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1. UtilitasMathematica
ISSN 0315-3681 Volume 120, 2023
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New Class of p-valent Functions Defined by Multiplier Transformations
Lafta Hussain Hassan1, Najah Ali Jiben Al-Ziadi2
1
Department of Mathematics, College of Education, University of Al-Qadisiyah, Diwaniya-Iraq,
laftaalnaeely@gmail.com
2
Department of Mathematics, College of Education, University of Al-Qadisiyah, Diwaniya-Iraq,
najah.ali@qu.edu.iq
Abstract
The object of this paper to study the new class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) of p-valent functions
defined by multiplier transformations 𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)(𝛿, 𝑙 ≥ 0, 𝛽 ≥ 𝜈 ≥ 0; 𝑝 ∈ ℕ) in the open unit
disk ∆ = {𝑡 ∶ 𝑡 ∈ ℂ and |𝑡| < 1}. We obtain some geometric properties for this class, like,
coefficient estimate, radii of convexity, starlikeness and close-to-convexity, extreme points,
closure theorems, integral operators and integral means inequalities.
Keywords: Holomorphic function, p-valent functions, multiplier transformations, coefficient
inequality, radii of convexity and starlikeness, extreme points, closure theorem.
1. Introduction
Let 𝒜(𝑝, 𝑘) symbolize the class of functions normalized by
𝑔(𝑡) = 𝑡𝑝
+ ∑ 𝑏𝑛
∞
𝑛=𝑘+𝑝
𝑡𝑛 (𝑡 ∈ ∆; 𝑝, 𝑘 ∈ ℕ
= { 1, 2, 3, … }), (1.1)
which are holomorphic and p-valent in the open unit disk ∆ = {𝑡 ∶ 𝑡 ∈ ℂ and |𝑡| < 1}.
Let ℳ(𝑝, 𝑘) symbolize the function subclass of 𝒜(𝑝, 𝑘) consisting of functions of the shape:
𝑔(𝑡) = 𝑡𝑝
− ∑ 𝑏𝑛
∞
𝑛=𝑘+𝑝
𝑡𝑛 (𝑡 ∈ ∆; 𝑏𝑛 ≥ 0; 𝑛 ≥ 𝑘 + 𝑝; 𝑝, 𝑘 ∈ ℕ
= {1, 2, 3, … }). (1.2)
For function 𝑔(𝑡) ∈ ℳ(𝑝, 𝑘), given by (1.2), and ℎ(𝑡) ∈ ℳ(𝑝, 𝑘) given by

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ℎ(𝑡) = 𝑡𝑝
− ∑ 𝑐𝑛
∞
𝑛=𝑘+𝑝
𝑡𝑛 (𝑡 ∈ ∆; 𝑐𝑛 ≥ 0; 𝑛 ≥ 𝑘 + 𝑝; 𝑝, 𝑘 ∈ ℕ
= {1, 2, 3, … }), (1.3)
the Hadamard product (or convolution) of 𝑔(𝑡) and ℎ(𝑡) is defined by
(𝑔 ∗ ℎ)(𝑡) = 𝑡𝑝
− ∑ 𝑏𝑛𝑐𝑛
∞
𝑛=𝑘+𝑝
𝑡𝑛
= (ℎ ∗ 𝑔)(𝑡). (1.4)
A function 𝑔(𝑡) ∈ ℳ(𝑝, 𝑘) is called p-valent starlike of order 𝛼 (0 ≤ 𝛼 < 𝑝), if 𝑔(𝑡) fulfills the
condition:
𝑅𝑒 (
𝑡𝑔′(𝑡)
𝑔(𝑡)
) > 𝛼 (𝑡 ∈ ∆; 0 ≤ 𝛼 < 𝑝; 𝑝, 𝑘 ∈ ℕ = {1, 2, 3, … }). (1.5)
Also, a function 𝑔(𝑡) ∈ ℳ(𝑝, 𝑘) is called p-valent convex of order 𝛼 (0 ≤ 𝛼 < 𝑝), if 𝑔(𝑡)
fulfills the condition:
𝑅𝑒 (1 +
𝑡𝑔′′(𝑡)
𝑔′(𝑡)
) > 𝛼 (𝑡 ∈ ∆; 0 ≤ 𝛼 < 𝑝; 𝑝, 𝑘 ∈ ℕ = {1, 2, 3, … }). (1.6)
Symbolize by 𝑆𝑛
∗(𝑝, 𝛼) the class of p-valent starlike functions of order 𝛼. Symbolize by
𝐶𝑛(𝑝, 𝛼) the class of p-valent convex functions of order𝛼, which were studied by Owa [9]. It is
noted that
𝑔(𝑡) ∈ 𝐶𝑛(𝑝, 𝛼) if and only if
𝑡𝑔′(𝑡)
𝑝
∈ 𝑆𝑛
∗(𝑝, 𝛼).
A function 𝑔(𝑡) ∈ ℳ(𝑝, 𝑘) is called p-valent close to convex of order 𝛼 (0 ≤ 𝛼 < 𝑝), if 𝑔(𝑡)
fulfills the condition:
𝑅𝑒 (
𝑔′(𝑡)
𝑡𝑝−1
) > 𝛼 (𝑡 ∈ ∆; 0 ≤ 𝛼 < 𝑝; 𝑝, 𝑘 ∈ ℕ = {1, 2, 3, … }). (1.7)
Let the functions 𝑔(𝑡) and ℎ(𝑡) be holomorphic in Δ. We say that the function 𝑔(𝑡) is
subordinate to ℎ(𝑡), if there exists a Schwarz function 𝑤(𝑡) holomorphic in 𝛥 with 𝑤(0) = 0 and
|𝑤(𝑡)| < 1, 𝑡 ∈ 𝛥 such that 𝑔(𝑡) = ℎ(𝑤(𝑡)). This subordinate is denoted by 𝑔 ≺ ℎ or 𝑔(𝑡) ≺
ℎ(𝑡) (𝑡 ∈ 𝛥). It is well known that (see [8]), if the function 𝑔(𝑡) is univalent in 𝛥, then 𝑔(𝑡) ≺
ℎ(𝑡) if and only if 𝑔(0) = ℎ(0) and 𝑔(𝛥) ⊂ ℎ(𝛥).

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For a function 𝑔(𝑡) in 𝒜(𝑝, 𝑘), Deniz and Orhan [5] define the multiplier transformations
𝒥𝑝
𝛿(𝛽, 𝑣, 𝑙) as follows:
Definition (1.1)[ 5]: Let 𝑔(𝑡) ∈ 𝒜(𝑝, 𝑘). For the parameters 𝛿, 𝛽, 𝜈, 𝑙 ∈ ℝ, 𝛽 ≥ 𝜈 ≥ 0 and
𝛿, 𝑙 ≥ 0, we realize the multiplier transformations 𝒥𝑝
𝛿(𝛽, 𝑣, 𝑙) on 𝒜(𝑝, 𝑘) as
𝒥𝑝
0(𝛽, 𝜈, 𝑙)𝑔(𝑡) = 𝑔(𝑡)
(𝑝 + 𝑙)𝒥𝑝
1(𝛽, 𝜈, 𝑙)𝑔(𝑡) = 𝛽𝜈𝑡2
𝑔′′(𝑡) + (𝛽 − 𝜈 + (1 − 𝑝)𝛽𝜈)𝑡𝑔′(𝑡) + (𝑝(1 − 𝛽 + 𝜈) + 𝑙)𝑔(𝑡)
(𝑝 + 𝑙)𝒥𝑝
2(𝛽, 𝜈, 𝑙)𝑔(𝑡) = 𝛽𝜈𝑡2
[𝒥𝑝
1(𝛽, 𝜈, 𝑙)𝑔(𝑡)]
′′
+ (𝛽 − 𝜈 + (1 − 𝑝)𝛽𝜈)𝑡[𝒥𝑝
1(𝛽, 𝜈, 𝑙)𝑔(𝑡)]
′
+(𝑝(1 − 𝛽 + 𝜈) + 𝑙)𝒥𝑝
1(𝛽, 𝜈, 𝑙)𝑔(𝑡)
𝒥𝑝
δ1
(𝛽, 𝜈, 𝑙) (𝒥𝑝
δ2
(𝛽, 𝜈, 𝑙)𝑔(𝑡)) = 𝒥𝑝
δ2
(𝛽, 𝜈, 𝑙) (𝒥𝑝
δ1
(𝛽, 𝜈, 𝑙)𝑔(𝑡)) for 𝑡 ∈ 𝛥 and 𝑝, 𝑘 ∈ ℕ
= {1,2, … }. (1.8)
If 𝑔(𝑡) is given by (1.1), then from the definition of the multiplier transformations 𝒥𝑝
𝛿(𝛽, 𝑣, 𝑙),
we see that
𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡) = 𝑡𝑝
+ ∑ 𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)𝑏𝑛
∞
𝑛=𝑘+𝑝
𝑡𝑛
, (1.9)
where
𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙) = [
(𝑛 − 𝑝)(𝛽𝜈𝑛 + 𝛽 − 𝜈) + 𝑝 + 𝑙
𝑝 + 𝑙
]
𝛿
(𝛿, 𝛽, 𝜈, 𝑙 ∈ ℝ, 𝛽 ≥ 𝜈 ≥ 0 and 𝛿, 𝑙
≥ 0). (1.10)
With a view to derive our main outcomes, we have to recall here the following lemmas.
Lemma (1.1) [3]: Let 𝛼 ≥ 0. Then 𝑅𝑒(𝑤) > 𝛼 if and only if |𝑤 − (𝑝 + 𝛼)| < |𝑤 + (𝑝 − 𝛼)|,
where 𝑤 be any complex number.
Lemma (1.2) [7]: If 𝑔 and ℎ are holomorphic in ∆ with 𝑔 ≺ ℎ, then
∫ |𝑔(𝑟𝑒𝑖𝜃
)|
𝜂
2𝜋
0
𝑑𝜃 ≤ ∫ |ℎ(𝑟𝑒𝑖𝜃
)|
𝜂
2𝜋
0
𝑑𝜃,
where 𝜂 > 0, 𝑡 = 𝑟𝑒𝑖𝜃
and (0 < 𝑟 < 1).

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Definition (1.2): A function 𝑔(𝑡) ∈ ℳ(𝑝, 𝑘) of the shape (1.2), is told to be in the class
ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) if and only if satisfies the following condition:
𝑅𝑒 {
𝜆𝜇𝑡3
(𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′′′
+ (2𝜆𝜇 + 𝜆 − 𝜇)𝑡2
(𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙) 𝑔(𝑡))
′′
+ 𝑡 (𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′
𝜆𝜇𝑡2 (𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙) 𝑔(𝑡))
′′
+ (𝜆 − 𝜇)𝑡 (𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′
+ (1 − 𝜆 + 𝜇)𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡)
}
> 𝛼, (1.11)
where 𝑡 ∈ ∆ , 0 ≤ 𝛼 < 𝑝, 0 ≤ 𝜇 ≤ 𝜆 ≤ 1, 𝛿, 𝑙 ≥ 0, 𝛽 ≥ 𝜈 ≥ 0, 𝑝, 𝑘 ∈ ℕ = {1,2, 3, … }.
Noting that by customizing the parameter 𝛿, 𝜇, 𝜆, 𝑝 and 𝑘, we obtain the following various
subclasses investigated by different investigators:
1) If 𝛿 = 0 and 𝜇 = 0, the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) shortens to the class 𝒯
𝑛(𝑝, 𝛼, 𝜆)
which is introduced by Altintaş et al. [2].
2) If 𝛿 = 0, 𝜇 = 0 and 𝑝 = 1, the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) shortens to the class
Р(𝑛, 𝛼, 𝜆) which is introduced by Altintaş [1].
3) If 𝛿 = 0, 𝜇 = 0 and 𝜆 = 1, the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) shortens to the class
𝐶𝑛(𝑝, 𝛼) which is introduced by Owa [9].
4) If 𝛿 = 0, 𝜇 = 0, 𝜆 = 1, 𝑝 = 1 and 𝑘 = 1, the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) shortens to
the class 𝐶(𝛼) which is introduced Silverman [11].
5) If 𝛿 = 0, 𝜇 = 0 and 𝜆 = 0, the class ℳ𝒜(𝑝, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) shortens to the class 𝑆𝑛
∗(𝑝, 𝛼)
which is introduced by Owa [9].
6) If 𝛿 = 0, 𝜇 = 0, 𝜆 = 0, 𝑝 = 1 and 𝑘 = 1, the class ℳ𝒜(𝑝, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) shortens to
the class 𝒯∗(𝛼) which is introduced Silverman [11].
Several of the upcoming characteristics were researched for different classes in [4, 5, 6, 10, 12].
2. Coefficient Inequality
From the following theorem, we get the necessary and sufficient condition for the
function 𝑔(𝑡) to be in the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙).
Theorem (2.1): Let 𝑔(𝑡) be in the shape (1.2). Then 𝑔(𝑡) is in the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙)
if and only if

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∑ (𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙) 𝑏𝑛
∞
𝑛=𝑘+𝑝
≤ (𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1], (2.1)
where 𝑡 ∈ ∆, 0 ≤ 𝛼 < 𝑝, 0 ≤ 𝜇 ≤ 𝜆 ≤ 1, 𝛿, 𝑙 ≥ 0, 𝛽 ≥ 𝜈 ≥ 0, 𝑝, 𝑘 ∈ ℕ = {1, 2, 3, … }.
The result is sharp for the function 𝑔(𝑡) given by
𝑔(𝑡) = 𝑡𝑝
−
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑡𝑛
, (𝑛 ≥ 𝑘 + 𝑝; 𝑝 , 𝑘
∈ ℕ). (2.2)
Proof: Assume that 𝑔(𝑡) ∈ ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙), so we have
𝑅𝑒 {
𝜆𝜇𝑡3
(𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′′′
+ (2𝜆𝜇 + 𝜆 − 𝜇)𝑡2
(𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙) 𝑔(𝑡))
′′
+ 𝑡 (𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′
𝜆𝜇𝑡2 (𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙) 𝑔(𝑡))
′′
+ (𝜆 − 𝜇)𝑡 (𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′
+ (1 − 𝜆 + 𝜇)𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡)
}
> 𝛼.
Then
𝑅𝑒 {
𝑝[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1 ]𝑡𝑝
− ∑ 𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)𝑛[(𝑛 − 1)(𝜆𝜇 + 𝜆 − 𝜇) + 1]𝑏𝑛𝑡𝑛
∞
𝑛=𝑘+𝑝
[𝑝(𝜆𝜇𝑝 − 𝜆𝜇 + 𝜆 − 𝜇) + (1 − 𝜆 + 𝜇)]𝑡𝑝 − ∑ 𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)[𝑛(𝜆𝜇𝑛 − 𝜆𝜇 + 𝜆 − 𝜇)
∞
𝑛=𝑘+𝑝 + 1 − 𝜆 + 𝜇]𝑏𝑛𝑡𝑛
}
> 𝛼.
Or equivalently
𝑅𝑒 {
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]𝑡𝑝
− ∑ 𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝑏𝑛𝑡𝑛
∞
𝑛=𝑘+𝑝
[𝑝(𝜆𝜇𝑝 − 𝜆𝜇 + 𝜆 − 𝜇) + (1 − 𝜆 + 𝜇)]𝑡𝑝 − ∑ 𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)[𝑛(𝜆𝜇𝑛 − 𝜆𝜇 + 𝜆 − 𝜇)
∞
𝑛=𝑘+𝑝 + 1 − 𝜆 + 𝜇]𝑏𝑛𝑡𝑛
}
> 0.
This inequality is valid for 𝑡 ∈ ∆. Letting 𝑡 → 1−
yields
𝑅𝑒 {(𝑝 − 𝛼)[(𝑝 − 1)((𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1)]
− ∑ (𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)𝑏𝑛
∞
𝑛=𝑘+𝑝
} > 0.

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Therefore
∑ (𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)𝑏𝑛
∞
𝑛=𝑘+𝑝
≤ (𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1].
Conversely, let (2.1) hold. We will show that (1.11) is valid and then 𝑔(𝑡) ∈
ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙). By Lemma (1.1), we set
𝑤 =
𝜆𝜇𝑡3
(𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′′′
+ (2𝜆𝜇 + 𝜆 − 𝜇)𝑡2
(𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙) 𝑔(𝑡))
′′
+ 𝑡 (𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′
𝜆𝜇𝑡2 (𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙) 𝑔(𝑡))
′′
+ (𝜆 − 𝜇)𝑡 (𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′
+ (1 − 𝜆 + 𝜇)𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡)
Or show that
𝑇 =
1
|𝑁(𝑡)|
|𝜆𝜇𝑡3
(𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′′′
+ (2𝜆𝜇 + 𝜆 − 𝜇)𝑡2
(𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙) 𝑔(𝑡))
′′
+ 𝑡 (𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′
− (𝑝 + 𝛼)𝜆𝜇𝑡2
(𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙) 𝑔(𝑡))
′′
− (𝑝 + 𝛼)(𝜆 − 𝜇)𝑡 (𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′
− (𝑝 + 𝛼)(1 − 𝜆 + 𝜇)𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡)|
<
1
|𝑁(𝑡)|
|𝜆𝜇𝑡3
(𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′′′
+ (2𝜆𝜇 + 𝜆 − 𝜇)𝑡2
(𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙) 𝑔(𝑡))
′′
+ 𝑡 (𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′
+ (𝑝 − 𝛼)𝜆𝜇𝑡2
(𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙) 𝑔(𝑡))
′′
+ (𝑝 − 𝛼)(𝜆 − 𝜇)𝑡 (𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡))
′
+ (𝑝 − 𝛼)(1 − 𝜆 + 𝜇)𝒥𝑝
𝛿(𝛽, 𝜈, 𝑙)𝑔(𝑡)|
= 𝑄,
where
𝑁(𝑡) = 𝜆𝜇𝑡2
(𝒥𝑝
𝛿(𝛽, 𝜈, 𝜄) 𝑔(𝑡))
′′
+ (𝜆 − 𝜇)𝑡 (𝒥𝑝
𝛿(𝛽, 𝜈, 𝜄)𝑔(𝑡))
′
+ (1 − 𝜆 + 𝜇)𝒥𝑝
𝛿(𝛽, 𝜈, 𝜄)𝑔(𝑡)
and it is simple to prove that 𝑄 − 𝑇 > 0.
The proof is therefore complete.
Corollary (2.1): Let 𝑔(𝑡) be in the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙). Then
𝑏𝑛 ≤
(𝑝 − 𝛼)[(𝑝 − 1)((𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1)]
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
, (2.3)

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where 𝑡 ∈ ∆, 0 ≤ 𝛼 < 1, 0 ≤ 𝜇 ≤ 𝜆 ≤ 1, 𝛿, 𝑙 ≥ 0, 𝛽 ≥ 𝜈 ≥ 0, 𝑝, 𝑘 ∈ ℕ = {1, 2, 3, … }.
3. Radii of Convexity, Starlikeness and Close-to-Convexity
In the next theorems, we will find the radii of convexity, starlikeness and close-to-convexity for
the functions in the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙).
Theorem (3.1): Let 𝑔(𝑡) ∈ ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙). Then the function 𝑔(𝑡) is p-valent convex
of order 𝛾 (0 ≤ 𝛾 < 𝑝) in the disk |𝑡| < 𝑅1, where
𝑅1 = 𝑖𝑛𝑓
𝑛
[
𝑝(𝑝 − 𝛾)(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝜄)
𝑛(𝑛 − 𝛾)(𝑝 − 𝛼)[(𝑝 − 1)((𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1)]
]
1
𝑛−𝑝
,
(𝑛 ≥ 𝑘 + 𝑝; 𝑝, 𝑘 ∈ ℕ ).
The outcome is sharp for the function 𝑔(𝑡) given by (2.2).
Proof: It is sufficient to prove that
|1 +
𝑡𝑔′′(𝑡)
𝑔′(𝑡)
− 𝑝| ≤ 𝑝 − 𝛾 (0 ≤ 𝛾 < 𝑝),
for |𝑡| < 𝑅1, we have
|1 +
𝑡𝑔′′(𝑡)
𝑔′(𝑡)
− 𝑝| ≤
∑ 𝑛(𝑛 − 𝑝)𝑏𝑛 |𝑡|𝑛−𝑝
∞
𝑛=𝑘+𝑝
𝑝 − ∑ 𝑛𝑏𝑛 |𝑡|𝑛−𝑝
∞
𝑛=𝑘+𝑝
.
Thus
|1 +
𝑡𝑔′′(𝑡)
𝑔′(𝑡)
− 𝑝| ≤ 𝑝 − 𝛾,
if
∑
𝑛(𝑛 − 𝛾)
𝑝(𝑝 − 𝛾)
𝑏𝑛 |𝑡|𝑛−𝑝
≤ 1. (3.1)
∞
𝑛=𝑘+𝑝
Therefore, by Theorem (2.1), (3.1) will be true if
𝑛(𝑛 − 𝛾)
𝑝(𝑝 − 𝛾)
|𝑡|𝑛−𝑝
≤
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
,

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and hence
|𝑡| ≤ [
𝑝(𝑝 − 𝛾)(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑛(𝑛 − 𝛾)(𝑝 − 𝛼)[(𝑝 − 1)((𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1)]
]
1
𝑛−𝑝
, (𝑛 ≥ 𝑘 + 𝑝; 𝑝, 𝑘 ∈ ℕ ).
Setting |𝑡| = 𝑅1, we get the desired result.
Theorem (3.2): Let 𝑔(𝑡) ∈ ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙). Then the function 𝑔(𝑡) is p-valent starlike
of order 𝛾 (0 ≤ 𝛾 < 𝑝) in the disk |𝑡| < 𝑅2, where
𝑅2 = 𝑖𝑛𝑓
𝑛
[
(𝑝 − 𝛾)(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑛 − 𝛾)(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
]
1
𝑛−𝑝
,
(𝑛 ≥ 𝑘 + 𝑝; 𝑝, 𝑘 ∈ ℕ).
The outcome is sharp for the function 𝑔(𝑡) given by (2.2).
Proof: It is sufficient to prove that
|
𝑡𝑔′(𝑡)
𝑔(𝑡)
− 𝑝| ≤ 𝑝 − 𝛾 (0 ≤ 𝛾 < 𝑝),
for |𝑡| < 𝑅2, we have
|
𝑡𝑔′(𝑡)
𝑔(𝑡)
− 𝑝| ≤
∑ (𝑛 − 𝑝)𝑏𝑛 |𝑡|𝑛−𝑝
∞
𝑛=𝑘+𝑝
1 − ∑ 𝑏𝑛 |𝑡|𝑛−𝑝
∞
𝑛=𝑘+𝑝
.
Thus
|
𝑡𝑔′(𝑡)
𝑔(𝑡)
− 𝑝| ≤ 𝑝 − 𝛾,
if
∑
𝑛 − 𝛾
𝑝 − 𝛾
𝑏𝑛 |𝑡|𝑛−𝑝
≤ 1. (3.2)
∞
𝑛=𝑘+𝑝
Therefore, by Theorem (2.1), (3.2) will be true if
𝑛 − 𝛾
𝑝 − 𝛾
|𝑡|𝑛−𝑝
≤
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
.
and hence

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|𝑡| ≤ [
(𝑝 − 𝛾)(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑛 − 𝛾)(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
]
1
𝑛−𝑝
,
(𝑛 ≥ 𝑘 + 𝑝; 𝑝, 𝑘 ∈ ℕ ).
Setting |𝑡| = 𝑅2, we obtain the desired result.
Theorem (3.3): Let 𝑔(𝑡) ∈ ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙). Then the function 𝑔(𝑡) is p-valent close to
convex of order 𝛾 (0 ≤ 𝛾 < 𝑝) in the disk |𝑡| < 𝑅3, where
𝑅3 = 𝑖𝑛𝑓
𝑛
[
(𝑝 − 𝛾)(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑛(𝑝 − 𝛼)[(𝑝 − 1)((𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1)]
]
1
𝑛−𝑝
, (𝑛 ≥ 𝑘 + 𝑝; 𝑝, 𝑘
∈ ℕ ).
The result is sharp for the function 𝑔(𝑡) given by (2.2).
Proof: It is sufficient to prove that
|
𝑔′(𝑡)
𝑡𝑝−1
− 𝑝| ≤ 𝑝 − 𝛾 (0 ≤ 𝛾 < 𝑝),
for |𝑡| < 𝑅3, we have that
|
𝑔′(𝑡)
𝑡𝑝−1
− 𝑝| ≤ ∑ 𝑛𝑏𝑛|𝑡|𝑛−𝑝
.
∞
𝑛=𝑘+𝑝
Thus
|
𝑔′(𝑡)
𝑡𝑝−1
− 𝑝| ≤ 𝑝 − 𝛾,
if
∑
𝑛𝑏𝑛|𝑡|𝑛−𝑝
𝑝 − 𝛾
≤ 1. (3.3)
∞
𝑛=𝑘+𝑝
Therefore, by Theorem (2.1), (3.3) will be true if
𝑛
𝑝 − 𝛾
|𝑡|𝑛−𝑝
≤
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
,
and hence

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|𝑡| ≤ [
(𝑝 − 𝛾)(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑛(𝑝 − 𝛼)[(𝑝 − 1)((𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1)]
]
1
𝑛−𝑝
, (𝑛 ≥ 𝑘 + 𝑝; 𝑝, 𝑘
∈ ℕ ).
The result is sharp for the function 𝑔(𝑡) given by (2.2).
4. Extreme Points
We get here an extreme points of the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙).
Theorem (4.1): Let 𝑔𝑝(𝑡) = 𝑡𝑝
and
𝑔𝑛(𝑡)
= 𝑡𝑝
−
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 1) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑡𝑛
, (4.1)
where 𝑡 ∈ ∆, 0 ≤ 𝛼 < 𝑝, 0 ≤ 𝜇 ≤ 𝜆 ≤ 1, 𝛿, 𝑙 ≥ 0, 𝛽 ≥ 𝜈 ≥ 0, 𝑝, 𝑘 ∈ ℕ = {1, 2, 3, … }.
Then 𝑔(𝑡) ∈ ℳ𝒜(𝑝, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) if and only if It can be written as:
𝑔(𝑡) = 𝛾𝑝𝑡𝑝
+ ∑ 𝛾𝑛𝑔𝑛(𝑡),
∞
𝑛=𝑘+𝑝
(4.2)
where ( 𝛾𝑝 ≥ 0, 𝛾𝑛 ≥ 0, 𝑛 ≥ 𝑘 + 𝑝) and 𝛾𝑝 + ∑ 𝛾𝑛
∞
𝑛=𝑘+𝑝 = 1.
Proof: Consider this 𝑔(𝑡) is represented in the shape (4.2). Then
𝑔(𝑡) = 𝛾𝑝𝑡𝑝
+ ∑ 𝛾𝑛 [𝑡𝑝
−
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 1) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑡𝑛
]
∞
𝑛=𝑘+𝑝
= 𝑡𝑝
− ∑
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 1) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝛾𝑛𝑡𝑛
∞
𝑛=𝑘+𝑝
.
Hence
∑
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 1) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
∞
𝑛=𝑘+𝑝

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×
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]𝛾𝑛
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 1) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
= ∑ 𝛾𝑛
∞
𝑛=𝑘+𝑝
= 1 − 𝛾𝑝 ≤ 1.
Then 𝑔(𝑡) ∈ ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙).
Conversely, suppose that 𝑔(𝑡) ∈ ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙). We may set
𝛾𝑛 =
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 1) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
𝑏𝑛,
where 𝑏𝑛 is given by (2.3). Then
𝑔(𝑡) = 𝑡𝑝
− ∑ 𝑏𝑛
∞
𝑛=𝑘+𝑝
𝑡𝑛
= 𝑡𝑝
− ∑
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 1) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝛾𝑛 𝑡𝑛
∞
𝑛=𝑘+𝑝
= 𝑡𝑝
− ∑ [
∞
𝑛=𝑘+𝑝
𝑡𝑝
− 𝑔𝑛(𝑡)]𝛾𝑛 = (1 − ∑ 𝛾𝑛
∞
𝑛=𝑘+𝑝
) 𝑡𝑝
+ ∑ 𝛾𝑛𝑔𝑛(𝑡)
∞
𝑛=𝑘+𝑝
= 𝛾𝑝𝑡𝑝
+ ∑ 𝛾𝑛𝑔𝑛(𝑡).
∞
𝑛=𝑘+𝑝
This completes the proof of Theorem (4.1).
5. Closure Theorems
Theorem (5.1): Let the functions 𝑔𝑠 defined by
𝑔𝑠(𝑡) = 𝑡𝑝
− ∑ 𝑏𝑛,𝑠
∞
𝑛=𝑘+𝑝
𝑡𝑛
, (𝑏𝑛,𝑠 ≥ 0, 𝑛 ≥ 𝑘 + 𝑝, 𝑝, 𝑘 ∈ ℕ, 𝑠 = 1,2, … , 𝑞), ( 5.1 )
be in the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) for every 𝑠 = 1,2, … , 𝑞.

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Then the function 𝑚1(𝑡) defined by
𝑚1(𝑡) = 𝑡𝑝
− ∑ 𝑒𝑛
∞
𝑛=𝑘+𝑝
𝑡𝑛
, (𝑒𝑛 ≥ 0, 𝑛 ≥ 𝑘 + 𝑝, 𝑝, 𝑘 ∈ ℕ),
also belongs to the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙), where
𝑒𝑛 =
1
𝑞
∑ 𝑏𝑛,𝑠
𝑞
𝑠=1
, (𝑛 ≥ 𝑘 + 𝑝, 𝑝, 𝑘 ∈ ℕ).
Proof: Since 𝑔𝑠 ∈ ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) it follows from Theorem (2.1) that
∑ (𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)𝑏𝑛,𝑠
∞
𝑛=𝑘+𝑝
≤ (𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1],
for every 𝑠 = 1, 2, … , 𝑞. Hence
∑ [(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)]𝑒𝑛
∞
𝑛=𝑘+𝑝
= ∑ [(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)] (
1
𝑞
∑ 𝑏𝑛,𝑠
𝑞
𝑠=1
)
∞
𝑛=𝑘+𝑝
=
1
𝑞
∑
𝑞
𝑠=1
( ∑ [(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)]𝑏𝑛,𝑠
∞
𝑛=𝑘+𝑝
)
≤ (𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1].
By Theorem (2.1), it follows that 𝑚1(𝑡) ∈ ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙).
Theorem (5.2): Let the function 𝑔𝑠 defined by (5.1) be in the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) for
every 𝑠 = 1, 2, … , 𝑞. Then the function 𝑚2(𝑡) defined by
𝑚2(𝑡) = ∑ 𝑦𝑠
𝑞
𝑠=1
𝑔𝑠(𝑡)
is also in the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙), where

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∑ 𝑦𝑠 = 1,
𝑞
𝑠=1
(𝑦𝑠 ≥ 0).
Proof: By Theorem (2.1), for every 𝑠 = 1,2, … , 𝑞, we have
∑ (𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)𝑏𝑛,𝑠
∞
𝑛=𝑘+𝑝
≤ (𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1].
But
𝑚2(𝑡) = ∑ 𝑦𝑠
𝑞
𝑠=1
𝑔𝑠(𝑡) = ∑ 𝑦𝑠
𝑞
𝑠=1
(𝑡𝑝
− ∑ 𝑏𝑛,𝑠
∞
𝑛=𝑘+𝑝
𝑡𝑛
) = 𝑡𝑝
− ∑ (∑ 𝑦𝑠 𝑏𝑛,𝑠
𝑞
𝑠=1
)
∞
𝑛=𝑘+𝑝
𝑡𝑛
.
Therefore
∑ (𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙) (∑ 𝑦𝑠𝑏𝑛,𝑠
𝑞
𝑠=1
)
∞
𝑛=𝑘+𝑝
= ∑ 𝑦𝑠
𝑞
𝑠=1
( ∑ (𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)𝑏𝑛,𝑠
∞
𝑛=𝑘+𝑝
)
≤ ∑ 𝑦𝑠
𝑞
𝑠=1
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1] = (𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
The proof is therefore complete.
Corollary (5.1): The class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) is close under convex linear combination.
6. Integral Operators
In this segment, we consider integral transforms of functions in the class
ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙).
Theorem (6.1): Let 𝑔(𝑡) ∈ ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) be defined by (1.2) and 𝑐 be any real number
such that 𝑐 > −𝑝. Then the integral operator

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𝐺(𝑡) =
𝑐 + 𝑝
𝑡𝑐
∫ 𝑧𝑐−1
𝑔(𝑧)𝑑𝑧 ( 𝑐 > −𝑝 ),
𝑡
0
(6.1)
also in the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙).
Proof: By virtue of (6.1) it follows from (1.2) that
𝐺(𝑡) =
𝑐 + 𝑝
𝑡𝑐
∫ 𝑧𝑐−1
(𝑧𝑝
− ∑ 𝑏𝑛
∞
𝑛=𝑘+𝑝
𝑧𝑛
) 𝑑𝑧
𝑡
0
=
𝑐 + 𝑝
𝑡𝑐
∫ (𝑧𝑝+𝑐−1
− ∑ 𝑏𝑛
∞
𝑛=𝑘+𝑝
𝑧𝑛+𝑐−1
) 𝑑𝑧
𝑡
0
= 𝑡𝑝
− ∑ (
𝑐 + 𝑝
𝑐 + 𝑛
) 𝑏𝑛
∞
𝑛=𝑘+𝑝
𝑡𝑛
= 𝑡𝑝
− ∑ ℎ𝑛
∞
𝑛=𝑘+𝑝
𝑡𝑛
,
where ℎ𝑛 = (
𝑐+𝑝
𝑐+𝑛
) 𝑏𝑛.
But
∑ [(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)ℎ𝑛
∞
𝑛=𝑘+𝑝
= ∑ [(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙) (
𝑐 + 𝑝
𝑐 + 𝑛
) 𝑏𝑛
∞
𝑛=𝑘+𝑝
.
Since (
𝑐+𝑝
𝑐+𝑛
) ≤ 1 and by (2.1), the last expression is less than or equal to
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]. So the proof is ends.
7. Integral Means Inequalities
By using Theorem (2.1) and Lemma (1.2), we show the following theorems.
Theorem (7.1): Let 𝜂 > 0. If 𝑔(𝑡) ∈ ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) and suppose that 𝑔𝑠(𝑡) is defined
by

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𝑔𝑠(𝑡) = 𝑡𝑝
−
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑡𝑠
, (𝑠 ≥ 𝑘 + 𝑝; 𝑝, 𝑘 ∈ ℕ).
If there is a holomorphic function 𝑤(𝑡) defined by
(𝑤(𝑡))
𝑠−𝑝
=
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
∑ 𝑏𝑛 𝑡𝑛−𝑝
.
∞
𝑛=𝑘+𝑝
Then, for 𝑡 = 𝑟𝑒𝑖𝜃
and (0 < 𝑟 < 1),
∫ |𝑔(𝑡)|𝜂
𝑑𝜃 ≤
2𝜋
0
∫ |𝑔𝑠(𝑡)|𝜂
𝑑𝜃, (𝜂 > 0).
2𝜋
0
(7.1)
Proof: We must show that
∫ |1 − ∑ 𝑏𝑛𝑡𝑛−𝑝
∞
𝑛=𝑘+𝑝
|
𝜂
𝑑𝜃
2𝜋
0
≤ ∫ |1 −
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑡𝑠−𝑝
|
𝜂
𝑑𝜃
2𝜋
0
.
By using Lemma (1.2), it suffices to show that
1 − ∑ 𝑏𝑛𝑡𝑛−𝑝
∞
𝑛=𝑘+𝑝
≺ 1 −
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑡𝑠−𝑝
.
Put
1 − ∑ 𝑏𝑛𝑡𝑛−𝑝
∞
𝑛=𝑘+𝑝
= 1 −
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑤(𝑡))
𝑠−𝑝
.
We find that
(𝑤(𝑡))
𝑠−𝑝
=
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
∑ 𝑏𝑛 𝑡𝑛−𝑝
,
∞
𝑛=𝑘+𝑝
that yield easily 𝑤(0) = 0.
In addition by using (2.1), we get

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|𝑤(𝑡)|𝑠−𝑝
= |
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
∑ 𝑏𝑛 𝑡𝑛−𝑝
∞
𝑛=𝑘+𝑝
|
≤ |𝑡| | ∑
∞
𝑛=𝑘+𝑝
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
𝑏𝑛|
≤ |𝑡| < 1.
Next, the proof for the first derivative.
Theorem (7.2): Let 𝜂 > 0. If 𝑔(𝑡) ∈ ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) and
𝑔𝑠(𝑡) = 𝑡𝑝
−
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]Φ𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑡𝑠
, (𝑠 ≥ 𝑘 + 𝑝; 𝑝, 𝑘 ∈ ℕ).
Then, for 𝑡 = 𝑟𝑒𝑖𝜃
and (0 < 𝑟 < 1),
∫ |𝑔′(𝑡)|𝜂
𝑑𝜃 ≤
2𝜋
0
∫ |𝑔𝑠
′
(𝑡)|𝜂
𝑑𝜃, (𝜂 > 0). (7.
2𝜋
0
2)
Proof: It is sufficient to demonstrate that
1 − ∑
𝑛
𝑝
𝑏𝑛𝑡𝑛−𝑝
∞
𝑛=𝑘+𝑝
≺ 1 −
𝑠((𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1])
𝑝[(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑡𝑠−𝑝
.
This follows because
|𝑤(𝑡)|𝑠−𝑝
= |
𝑝[(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑠((𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1])
∑
𝑛
𝑝
𝑏𝑛 𝑡𝑛−𝑝
∞
𝑛=𝑘+𝑝
|
≤ |𝑡| | ∑
∞
𝑛=𝑘+𝑝
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
𝑏𝑛|
≤ |𝑡| < 1.
Theorem (7.3): Let ℎ(𝑡) = 𝑡𝑝
− ∑ 𝑐𝑛
∞
𝑛=𝑘+𝑝 𝑡𝑛 (𝑡 ∈ ∆; 𝑐𝑛 ≥ 0; 𝑛 ≥ 𝑘 + 𝑝; 𝑝, 𝑘 ∈ ℕ =
{1,2,3 … }) and 𝑔(𝑡) ∈ ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙) be of the form (1.2) and let for some 𝑠 ∈ ℕ,

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𝑄𝑠
𝑐𝑠
= min
𝑛≥𝑘+𝑝
𝑄𝑛
𝑐𝑛
,
where
𝑄𝑛 =
(𝑛 − 𝛼)[(𝑛 − 1)(𝜆𝜇𝑛 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
.
Also, let for such 𝑠 ∈ ℕ, the functions 𝑔𝑠 and ℎ𝑠 be defined by
𝑔𝑠(𝑡) = 𝑡𝑝
−
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑡𝑠
,
ℎ𝑠(𝑡) = 𝑡𝑝
− 𝑐𝑠𝑡𝑠
. (7.3)
If there is a holomorphic function 𝑤(𝑡) defined by
(𝑤(𝑡))
𝑠−𝑝
=
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]𝑐𝑠
∑ 𝑏𝑛𝑐𝑛 𝑡𝑛−𝑝
,
∞
𝑛=𝑘+𝑝
then, for 𝜂 > 0, 𝑡 = 𝑟𝑒𝑖𝜃
and (0 < 𝑟 < 1),
∫ |(𝑔 ∗ ℎ)(𝑡)|𝜂
𝑑𝜃 ≤
2𝜋
0
∫ |(𝑔𝑠 ∗ ℎ𝑠)(𝑡)|𝜂
𝑑𝜃, (𝜂 > 0).
2𝜋
0
Proof: Convolution of 𝑔(𝑡) and ℎ(𝑡) is defined by
(𝑔 ∗ ℎ)(𝑡) = 𝑡𝑝
− ∑ 𝑏𝑛 𝑐𝑛
∞
𝑛=𝑘+𝑝
𝑡𝑛
.
Similarly, from (7.3), we get
(𝑔𝑠 ∗ ℎ𝑠)(𝑡) = 𝑡𝑝
−
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]𝑐𝑠
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑡𝑠
.
To prove the theorem, we must show that for 𝜂 > 0, 𝑠 = 𝑟𝑒𝑖𝜃
and (0 < 𝑟 < 1),

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∫ |1 − ∑ 𝑏𝑛𝑐𝑛𝑡𝑛−𝑝
∞
𝑛=𝑘+𝑝
|
𝜂
𝑑𝜃
2𝜋
0
≤ ∫ |1 −
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]𝑐𝑠
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑡𝑠−𝑝
|
𝜂
𝑑𝜃
2𝜋
0
.
Therefore, using Lemma (1.2), it is sufficient to prove that
1 − ∑ 𝑏𝑛𝑐𝑛𝑡𝑛−𝑝
∞
𝑛=𝑘+𝑝
≺ 1 −
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]𝑐𝑠
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
𝑡𝑠−𝑝
. (7.4)
If the subordination (7.4) is valid, then there is a holomorphic function 𝑤(𝑡) with |𝑤(𝑡)| < 1
and 𝑤(0) = 0 such that
1 − ∑ 𝑏𝑛𝑐𝑛𝑠𝑛−𝑝
∞
𝑛=𝑘+𝑝
= 1 −
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]𝑐𝑠
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑤(𝑡))
𝑠−𝑝
.
According to the assumption of the theorem, there is a holomorphic function 𝑤(𝑡) given by
(𝑤(𝑡))
𝑠−𝑝
=
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]𝑐𝑠
∑ 𝑏𝑛𝑐𝑛𝑡𝑛−𝑝
,
∞
𝑛=𝑘+𝑝
which readily yield 𝑤(0) = 0. So for such function 𝑤(𝑡), using the assumption in the coefficient
inequality for the class ℳ𝒜(𝑝, 𝑘, 𝛼, 𝜆, 𝜇, 𝛽, 𝜈, 𝑙), we have
|𝑤(𝑡)|𝑠−𝑝
= |
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]𝑐𝑠
∑ 𝑏𝑛𝑐𝑛𝑡𝑛−𝑝
∞
𝑛=𝑘+𝑝
|
≤ |𝑡| |
(𝑠 − 𝛼)[(𝑠 − 1)(𝜆𝜇𝑠 + 𝜆 − 𝜇) + 1]𝛷𝑝
𝑛(𝛿, 𝛽, 𝜈, 𝑙)
(𝑝 − 𝛼)[(𝑝 − 1)(𝜆𝜇𝑝 + 𝜆 − 𝜇) + 1]𝑐𝑠
∑ 𝑏𝑛𝑐𝑛
∞
𝑛=𝑘+𝑝
|
≤ |𝑡| < 1.
Therefore, the subordination (7.4) holds true.

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