On a subclass of multivalent functions defined by generalized multiplier transformation

Ma’moun I. Y. Alharayzeh

doi:10.1515/math-2024-0088

Article Open Access

On a subclass of multivalent functions defined by generalized multiplier transformation

Ma’moun I. Y. Alharayzeh

Published/Copyright: April 23, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

The objective of this study is to establish a novel category of multivalent analytic functions by utilizing generalized multiplier transformations. The subclass is further examined for coefficient estimations, growth and distortion theorems, extreme points, and the radius of starlikeness and convexity.

Keywords: new subclass; multivalent functions; generalized multiplier transformations

MSC 2010: 30C45; 30C50

1 Introduction and definition

Assume A represents the class of all analytic functions that may be expressed in the following form:

f ( z ) = z + ∑ n = 2 ∞ a n z n ,

where a n is complex number. These functions are defined on the open unit disk D , which is the set of all complex numbers z such that ∣ z ∣ < 1 . This disk D lies within the complex plane C .

Consider A ( p ) , where p ∈ N = { 1 , 2 , 3 , … } as the set of all functions f given by the following expression:

f ( z ) = z p + ∑ n = p + 1 ∞ a n z n ,

with a n being complex numbers. These functions are analytic and p -valent within the open disk D in the complex plane C . For simplicity, we see that A ( 1 ) = A .

Let S p ∗ ( α ) and C p ( α ) be subclasses of p -valent functions that are, respectively, starlike of order α and convex of order α , for 0 ≤ α < p . These subclasses provide a structured way to study p -valent functions by imposing geometric constraints based on the parameter α . Specifically, a function in S p ∗ ( α ) maps the unit disk D onto a star-shaped region with respect to the origin, with the parameter α adjusting the degree of starlikeness. Similarly, a function in C p ( α ) maps D onto a convex region, with α modulating the degree of convexity. In the special cases where α = 0 , the classes reduce to S p ∗ ( 0 ) = S p ∗ and C p ( 0 ) = C p , which are the traditional subclasses of starlike and convex p -valent functions, respectively. These foundational subclasses play a crucial role in the broader study of multivalent functions, offering insights into their geometric properties and behaviors. In this context, let T ( p ) (where p ∈ N = { 1 , 2 , 3 , … } ) be the subclass of A ( p ) , consisting of functions of the following form:

(1.1) f ( z ) = z p − ∑ n = p + 1 ∞ a n z n , a n > 0

defined within the open unit disk D = { z ∈ C : ∣ z ∣ < 1 } . A function f belonging to the set T ( p ) is referred to as a p -valent function with negative coefficients. The subclasses of T ( p ) are represented by S T , p ∗ ( α ) and C T , p ( α ) , where 0 ≤ α < p . These subclasses consist of p -valent functions that are starlike of order α and convex of order α , respectively. The class denoted as T ( 1 ) = T was initially introduced and examined by Silverman [1]. In his study, Silverman examined the subclasses of T ( 1 ) , specifically referred to as S T , 1 * ( α ) = S T * ( α ) and C T , 1 ( α ) = C T ( α ) . For values of α such that 0 ≤ α < 1 , there are starlike sets of order α and convex sets of order α .

Let us consider the subclass of A ( 1 ) , denoted as M ( α , β , γ ) , which is composed of functions f ∈ A ( 1 ) that fulfill the inequality

z f ′ ( z ) − f ( z ) α z f ′ ( z ) + ( 1 − γ ) f ( z ) < β ,

for any z that are contained within the set D , where 0 ≤ α ≤ 1 , 0 < β ≤ 1 , and where 0 ≤ γ < 1 . Darus [2] conducted research on this particular category of functions.

In a recent publication, Catas [3] introduced the concept of the generalized multiplier transformation Φ p m ( λ , i ) on A ( p ) , which is defined as an infinite series.

Definition 1.1

Let λ > 0 , i ⩾ 0 , m ∈ N 0 = { 0 , 1 , 2 , … } and p ∈ N = { 1 , 2 , 3 , … } . For f ∈ A ( p ) , the generalized multiplier transformation Φ p m ( λ , i ) is defined by Φ p m ( λ , i ) : A ( p ) → A ( p )

Φ p m ( λ , i ) f ( z ) = z p + ∑ n = p + 1 ∞ p + λ ( n − p ) + i p + i m a n z n , ( z ∈ D ) .

We note that Φ p 0 ( 1,0 ) f ( z ) = f ( z ) , Φ p 1 ( 1,0 ) f ( z ) = z f ′ ( z ) p .

The generalized multiplier transformation Φ p m ( λ , i ) reduces several familiar operators by specializing the parameters m , λ , i , and p . For the choice of λ = 1 , the operator defined by Φ p m ( λ , i ) reduces the operator I p ( m , i ) studied by Srivastava et al. [4] and Sivaprasad Kumar et al. [5]. The generalized multiplier transformation Φ p m ( λ , i ) , which was researched by Cho and Kim [6] and Cho and Srivastava [7], results in the operator I i m . This operator is obtained by taking λ = p = 1 . When λ is equal to one and i is equal to zero, the operator Φ p m ( λ , i ) reduces the differential operator D P m that was investigated by Kamali and Orhan [8] and Orhan and Kiziltunc [9]. In addition, when λ = p = 1 and i = 0 , it results in the differential operator D m that was presented by Salagean [10]. The operator Φ p m ( λ , i ) , which is a special instance of this operator, lowers the generalized Salagean operator D λ m , which was explored by Al-Oboudi [11]. In addition, it gives the operator I m , which was investigated by Uralegaddi and Somanatha [12]. This may be seen as a special example of this operator.

Now, we will construct a new subclass of functions in the A ( p ) space by utilizing the generalized multiplier transformation Φ p m ( λ , i ) in the following manner.

Definition 1.2

For 0 ≤ α ≤ 1 , 0 ≤ β < 1 , 0 ≤ γ < 1 , k ≥ 0 , λ > 0 , i ⩾ 0 , m ∈ N 0 = { 0 , 1 , 2 , … } and p that belongs to N = { 1 , 2 , 3 , … } , we assume Ω ( α , β , γ , k , λ , i , m , p ) consist of functions f that belongs to T ( P ) satisfying the condition

(1.2) Re z ( Φ p m ( λ , i ) f ( z ) ) ′ − ( Φ p m ( λ , i ) f ( z ) ) α z ( Φ p m ( λ , i ) f ( z ) ) ′ + ( 1 − γ ) ( Φ p m ( λ , i ) f ( z ) ) > k z ( Φ p m ( λ , i ) f ( z ) ) ′ − ( Φ p m ( λ , i ) f ( z ) ) α z ( Φ p m ( λ , i ) f ( z ) ) ′ + ( 1 − γ ) ( Φ p m ( λ , i ) f ( z ) ) − 1 + β .

The geometric significance of the class defined by the given inequality stems from its role in characterizing and controlling the geometric properties of functions in the complex plane. This is particularly relevant to properties such as univalence, starlikeness, and convexity. By incorporating the generalized multiplier transformation, this class ensures that the functions it includes exhibit certain desirable geometric behaviors and constraints. The inequality serves as a mechanism to ensure that the transformed function Φ p m ( λ , i ) f ( z ) retains specific geometric characteristics. By involving terms such as the derivative and the function itself, it imposes control over how the function maps the complex plane, particularly regarding its growth, distortion, and boundary behavior. The inequality can be employed to derive bounds on the coefficients of the Taylor series expansion of f ( z ) . These bounds are crucial for understanding the growth and distortion of the function. By controlling the real part and magnitude of the transformed function, it is possible to obtain valuable estimates that offer insights into the function’s behavior within the unit disk.

The first thing that we have discovered is the estimation of the coefficients for functions f that belong to the set Ω ( α , β , γ , k , λ , i , m , p ) . Other results include the growth and distortion theorem, and we also find the extreme points. In the end, we were able to ascertain the radius of starlikeness and convexity for the function that is a member of the class Ω ( α , β , γ , k , λ , i , m , p ) . And the technique is studied by Aqlan et al. [13] and also in [14,15].

First things first, let us have a look at the estimations of the coefficients.

2 Coefficient estimates

In the following section, we will acquire the estimates of the coefficients for the function f that belongs to the class Ω ( α , β , γ , k , λ , i , m , p ) . This is the first result that we have obtained.

Theorem 2.1

Let 0 ≤ α ≤ 1 , 0 ≤ β < 1 , 0 ≤ γ < 1 , k ≥ 0 , λ > 0 , i ⩾ 0 , m ∈ N 0 = { 0 , 1 , 2 , … } and p is a positive integer number. If a function f given by (1.1) belongs to the class Ω ( α , β , γ , k , λ , i , m , p ) , then

(2.1) ∑ n = p + 1 ∞ T n a n ≤ T p ,

where

(2.2) T n = [ ( k + 1 ) ∣ 2 − γ − n ( 1 − α ) ∣ − ( 1 − β ) ( 1 − γ + α n ) ] 1 + λ ( n − p ) p + i m , T p = ( k + 1 ) ∣ 2 − γ − p ( 1 − α ) ∣ − ( 1 − β ) ( 1 − γ + α p ) , T p + 1 = [ ( k + 1 ) ∣ 2 − γ − ( p + 1 ) ( 1 − α ) ∣ − ( 1 − β ) ( 1 − γ + α ( p + 1 ) ) ] 1 + λ p + i m .

Proof

We start by f ∈ Ω ( α , β , γ , k , λ , i , m , p ) if and only if inequality (1.2) is satisfied. Now assume

w = z ( Φ p m ( λ , i ) f ( z ) ) ′ − ( Φ p m ( λ , i ) f ( z ) ) α z ( Φ p m ( λ , i ) f ( z ) ) ′ + ( 1 − γ ) ( Φ p m ( λ , i ) f ( z ) ) ,

depending on the fact that

Re ( w ) ≥ k ∣ w − 1 ∣ + β ⇔ ( k + 1 ) ∣ w − 1 ∣ ≤ 1 − β .

Now

( k + 1 ) ∣ w − 1 ∣ = ( k + 1 ) ( p − 1 ) z p + ∑ n = p + 1 ∞ ( 1 − n ) 1 + λ ( n − p ) p + i m a n z n ( 1 + p α − γ ) z p − ∑ n = p + 1 ∞ ( α n + 1 − γ ) 1 + λ ( n − p ) p + i m a n z n − 1 ≤ 1 − β

is equivalent to

( k + 1 ) p − 1 + ∑ n = p + 1 ∞ ( 1 − n ) 1 + λ ( n − p ) p + i m a n z n − p ( 1 + p α − γ ) − ∑ n = p + 1 ∞ ( α n + 1 − γ ) 1 + λ ( n − p ) p + i m a n z n − p − 1 ≤ 1 − β .

(2.3) ( k + 1 ) ∑ n = p + 1 ∞ ( 2 − γ − n ( 1 − α ) ) 1 + λ ( n − p ) p + i m a n z n − p − ( 2 − γ − p ( 1 − α ) ) ( 1 + p α − γ ) − ∑ n = p + 1 ∞ ( α n + 1 − γ ) 1 + λ ( n − p ) p + i m a n z n − p ≤ 1 − β .

After that,

(2.4) ( k + 1 ) ∑ n = p + 1 ∞ ( 2 − γ − n ( 1 − α ) ) 1 + λ ( n − p ) p + i m a n z n − p − ( 2 − γ − p ( 1 − α ) ) ≤ ( 1 − β ) ( 1 − γ + α p ) − ∑ n = p + 1 ∞ ( α n + 1 − γ ) 1 + λ ( n − p ) p + i m a n z n − p .

We let

S n = ( k + 1 ) ( 2 − γ − n ( 1 − α ) ) 1 + λ ( n − p ) p + i m S P = ( k + 1 ) ( 2 − γ − p ( 1 − α ) ) R n = ( 1 − β ) ( 1 − γ + α n ) 1 + λ ( n − p ) p + i m R p = ( 1 − β ) ( 1 − γ + α p ) ,

now from (2.4), we have ∑ n = p + 1 ∞ S n a n z n − p − S p ⩽ R p − ∑ n = p + 1 ∞ R n a n z n − p . By using the fact ∣ A ∣ − ∣ B ∣ ⩽ ∣ A − B ∣ , the inequality becomes

∑ n = p + 1 ∞ ∣ S n ∣ a n ∣ z ∣ n − p − ∣ S p ∣ ⩽ ∑ n = p + 1 ∞ S n a n z n − p − S p ⩽ R p − ∑ n = p + 1 ∞ R n a n z n − p = ∑ n = p + 1 ∞ R n a n ∣ z ∣ n − p − R P ,

after that we obtain the inequality

∑ n = p + 1 ∞ ∣ S n ∣ a n − ∣ S p ∣ ⩽ ∑ n = p + 1 ∞ R n a n − R P where ∣ z ∣ < 1 ,

and then ∑ n = p + 1 ∞ ( ∣ S n ∣ − R n ) a n ⩽ ∣ S p ∣ − R P which yield to (2.1).□

Theorem 2.2

Let 0 ≤ α ≤ 1 , 0 ≤ β < 1 , 0 ≤ γ < 1 , k ≥ 0 , λ > 0 , i ⩾ 0 , m ∈ N 0 = { 0 , 1 , 2 , … } and p is positive integer number. If the function f given by (1.1) belongs to the class Ω ( α , β , γ , k , λ , i , m , p ) , then

(2.5) a n ≤ T p T n , n = p + 1 , p + 2 , p + 3 , … ,

where T n and T p given by (2.2).

The functions represented by the form,

(2.6) f ( z ) = z P − T p z n T n ,

satisfy the condition of equality.

Proof

From Theorem 2.1, we have the result, if f ∈ Ω ( α , β , γ , k , λ , i , m , p ) , then ∑ n = p + 1 ∞ T n a n ≤ T p , after that a n ≤ T p T n . The function defined by equation (2.6) fulfills inequality (2.5), indicating that the function f described by equation (2.6) belongs to the set Ω ( α , β , γ , k , λ , i , m , p ) . It is evident that this result is clearly sharp for this particular function.□

3 Growth and distortion theorems for the subclass Ω ( α , β , γ , k , λ , i , m , p )

This section will discuss the growth and distortion theorem, as well as provide the covering property for functions in the class Ω ( α , β , γ , k , λ , i , m , p ) .

Theorem 3.1

Let 0 ≤ α ≤ 1 , 0 ≤ β < 1 , 0 ≤ γ < 1 , k ≥ 0 , λ > 0 , i ⩾ 0 , m ∈ N 0 = { 0 , 1 , 2 , … } , and p positive integer number. If the function f represented by (1.1) belong to the class Ω ( α , β , γ , k , λ , i , m , p ) , then for 0 < ∣ z ∣ = r < 1 , we have

(3.1) r p − T p T p + 1 r p + 1 ⩽ ∣ f ( z ) ∣ ⩽ r p + T p T p + 1 r p + 1 .

Equality obtained for the following function,

f ( z ) = z p − T p T p + 1 z p + 1 , ( z = ± r , ± i r ) ,

where T p and T p + 1 can be found by (2.2).

Proof

We solely establish the validity of the inequality on the right-hand side in equation (3.1), as the other inequality can be substantiated using analogous reasoning. From Theorem 2.1, we have the result, if f ∈ Ω ( α , β , γ , k , λ , i , m , p ) , then ∑ n = p + 1 ∞ T n a n ≤ T p . Now

T p + 1 ∑ n = p + 1 ∞ a n = ∑ n = p + 1 ∞ T p + 1 a n ≤ ∑ n = p + 1 ∞ T n a n ≤ T p .

And therefore,

(3.2) ∑ n = p + 1 ∞ a n ⩽ T p T p + 1 ,

since f ( z ) = z p − ∑ n = p + 1 ∞ a n z n , we have

∣ f ( z ) ∣ = z p − ∑ n = p + 1 ∞ a n z n ≤ ∣ z ∣ p + ∣ z ∣ p + 1 ∑ n = p + 1 ∞ a n ∣ z ∣ n − ( p + 1 ) ≤ r p + r p + 1 ∑ n = p + 1 ∞ a n .

By using inequality (3.2), we may derive the inequality on the right-hand side of (3.1).□

Theorem 3.2

If the function f defined by (1.1) belongs to the class Ω ( α , β , γ , k , λ , i , m , p ) for 0 < ∣ z ∣ = r < 1 , then it follows that

(3.3) p r p − 1 − ( p + 1 ) T p T p + 1 r p ⩽ ∣ f ′ ( z ) ∣ ⩽ p r p − 1 + ( p + 1 ) T p T p + 1 r p .

The property of equality holds for the function f given by

f ( z ) = z p − T p T p + 1 z p + 1 , ( z = ± r , ± i r ) ,

where T p and T p + 1 can be found by (2.2).

Proof

From Theorem 2.1, we have the result, if f ∈ Ω ( α , β , γ , k , λ , i , m , p ) , then ∑ n = p + 1 ∞ T n a n ≤ T p . Now

T p + 1 ∑ n = p + 1 ∞ n a n ≤ ( p + 1 ) ∑ n = p + 1 ∞ T n a n ≤ ( p + 1 ) T p .

Hence,

(3.4) ∑ n = p + 1 ∞ n a n ⩽ ( p + 1 ) T p T p + 1 ,

since

f ′ ( z ) = p z p − 1 − ∑ n = p + 1 ∞ n a n z n − 1 .

Then, we have

p ∣ z ∣ p − 1 − ∣ z ∣ p ∑ n = p + 1 ∞ n a n ∣ z ∣ n − 1 − p ⩽ ∣ f ′ ( z ) ∣ ⩽ p ∣ z ∣ p − 1 + ∣ z ∣ p ∑ n = p + 1 ∞ n a n ∣ z ∣ n − 1 − p , where ∣ z ∣ < 1 .

By utilizing inequality (3.4), we derive Theorem 3.2, therefore concluding the proof.□

Theorem 3.3

If the function f defined by (1.1) belongs to the class Ω ( α , β , γ , k , λ , i , m , p ) , then f is starlike of order δ , where δ is provided by the expression

δ = 1 − T p p − T p + T p + 1 .

The result is sharp with

f ( z ) = z p − T p T p + 1 z p + 1 .

The T p and T p + 1 can be determined using (2.2).

Proof

It is sufficient to show that (2.1) implies

(3.5) ∑ n = p + 1 ∞ a n ( n − δ ) ≤ 1 − δ .

That is,

(3.6) n − δ 1 − δ ≤ T n T p , n ≥ p + 1 .

The aforementioned inequality is equivalent to

δ ⩽ 1 − T p ( n − 1 ) − T p + T n = ψ ( n ) ,

where n ≥ p + 1 .

The inequality ψ ( n ) ≥ ψ ( p + 1 ) , as stated in (3.6) is valid for any values of 0 ≤ α ≤ 1 , 0 ≤ β < 1 , 0 ≤ γ < 1 , k ≥ 0 , λ > 0 , i ⩾ 0 , m ∈ N 0 = { 0 , 1 , 2 , … } and p is positive integer number. The proof of Theorem 3.3 is now finished.□

4 Extreme points of the class Ω ( α , β , γ , k , λ , i , m , p )

The extreme points of the class Ω ( α , β , γ , k , λ , i , m , p ) are given by the following theorem.

Theorem 4.1

Let f p ( z ) = z p , and

f n ( z ) = z p − T p T n z n , n = p + 1 , p + 2 , p + 3 , … ,

where T n and T p given by (2.2). If f ∈ Ω ( α , β , γ , k , λ , i , m , p ) , then f can be represented in the form

(4.1) f ( z ) = ∑ n = p ∞ y n f n ( z ) ,

where y n ≥ 0 a n d ∑ n = p ∞ y n = 1 .

Proof

Assume the function f ∈ Ω ( α , β , γ , k , λ , i , m , p ) our goal to obtain (4.1). From (4.1), we obtain

f ( z ) = ∑ n = p ∞ y n f n ( z ) = y p f p ( z ) + ∑ n = p + 1 ∞ y n f n ( z ) = y p f p ( z ) + ∑ n = p + 1 ∞ y n z p − T p T n z n = y p f p ( z ) + ∑ n = p + 1 ∞ y n z p − ∑ n = p + 1 ∞ y n T p T n z n = ∑ n = p ∞ y n z p − ∑ n = p + 1 ∞ y n T p T n z n = z p − ∑ n = p + 1 ∞ y n T p T n z n .

Our goal is prove that a n = y n T p T n , n ⩾ p + 1 .

Since f ∈ Ω ( α , β , γ , k , λ , i , m , p ) , then by previous Theorem 2.2.

a n ⩽ T p T n , n ⩾ p + 1 .

That is,

T n a n T p ⩽ 1 .

Now y n ⩾ 0 and ∑ n = p ∞ y n = 1 , then we see ∑ n = p ∞ y n = y p + ∑ n = p + 1 ∞ y n = 1 ; hence, ∑ n = p + 1 ∞ y n = 1 − y p ⩽ 1 . And since ∑ n = p + 1 ∞ y n ⩽ 1 , we obtain y n ⩽ 1 for each n = p + 1 , p + 2 , p + 3 , … and p = 1 , 2 , 3 , … . Setting y n = T n T p a n , thus the desired result is that a n = y n T p T n . This completes the proof of the theorem.□

Corollary 4.2

The extreme point of the class Ω ( α , β , γ , k , λ , i , m , p ) is the function

f p ( z ) = z p

and

f n ( z ) = z p − T p T n z n , n = p + 1 , p + 2 , p + 3 , … ,

where T n and T p given by (2.2).

In this study, we examine the radius of starlikeness and convexity.

5 Radius of starlikeness and convexity

The next theorems provide the values for the radius of starlikeness and convexity for the class Ω ( α , β , γ , k , λ , i , m , p ) .

Theorem 5.1

If the function f represented by (1.1) is in the set Ω ( α , β , γ , k , λ , i , m , p ) , then f is starlike of order δ ( 0 ≤ δ < p ) , in the disk ∣ z ∣ < R , where

(5.1) R = inf T n T p p − δ n − δ 1 n − p , n = p + 1 , p + 2 , p + 3 , … ,

where T n and T p given by (2.2).

Proof

Here, (5.1) implies

T p ( n − δ ) ∣ z ∣ n − P ≤ T n ( p − δ ) .

It suffices to show that

z f ′ ( z ) f ( z ) − p ≤ p − δ ,

for ∣ z ∣ < R , we have

(5.2) z f ′ ( z ) f ( z ) − p ≤ ∑ n = p + 1 ∞ ( n − p ) a n ∣ z ∣ n − p 1 − ∑ n = p + 1 ∞ a n ∣ z ∣ n − p .

From (2.5), we have

z f ′ ( z ) f ( z ) − p ≤ ∑ n = p + 1 ∞ T p ( n − p ) ∣ z ∣ n − p T n 1 − ∑ n = p + 1 ∞ T p ∣ z ∣ n − p T n .

The final phrase is bounded above by p − δ if

∑ n = p + 1 ∞ T p ( n − p ) ∣ z ∣ n − p T n ≤ 1 − ∑ n = p + 1 ∞ T p ∣ z ∣ n − p T n ( p − δ ) ,

and it follows that

∣ z ∣ n − p ⩽ T n T p p − δ n − δ , n ⩾ p + 1 .

This is equivalent to our condition (5.1) stated in the theorem.□

Theorem 5.2

If the function f defined by (1.1) belongs to the class Ω ( α , β , γ , k , λ , i , m , p ) , then f is a convex function of order ε ( 0 ≤ ε < p ) , within the disk ∣ z ∣ < w , where

(5.3) w = inf T n T p p ( p − ε ) n ( n − ε ) 1 n − p , n = p + 1 , p + 2 , p + 3 , … ,

where T n and T p given by (2.2).

Proof

By employing the identical methodology utilized in the demonstration of Theorem 5.1, we can demonstrate that

z f ″ ( z ) f ′ ( z ) − ( p − 1 ) ≤ p − ε , for ∣ z ∣ ≤ w ,

from (5.3), we write it T p n ( n − ε ) ∣ z ∣ n − p ⩽ T n p ( p − ε ) and then ∣ z ∣ n − p ⩽ T n p ( p − ε ) T p n ( n − ε ) . Now

z f ″ ( z ) f ′ ( z ) − ( p − 1 ) = ∑ n = p + 1 ∞ n ( p − n ) a n z n − p p − ∑ n = p + 1 ∞ n a n z n − p ⩽ ∑ n = p + 1 ∞ n ( n − p ) a n ∣ z ∣ n − p p − ∑ n = p + 1 ∞ n a n ∣ z ∣ n − p ,

from (2.5), we obtain

z f ″ ( z ) f ′ ( z ) − ( p − 1 ) ⩽ ∑ n = p + 1 ∞ T p n ( n − p ) ∣ z ∣ n − p T n p − ∑ n = p + 1 ∞ T p n ∣ z ∣ n − p T n .

Now the last inequality is bounded above by ( p − ε ) if

∑ n = p + 1 ∞ T p n ( n − p ) ∣ z ∣ n − p T n ≤ p − ∑ n = p + 1 ∞ T p n ∣ z ∣ n − p T n ( p − ε ) .

After that ∣ z ∣ n − p ⩽ T n p ( p − ε ) T p n ( n − ε ) , n ⩾ p + 1 , which is equivalent expression for condition (5.3) in our theorem.□

6 Conclusion

The class defined by the given inequality is geometrically significant because it governs and characterizes the geometric properties of complex functions, ensuring they maintain or achieve univalence, starlikeness, convexity, and other desirable attributes. By utilizing generalized multiplier transformations and associated parameters, this class enables precise manipulation and analysis of functions, making it a potent tool in both theoretical and applied complex analysis. We remark that some extended classes of multivalent analytic functions can be derived with generalized multiplier transformation and studied their coefficients characterization.

As a future research, some extended classes of multivalent analytic functions can be derived with ( r , q ) derivative operator and studied their properties, as well as these results can be extended to study problems related to the Fekete-Szego theorem (Alharayzeh and Darus [16,17]).

Acknowledgements

The author is grateful to the referee for his helpful comments. The author thank Dr. Habis S. Al-zboon for his time and invaluable contribution.

Funding information: The author states no funding involved.
Author contributions: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.
Conflict of interest: The author states no conflict of interest.
Data availability statement: The datasets generated and analyzed during the current study are available from the author on reasonable request.

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Received: 2023-07-09

Revised: 2024-07-14

Accepted: 2024-10-10

Published Online: 2025-04-23

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2024-0088

Keywords for this article

new subclass; multivalent functions; generalized multiplier transformations

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