣2
∞
∑
𝑛=2
𝑏𝑛>∣𝑧∣ − 1−𝛽
(2 +𝑘−𝛽)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚)∣𝑧∣2 (𝑧∈𝕌).
Thus the assertion (3.6) of Theorem 5 follows at once.
In a similar manner, for the derivative𝑔′(𝑧), the following inequalities:
∣𝑔′(𝑧)∣≦1 +
∞
∑
𝑛=2
𝑛𝑏𝑛∣𝑧∣𝑛−1<1 +∣𝑧∣
∞
∑
𝑛=2
𝑛𝑏𝑛 (𝑧∈𝕌)
and ∞
∑
𝑛=2
𝑛𝑏𝑛 < 2(1−𝛽)
(2 +𝑘−𝛽)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚)
lead us immediately to the assertion (3.7) of Theorem 5. This completes the proof
of Theorem 5. □
Theorem 6. If 𝑓 ∈ 𝒰 𝒞−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽),then
∣𝑧∣ − 1−𝛾
2(1 +𝑘)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚) (
1 + (𝑘+𝛾)(1−𝛽) (1−𝛾)(2 +𝑘−𝛽)
)
∣𝑧∣2<∣𝑓(𝑧)∣
<∣𝑧∣+ 1−𝛾
2(1 +𝑘)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚) (
1 + (𝑘+𝛾)(1−𝛽) (1−𝛾)(2 +𝑘−𝛽)
)
∣𝑧∣2 (𝑧∈𝕌) (3.9) and
1− 1−𝛾
(1 +𝑘)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚) (
1 + (𝑘+𝛾)(1−𝛽) (1−𝛾)(2 +𝑘−𝛽)
)
∣𝑧∣<∣𝑓′(𝑧)∣
<1 + 1−𝛾
(1 +𝑘)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚) [
1 + (𝑘+𝛾)(1−𝛽) (1−𝛾)(2 +𝑘−𝛽)
]
∣𝑧∣ (𝑧∈𝕌). (3.10) Proof. For𝑓 ∈ 𝒰 𝒞−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽) given by (1.2), by using Theorem 1, we obtain
2(1 +𝑘)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚)
∞
∑
𝑛=2
𝑎𝑛<
∞
∑
𝑛=2
𝑛(1 +𝑘)𝑎𝑛𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)
<1−𝛾+
∞
∑
𝑛=2
(𝑘+𝛾)𝑏𝑛𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚), (3.11) which immediately yields
∞
∑
𝑛=2
𝑎𝑛< 1−𝛾
2(1 +𝑘)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚)
+ 𝑘+𝛾
2(1 +𝑘)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚)
∞
∑
𝑛=2
𝑏𝑛𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚). (3.12) Also, by applying Corollary 1, we have
∞
∑
𝑛=2
𝑏𝑛𝜙𝜏𝑛(𝛼1, 𝜆, 𝑙, 𝑚)< 1−𝛽 2 +𝑘−𝛽, so that
∣𝑓(𝑧)∣≦∣𝑧∣+∣𝑧∣2
∞
∑
𝑛=2
𝑎𝑛
<∣𝑧∣+ 1−𝛾
2(1 +𝑘)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚) (
1 + (𝑘+𝛾)(1−𝛽) (1−𝛾)(2 +𝑘−𝛽)
)
∣𝑧∣2 (𝑧∈𝕌).
Similarly, we can show that
∣𝑓(𝑧)∣≧∣𝑧∣ − ∣𝑧∣2
∞
∑
𝑛=2
𝑎𝑛
>∣𝑧∣ − 1−𝛾
2(1 +𝑘)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚) (
1 + (𝑘+𝛾)(1−𝛽) (1−𝛾)(2 +𝑘−𝛽)
)
∣𝑧∣2 (𝑧∈𝕌).
We thus have proved the assertion (3.9) of Theorem 6.
In a similar manner, for the derivative𝑓′(𝑧), the following inequalities:
∣𝑓′(𝑧)∣≦1 +
∞
∑
𝑛=2
𝑛𝑎𝑛∣𝑧∣𝑛−1<1 +∣𝑧∣
∞
∑
𝑛=2
𝑛𝑎𝑛 (𝑧∈𝕌) and
∞
∑
𝑛=2
𝑛𝑎𝑛< 1−𝛾
(1 +𝑘)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚) [
1 + (𝑘+𝛾)(1−𝛽) (1−𝛾)(2 +𝑘−𝛽)
]
lead us to the assertion (3.12) of Theorem 6. This evidently completes the proof of
Theorem 6. □
It is not difficult to deduce Corollary 4 below.
Corollary 4. Let 𝑓 ∈ 𝒰 𝒞−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽). Then {
𝜔:∣𝜔∣<1− 1−𝛾
(1 +𝑘)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚) (
1 + (𝑘+𝛾)(1−𝛽) (1−𝛾)(2 +𝑘−𝛽)
)}
⊂𝑓(𝕌)
⊂ {
𝜔:∣𝜔∣<1 + 1−𝛾
(1 +𝑘)𝜙𝜏2(𝛼1, 𝜆, 𝑙, 𝑚) (
1 + (𝑘+𝛾)(1−𝛽) (1−𝛾)(2 +𝑘−𝛽)
)}
. (3.13) Theorem 7 below follows easily from Corollary 1. In fact, the proof of Theorem 7 is essentially analogous to that of Theorem 8, which we have chosen to present here in detail.
Theorem 7. Let
𝑔𝑚(𝑧) =𝑧−
∞
∑
𝑛=2
𝑏𝑗,𝑚𝑧𝑗 ∈ 𝒰 𝒮−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽) (𝑚= 1,2).
Then
𝑔(𝑧) = (1−𝜉)𝑔1(𝑧) +𝜉𝑔2(𝑧) =𝑧−
∞
∑
𝑗=2
𝑏𝑗𝑧𝑗
∈ 𝒰 𝒮−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽) (0≦𝜉≦1). (3.14) Theorem 8. Let
𝑓𝑚(𝑧) =𝑧−
∞
∑
𝑛=2
𝑎𝑗,𝑚𝑧𝑗∈ 𝒰 𝒞−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽) (𝑚= 1,2).
Then
𝑓(𝑧) = (1−𝜉)𝑓1(𝑧) +𝜉𝑓2(𝑧) =𝑧−
∞
∑
𝑗=2
𝑎𝑗𝑧𝑗
∈ 𝒰 𝒞−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽) (0≦𝜉≦1). (3.15) Proof. Since
𝑓𝑚(𝑧)∈ 𝒰 𝒞−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽) (𝑚= 1,2), by using Theorem 2, we get the following coefficient inequalities:
∞
∑
𝑗=2
[(1 +𝑘)𝑗𝑎𝑗,1𝜙𝜏𝑗(𝛼1, 𝜆, 𝑙, 𝑚)−(𝑘+𝛾)𝑏𝑗,1𝜙𝜏𝑗(𝛼1, 𝜆, 𝑙, 𝑚)]<1−𝛾 and
∞
∑
𝑗=2
[(1 +𝑘)𝑗𝑎𝑗,2𝜙𝜏𝑗(𝛼1, 𝜆, 𝑙, 𝑚)−(𝑘+𝛾)𝑏𝑗,2𝜙𝜏𝑗(𝛼1, 𝜆, 𝑙, 𝑚)]<1−𝛾.
Furthermore, in view of the following obvious relationships:
𝑎𝑗= (1−𝜉)𝑎𝑗,1+𝜉𝑎𝑗,2 and 𝑏𝑗= (1−𝜉)𝑏𝑗,1+𝜉𝑏𝑗,2
(𝑗∈ℕ∖ {1}; 0≦𝑥𝑖≦1), we thus find that
∞
∑
𝑗=2
[(1 +𝑘)𝑗𝑎𝑗𝜙𝜏𝑗(𝛼1, 𝜆, 𝑙, 𝑚)−(𝑘+𝛾)𝑏𝑗𝜙𝜏𝑗(𝛼1, 𝜆, 𝑙, 𝑚)]
=
∞
∑
𝑗=2
(1 +𝑘)𝑗𝜙𝜏𝑗(𝛼1, 𝜆, 𝑙, 𝑚) [(1−𝜉)𝑎𝑗,1(𝑧) +𝜉𝑎𝑗,2(𝑧)]
−
∞
∑
𝑗=2
(𝑘+𝛾)𝑏𝑗𝜙𝜏𝑗(𝛼1, 𝜆, 𝑙, 𝑚) [(1−𝜉)𝑏𝑗,1(𝑧) +𝜉𝑏𝑗,2(𝑧)]
=
∞
∑
𝑗=2
(1−𝜉)[
(1 +𝑘)𝑗𝑎𝑗,1𝜙𝜏𝑗(𝛼1, 𝜆, 𝑙, 𝑚)−(𝑘+𝛽)𝑏𝑗,1𝜙𝜏𝑗(𝛼1, 𝜆, 𝑙, 𝑚)]
+
∞
∑
𝑗=2
𝜉[(1 +𝑘)𝑗𝑎𝑗,2𝜙𝜏𝑗(𝛼1, 𝜆, 𝑙, 𝑚)−(𝑘+𝛾)𝑏𝑗,2𝜙𝜏𝑗(𝛼1, 𝜆, 𝑙, 𝑚)]
≦(1−𝜉)(1−𝛾) +𝜉(1−𝛾) = 1−𝛾.
Thus, by using Theorem 2 again, we finally obtain 𝑓(𝑧)∈ 𝒰 𝒞−𝑙,𝑚(𝜏, 𝜆, 𝑘, 𝛾, 𝛽),
which completes the proof of Theorem 8. □
We remark in conclusion that, by suitably specializing the parameters involved in the results presented in this paper, we can deduce numerousfurthercorollaries and consequences of each of these results.
Acknowledgments. The present investigation was partially supported by the Natural Science Foundation of Inner Mongolia under Grant 2009MS0113 and by theHigher School Research Foundation of Inner Mongoliaunder Grant NJzy08150.
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H. M. Srivastava
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
E-mail address:[email protected]
Shu-Hai Li
Department of Mathematics, Chifeng University, Chifeng, Inner Mongolia 024000, People’s Republic of China
E-mail address:[email protected]
Huo Tang
Department of Mathematics, Chifeng University, Chifeng, Inner Mongolia 024000, People’s Republic of China
E-mail address:[email protected]
