A new family of multivalent functions defined by certain forms of the quantum integral operator

Ajmal Khan; Isra Al-shbeil; Amani Shatarah; Norah Mousa Alrayes; Shahid Khan; Wasim ul Haq

doi:10.1515/dema-2025-0128

Artikel Open Access

A new family of multivalent functions defined by certain forms of the quantum integral operator

Ajmal Khan , Isra Al-shbeil , Amani Shatarah , Norah Mousa Alrayes , Shahid Khan und Wasim ul Haq

Veröffentlicht/Copyright: 29. Juli 2025

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Abstract

In this work, using the concepts of q -calculus, we first define the q -Jung-Kim-Srivastava and q -Bernardi integral operators for multivalent functions. Then, we use these operators to establish the generalized integral operator ℬ q , p − m − λ f ( z ) for multivalent functions. By using the newly defined operator ℬ q , p − m − λ f ( z ) , we define a new class of multivalent analytic functions and diskuss various interesting features of the functions belonging to this class. Certain interesting examples of our developments are also considered. Furthermore, we explore the applications of the Carathéodory lemma in the form of specific results.

Keywords: analytic functions; q-Bernardi integral operator; multivalent analytic functions; q-difference operator; fractional q-integral operator; q-Jack’s lemma

MSC 2010: 30C45; 30C50; 11B65; 47B38

1 Introduction and definitions

Differential and integral operators have been the focus of study since the beginning of analytic function theory, when Alexander [1] presented the first integral operator in 1915. Differential and integral operators are continually being combined in new and interesting ways (for examples, see [2,3]). Recent studies of differential and integral operators from a wide range of ideas, including quantum calculus, have produced incredible findings with broad implications for physics and mathematics. A new survey-cum-expository review research [4] focuses on several intriguing applications of differential and integral operators.

The study of geometric functions gained influence from the theory of real and complex-order integrals and derivatives, which also demonstrated potential in mathematical modeling and analysis of real-world problems in the applied sciences. Examples of works that fall within the aforementioned area include an analysis of the dynamics of dengue transmission [5] and a new model of the human liver [6]. Particularly significant for the study of both pure and practical mathematics is the family of integral operators associated with the first-kind Lommel functions, which was presented in [7]. Because of differential and integral operators, functional analysis and operator theory may be used in the study of differential equations. In order to solve differential equations using the operator method, we make use of the properties of differential operators, and it can be demonstrated that such operators are involved in the solution of partial differential equations; however, this requires further research. A number of intriguing geometric and mapping properties are derived for the integral operators given here.

We suppose that class ℋ includes a set of analytic functions f ( z ) in open unit disk

E = { z ∈ C : ∣ z ∣ < 1 }

and takes the form given by

(1) f ( z ) = z + ∑ j = 2 ∞ a j z j .

Let ℋ ( p ) contain multivalent functions f ( z ) in the open unit disk E = { z ∈ C : ∣ z ∣ < 1 } and of the form

(2) f ( z ) = z p + ∑ k = p + j ∞ a k z k , p , j ∈ N = { 1 , 2 , 3 … } .

Numerous mathematicians, physicists, and engineers have devoted a great deal of effort and time to studying the standard quantum calculus, and the field has benefited much from its practical applications in fields as diverse as engineering, economics, mathematics, and others (see, for example, [8–11]). Taking into consideration the aforementioned information concerning q -calculus, it is reasonable to conclude that, over the last three decades, q -calculus has served as the bridge between mathematics and physics. Remarkably, the q -calculus operator, q -integral operator, and q -derivative operator are used to construct several classes of regular functions and find applications in many areas of mathematics, including general relativity, the calculus of variations, orthogonal polynomials, and fundamental hypergeometric functions. The q -derivative and q -integral were defined by Jackson [12] and were the first to be used. Purohit [13] was the first mathematician who used certain fractional q -derivative operators, to introduce and examine the class of multivalent functions in an open unit disk. He made a significant contribution by giving q -extension to a number of findings in analytic function theory. In a book chapter (see, for details, [14] pp. 347 et seq.; see also [15]) by Srivastava, the basic (or q -) hypergeometric functions were used for the first time in geometric function theory (GFT) and also provided a highly significant utilization of the q -calculus in the context of GFT. Ismail [16] earlier introduced a class of q -starlike functions by using the q -differential operator. Some differential equations were solved using the q -derivative by Akça et al. [17]. This study also makes use of q -calculus and introduces many new, significant q -analogs of differential and integral operators. The q -operators are defined with a number of interesting consequences by using the convolution of normalized analytic functions in the field of GFT (see [18,19]). By utilizing the idea of fractional q -calculus, Selvakumaran et al. [20] recently presented the q -integral operator for particular analytic functions in the open unit disk. They also investigated the various convexity characteristics in [20] for certain classes of analytic functions formed by a linear multiplier and a fractional q -differential integral operator. Serivastava et al. [21] defined a new fractional differintegral operator by subordination and diskussed how it relates to several classes of analytic functions. Also, some recent findings and studies related to the q -derivatives and q -integral operators have been presented (see, for example, [22,23,24,25,26,27,28,29]).

The following is the definition of the quantum difference operator, also known as the q -derivative operator:

Definition 1

[12]. Let f ∈ ℋ , the q -difference operator ( q -derivative operator) is defined is as follows:

(3) ∂ q f ( z ) = f ( z ) − f ( q z ) ( 1 − q ) z , z ≠ 0 , q ∈ ( 0 , 1 ) .

Using (1) in (3), and after some simple calculations, we obtain the series form of the q -difference operator as

∂ q f ( z ) = 1 + ∑ j = 2 ∞ [ j ] q a j z j − 1 , z ∈ E ,

where

(4) [ j ] q = 1 − q j 1 − q , q ∈ ( 0 , 1 ) .

We note that, for q → 1 −

∂ q f ( z ) ⟶ f ′ ( z )

and

∂ q f ( 0 ) = f ′ ( 0 ) .

More specifically,

(5) ∂ q z j = [ j ] q z j − 1 .

When q → 1 − , then

[ j ] q ⟶ j .

Remark 1

Similarly using (2) in (3), then we have the following series form for f ∈ ℋ ( p ) :

∂ q f ( z ) = [ p ] q z p − 1 + ∑ k = p + j ∞ [ k ] q a k z k − 1 , p , j ∈ N = { 1 , 2 , 3 , … } .

Definition 2

The definition of the q -gamma function Γ q is

(6) Γ q ( x ) = ( 1 − q ) 1 − x ∏ j = 0 ∞ 1 − q j + 1 1 − q j + x , x > 0 ,

which contains the following properties:

(7) Γ q ( x + 1 ) = [ x ] q Γ q ( x )

and

(8) Γ q ( x + 1 ) = [ x ] q !

where x ∈ N and [ x ] q ! are defined as

(9) [ x ] q ! = [ x ] q [ x − 1 ] q … [ 2 ] q [ 1 ] q if x ∈ N 1 if x = 0 .

Definition 3

The q -beta function B q is defined as

(10) B q ( v 0 , v 1 ) = ∫ 0 1 x v 0 − 1 ( 1 − q x ) q v − 1 d q x , ( v 0 , v 1 > 0 , 0 < q < 1 )

and

(11) B q ( v 0 , v 1 ) = Γ q ( v 0 ) Γ q ( v 1 ) Γ q ( v 0 + v 1 ) ,

where Γ q is given by (6).

The purpose of this article is to construct a new q -analogous integral operator for multivalent functions by combining the strengths of the q -Jung-Kim-Srivastava and q -Bernardi integral operators and subsequently investigate its applications and implications.

We begin by introducing the q -Bernardi integral operator for multivalent functions, which is defined as follows:

Definition 4

For functions f ( z ) ∈ ℋ ( p ) , we define the expression ℬ q , p − 1 f ( z )

(12) ℬ q , p − 1 f ( z ) = [ p + γ ] q z γ ∫ 0 z t γ − 1 f ( t ) d q t = z p + ∑ k = p + j ∞ [ p + γ ] q [ k + γ ] q a k z k , γ > − 1 , p , j ∈ N .

We consider

ℬ q , p − 2 f ( z ) = ℬ q , p − 1 ( ℬ q , p − 1 f ( z ) ) = z p + ∑ k = p + j ∞ [ p + γ ] q [ k + γ ] q 2 a k z k , γ > − 1 , j ∈ N .

Similarly, continuing this process, we obtain the generalizations of the q -Bernardi integral operator for multivalent functions:

(13) ℬ q , p − m f ( z ) = ℬ q , p − 1 ( ℬ q , p − m + 1 f ( z ) ) = z p + ∑ k = p + j ∞ [ p + γ ] q [ k + γ ] q m a k z k , γ > − 1 , j ∈ N ,

where m ∈ N and ℬ q , p − 0 f ( z ) = f ( z ) .

Remark 2

For m = 1 , and p = 1 , in (12), then we obtain the q -Bernardi integral operator for f ∈ ℋ , and defined in [30] as follows:

(14) ℬ q − 1 f ( z ) = [ 1 + γ ] q z γ ∫ 0 z t γ − 1 f ( t ) d q t = z + ∑ k = j + 1 ∞ [ 1 + γ ] q [ k + γ ] q a k z k , γ > − 1 , j ∈ N .

Remark 3

For q → 1 − and p = 1 , in (12), then we have the Bernardi integral operator for f ∈ ℋ and given in [31] and defined as follows:

(15) ℬ − 1 f ( z ) = 1 + γ z γ ∫ 0 z t γ − 1 f ( t ) d t = z + ∑ k = j + 1 ∞ 1 + γ k + γ a k z k , γ > − 1 , j ∈ N .

Remark 4

Furthermore, taking p = 1 , γ = 1 , and q → 1 − in (12) for f ( z ) ∈ ℋ , we have the Libera integral operator given in [32].

Definition 5

[33] For λ > 0 , the fractional q -integral operator is defined as

D q − λ f ( z ) = 1 Γ q ( λ ) ∫ 0 z 1 ( z − t q ) 1 − λ f ( t ) d q ( t )

and ( z − t q ) 1 − λ is defined by

( z − t q ) 1 − λ = z 1 − λ Φ 0 1 ( q − λ + 1 , − , q , t q λ ⁄ z ) .

The representation of series Φ 0 1 is given by

Φ 0 1 ( a , − , q , z ) = 1 + ∑ j = 1 ∞ ( a , q ) j ( q , q ) j z j , ( ∣ q ∣ < 1 , ∣ z ∣ < 1 ) ,

where

( a , q ) j = ∏ k = 0 j − 1 ( 1 − a q k ) , j ∈ N .

With the above definitions, and for the function of the form (2), we know that

D q − λ f ( z ) = Γ q ( p + 1 ) Γ q ( p + 1 + λ ) z p + λ + ∑ k = j + p ∞ Γ q ( k + 1 ) Γ q ( k + 1 + λ ) a k z k + λ

for λ > 0 , and f ( z ) ∈ ℋ ( p ) .

Definition 6

Using the fractional q -integral operator for f ∈ ℋ ( p ) , we define the q -Jung-Kim-Srivastava integral operator as follows:

(16) ℬ q , p − λ f ( z ) = Γ q ( p + γ + λ ) Γ q ( p + γ ) z 1 − γ − λ D q − λ ( z γ − 1 f ( z ) ) = z p + ∑ k = p + j ∞ Γ q ( p + γ + λ ) Γ q ( k + γ ) Γ q ( p + γ ) Γ q ( k + γ + λ ) a k z k ,

where 0 ≤ λ ≤ 1 , and γ > − 1 .

Remark 5

(i) For p = 1 and q → 1 − in (16), we obtain the Jung-Kim-Srivastava integral operator for f ∈ ℋ defined in [34] as

ℬ − λ f ( z ) = z + ∑ k = j + 1 ∞ Γ ( j + γ ) Γ ( 1 + γ + λ ) Γ ( j + γ + λ ) Γ ( 1 + γ ) a k z k ,

where 0 ≤ λ ≤ 1 , j ∈ N , and γ > − 1 .

(ii) If λ = 0 , in (16), then ℬ 0 f ( z ) = f ( z ) , and if λ = 1 , in (16), then we have the q -Bernardi integral operator for f ∈ ℋ ( p ) given by (12).

(iii) For q → 1 − , λ = 1 and p = 1 , then we have the Bernardi integral operator for f ∈ ℋ defined by Bernardi [30].

Definition 7

Using the operator ℬ q , p − λ f ( z ) given by (16) and ℬ q , p − m f ( z ) defined in (13), we define the operator ℬ q , p − m − λ f ( z ) as follows:

(17) ℬ q , p − m − λ f ( z ) = ℬ q , p − m ( ℬ q , p − λ f ( z ) ) = z p + ∑ k = p + j ∞ [ p + γ ] q [ k + γ ] q m Γ q ( k + γ ) Γ q ( p + γ + λ ) Γ q ( k + γ + λ ) Γ q ( p + γ ) a k z k ,

where 0 ≤ λ ≤ 1 , γ > − 1 , and m ∈ N .

Remark 6

(i) The operator ℬ q , p − m − λ f ( z ) is the generalized form of the q -Bernardi integral and q -Jung-Kim-Srivastava integral operator.

(ii) The following identity can be easily satisfied for the operator ℬ q , p − m − λ f ( z ) :

(18) ℬ q , p − m − λ + 1 f ( z ) = [ γ ] q [ p + γ ] q ℬ q , p − m − λ f ( z ) + q γ [ p + γ ] q z ∂ q ( ℬ q , p − m − λ f ( z ) ) .

(iii) From the definition ℬ q , p − m − λ f ( z ) , we observe that

ℬ q , p − m − λ f ( z ) = ℬ q , p − m ( ℬ q , p − λ f ( z ) ) = ℬ q , p − λ ( ℬ q , p − m f ( z ) ) .

(iv) For s different boundary points z l , l = 1 , 2 , 3 , … , s , with ∣ z l ∣ = 1 , we assume

(19) α s = 1 s ∑ l = 1 s ℬ q , p − m − λ f ( z l ) z l p ,

where

α s ∈ e i β ℬ q , p − m − λ f ( E ) ,

α s ≠ 1 , − π 2 ≤ β ≤ π 2

and E is the open unit disk.

Definition 8

Let − π 2 ≤ β ≤ π 2 , ρ > 0 , λ > 0 , γ > − 1 , and m ∈ N . If f ( z ) ∈ ℋ ( p ) satisfies

(20) e i β ( ℬ q , p − m − λ f ( z ) ) z p − α s e i β − α s − 1 < ρ , z ∈ E ,

then f ( z ) ∈ T p , j ( q , α s , β , ρ , λ , m ) .

Class T p , j ( q , α s , β , ρ , λ , m ) represented by the function f ( z ) ∈ ℋ ( p ) under the assumption

(21) ℬ q , p − m − λ f ( z ) z p − 1 < ρ ∣ e i β − α s ∣ , z ∈ E ,

is satisfied. If we consider the function f ( z ) ∈ ℋ ( p ) given by

(22) f ( z ) = z p + [ p + j + γ ] q [ k + γ ] q m Γ q ( p + γ ) Γ q ( p + j + γ + λ ) Γ q ( p + j + γ ) Γ q ( p + γ + λ ) ρ ( e i β − α s ) z p + j ,

where k = j + 1 , and j ∈ N , then f ( z ) satisfies

(23) ℬ q , p − m − λ f ( z ) z p − 1 = ρ ∣ e i β − α s ∣ ∣ z ∣ j < ρ ∣ e i β − α s ∣ , z ∈ E .

Hence, f ( z ) from (22) belongs to the class T p , j ( q , α s , β , ρ , λ , m ) .

In order to diskuss our problems with f ( z ) ∈ T p , j ( q , α s , β , ρ , λ , m ) , we must first consider the following q -Jack’s lemma proved by Çetinkaya and Polatoĝlu (see [35], Lemma 2.3, page 818).

Lemma 1

[35] Let u ( z ) be analytic in the unit disk E, with u ( 0 ) = 0 . If ∣ u ( z ) ∣ attains its maximum value on the circle. ∣ z ∣ = r at a point z 0 ∈ E , then

(24) z 0 ∂ q u ( z 0 ) u ( z 0 ) = j .

This article is structured into four sections. Section 1 provides a foundation by reviewing key concepts in GFT and examining applications of existing differential and integral operators. It also introduces innovative q -analogs of integral operators for multivalent functions and defines new classes of multivalent functions using the generalized operator ℬ q , p − m − λ f ( z ) , which is vital for our main diskoveries. Section 2 presents the core results supported by examples that illustrate their practical applications. Section 3 leverages the Carathéodory lemma to derive additional results. Section 4 offers concluding thoughts, distilling the essential insights from our investigation.

2 Main results and examples for class T p , j ( q , α s , β , ρ , λ , m )

In order to make a function f a member of T p , j ( q , α s , β , ρ , λ , m ) , we first investigate a sufficient condition for it.

Theorem 1

If f ( z ) ∈ ℋ ( p ) satisfies

(25) ℬ q , p − m − λ + 1 f ( z ) ℬ q , p − m − λ f ( z ) − 1 < q γ ∣ e i β − α s ∣ j ρ [ p + γ ] q ( 1 + ∣ e i β − α s ∣ ρ ) , z ∈ E ,

for some α s given by (19) with α s ≠ 1 such that z l ∈ ∂ E ( l = 1 , 2 , 3 , … , s ), and for some real ρ > 1 , then

ℬ q , p − m − λ f ( z ) z p − 1 < ρ ∣ e i β − α s ∣ , z ∈ E

that is f ( z ) ∈ T p , j ( q , α s , β , ρ , λ , m ) .

Proof

Let the function u ( z ) defined by

(26) u ( z ) = e i β ( ℬ q , p − m − λ f ( z ) ) z p − α s e i β − α s − 1 = e i β e i β − α s ∑ k = p + j ∞ [ p + γ ] q [ k + γ ] q m Γ q ( k + γ ) Γ q ( p + γ + λ ) Γ q ( k + γ + λ ) Γ q ( p + γ ) a k z k − p .

Then

(27) ℬ q , p − m − λ f ( z ) z p = 1 + ( 1 − e − i β α s ) u ( z ) .

Using identity (18), we have

(28) ℬ q , p − m − λ + 1 f ( z ) = [ γ ] q [ p + γ ] q ℬ q , p − m − λ f ( z ) + q γ [ p + γ ] q z ∂ q ( ℬ q , p − m − λ f ( z ) ) .

Divided both sides of (28) by ℬ q , p − m − λ f ( z ) , and using the fact, [ γ ] q + q γ [ p ] q = [ p + γ ] q , we have

(29) ℬ q , p − m − λ + 1 f ( z ) ℬ q , p − m − λ f ( z ) − 1 = q γ ( 1 − e − i β α s ) z ∂ q u ( z ) [ p + γ ] q ( 1 + ( 1 − e − i β α s ) u ( z ) ) .

Taking the modulus on both sides of (29), we have

ℬ q , p − m − λ + 1 f ( z ) ℬ q , p − m − λ f ( z ) − 1 = q γ [ p + γ ] q ( 1 − e − i β α s ) z ∂ q ( u ( z ) ) ( 1 + ( 1 − e − i β α s ) u ( z ) ) < q γ ∣ e i β − α s ∣ j ρ [ p + γ ] q ( 1 + ∣ e i β − α s ∣ ρ )

by employing (25). Assume, to arrive at a contradiction, that there exists a point z 0 , ( 0 < ∣ z 0 ∣ < 1 ) such that

max { ∣ u ( z ) ∣ ; ∣ z ∣ ≤ ∣ z 0 ∣ } = ∣ u ( z 0 ) ∣ = ρ > 1 .

Then, we can write that u ( z 0 ) = ρ e i θ , ( 0 ≤ θ ≤ 2 π ) and z 0 ∂ q u ( z 0 ) = k u ( z 0 ) , k ≥ j by Lemma 1. For the point z 0 ∈ E , we have

ℬ q , p − m − λ + 1 f ( z 0 ) ℬ q , p − m − λ f ( z 0 ) − 1 = q γ [ p + γ ] q ( 1 − e − i β α s ) z 0 ∂ q u ( z 0 ) ( 1 + ( 1 − e − i β α s ) u ( z 0 ) ) = q γ [ p + γ ] q ( 1 − e − i β α s ) k ρ ( 1 + ( 1 − e − i β α s ) ρ e i θ ) ≥ q γ ∣ 1 − e − i β α s ∣ j ρ [ p + γ ] q ( 1 + ∣ 1 − e − i β α s ∣ ρ ) = q γ ∣ e i β − α s ∣ j ρ [ p + γ ] q ( 1 + ∣ e i β − α s ∣ ρ ) .

Since this contradicts our condition (25), we see that there is no z 0 , ( 0 < ∣ z 0 ∣ < 1 ) such that

∣ u ( z 0 ) ∣ = ρ > 1 .

This shows us that

∣ u ( z ) ∣ = e i β ( ℬ q , p − m − λ f ( z ) ) z p − 1 e i β − α s < ρ ,

that is,

ℬ q , p − m − λ f ( z ) z p − 1 < ρ ∣ e i β − α s ∣ , z ∈ E .

The theorem has now been proven.□

Example 1

Using the function f ( z ) ∈ ℋ ( p ) provided by

(30) f ( z ) = z p + a p + j z p + j , z ∈ E , j ∈ N ,

with 0 < ∣ a p + j ∣ < 1 2 T ( γ , λ , q ) , where

(31) T ( γ , λ , q ) = [ p + γ ] q [ p + j + γ ] q m Γ q ( p + j + γ ) Γ q ( p + γ + λ ) Γ q ( p + j + γ + λ ) Γ q ( p + γ ) .

For such f ( z ) , we have

ℬ q , p − m − λ + 1 f ( z ) ℬ q , p − m − λ f ( z ) = 1 + [ p + j + γ ] q [ p + γ ] q T ( γ , λ , q ) a p + j z j 1 + T ( γ , λ , q ) a p + j z j .

Taking modulus, we have

ℬ q , p − m − λ + 1 f ( z ) ℬ q , p − m − λ f ( z ) − 1 < [ p + j + γ ] q [ p + γ ] q − 1 T ( γ , λ , q ) ∣ a p + j ∣ 1 − T ( γ , λ , q ) ∣ a p + j ∣ , z ∈ E .

We now take into account five boundary points so that

z 1 = e − i arg ( a p + j ) j , z 2 = e i π − 6 arg ( a p + j ) 6 j , z 3 = e i π − 4 arg ( a p + j ) 4 j z 4 = e i π − 3 arg ( a p + j ) 3 j , z 5 = e i π − 2 arg ( a p + j ) 2 j .

We know that for these five boundary points

ℬ q , p − m − λ f ( z 1 ) z 1 p = 1 + T ( γ , λ , q ) a p + j e − i arg ( a p + j ) = 1 + T ( γ , λ , q ) ∣ a p + j ∣ ,

ℬ q , p − m − λ f ( z 2 ) z 2 p = 1 + T ( γ , λ , q ) a p + j e i π 6 − arg ( a p + j ) = 1 + ( 3 + i ) 2 T ( γ , λ , q ) ∣ a p + j ∣ ,

ℬ q , p − m − λ f ( z 3 ) z 3 p = 1 + T ( γ , λ , q ) a p + j e i π 4 − arg ( a p + j ) = 1 + 2 ( 1 + i ) 2 T ( γ , λ , q ) ∣ a p + j ∣ ,

ℬ q , p − m − λ f ( z 4 ) z 4 p = 1 + T ( γ , λ , q ) a p + j e i π 3 − arg ( a p + j ) = 1 + ( 1 + 3 i ) 2 T ( γ , λ , q ) ∣ a p + j ∣ ,

and

ℬ q , p − m − λ f ( z 5 ) z 5 p = 1 + T ( γ , λ , q ) a p + j e i π 2 − arg ( a p + j ) = 1 + i T ( γ , λ , q ) ∣ a p + j ∣ .

Thus, from (19), we obtain α 5 and is given by

α 5 = 1 5 ℬ q , p − m − λ f ( z 1 ) z 1 p + ℬ q , p − m − λ f ( z 2 ) z 2 p + ℬ q , p − m − λ f ( z 3 ) z 3 p + ℬ q , p − m − λ f ( z 4 ) z 4 p + ℬ q , p − m − λ f ( z 5 ) z 5 p = 1 + ( 1 + i ) ( 3 + 2 + 3 ) 10 T ( γ , λ , q ) ∣ a p + j ∣ .

This gives us that

∣ 1 − e − i β α 5 ∣ = 2 ( 3 + 2 + 3 ) 10 T ( γ , λ , q ) ∣ a p + j ∣ .

For such α 5 and β , we take ρ > 1 with

[ p + j + γ ] q [ p + γ ] q − 1 T ( γ , λ , q ) ∣ a p + j ∣ 1 − T ( γ , λ , q ) ∣ a p + j ∣ ≤ q γ ∣ e i β − α 5 ∣ j ρ [ p + γ ] q ( 1 + ∣ e i β − α 5 ∣ ρ ) .

It follows from the above that

ρ ≥ 10 2 ( 3 + 2 + 3 ) ( 1 − 2 T ( γ , λ , q ) ∣ a p + j ∣ ) > 10 2 ( 3 + 2 + 3 ) > 1 .

For such α 5 and ρ > 1 , f ( z ) satisfies

ℬ q , p − m − λ f ( z ) z p − 1 = T ( γ , λ , q ) ∣ a p + j ∣ < ρ ∣ e i β − α 5 ∣ , z ∈ E .

Theorem 2

If f ( z ) ∈ ℋ ( p ) satisfies

(32) ℬ q , p − m − λ + 1 f ( z ) ℬ q , p − m − λ f ( z ) − 1 ℬ q , p − m − λ f ( z ) z p − 1 < q γ ∣ e i β − α s ∣ 2 j ρ 2 [ p + γ ] q ( 1 + ∣ e i β − α s ∣ ρ ) , z ∈ E ,

for some α s given by (19), with α s ≠ 1 such that z l ∈ ∂ E ( l = 1 , 2 , 3 , … , s ), and for some real ρ > 1 , then

ℬ q , p − m − λ f ( z ) z p − 1 < ρ ∣ e i β − α s ∣ , z ∈ E ,

that is f ( z ) ∈ T p , j ( q , α s , β , ρ , λ , m ) .

Proof

Given u ( z ) in (26). Using (27) and (29), we have

ℬ q , p − m − λ + 1 f ( z ) ℬ q , p − m − λ f ( z ) − 1 ℬ q , p − m − λ f ( z ) z p − 1 = q γ [ p + γ ] q ( 1 − e − i β α s ) 2 z u ( z ) ∂ q u ( z ) ( 1 + ( 1 − e − i β α s ) u ( z ) ) .

We suppose that there exists a point z 0 , ( 0 < ∣ z 0 ∣ < 1 ) such that

max { ∣ u ( z ) ∣ ; ∣ z ∣ ≤ ∣ z 0 ∣ } = ∣ u ( z 0 ) ∣ = ρ > 1 .

Then, we can write that

u ( z 0 ) = ρ e i θ , ( 0 ≤ θ ≤ 2 π )

and

z 0 ∂ q u ( z 0 ) = k u ( z 0 ) , k ≥ j

by Lemma 1. From the above, it follows that

ℬ q , p − m − λ + 1 f ( z 0 ) ℬ q , p − m − λ f ( z 0 ) − 1 ℬ q , p − m − λ f ( z 0 ) z 0 p − 1 = q γ [ p + γ ] q ( 1 − e − i β α s ) 2 z 0 u ( z 0 ) ∂ q u ( z 0 ) ( 1 + ( 1 − e − i β α s ) u ( z 0 ) ) ≥ q γ [ p + γ ] q ∣ e i β − α s ∣ 2 ρ 2 j 1 + ∣ ( e i β − α s ) ∣ ρ .

This is opposite to our condition (32) for f ( z ) . Thus, there is no z 0 , ( 0 < ∣ z 0 ∣ < 1 ) such that

∣ u ( z 0 ) ∣ = ρ > 1 .

This means that

□ ℬ q , p − m − λ f ( z ) z p − 1 < ρ ∣ e i β − α s ∣ , z ∈ E .

Example 2

Take a function f ( z ) defined by (30) with 0 < ∣ a p + j ∣ < 1 T ( γ , λ , q ) , where T ( γ , λ , q ) is defined by (31). For this function f ( z ) , we have

ℬ q , p − m − λ + 1 f ( z ) ℬ q , p − m − λ f ( z ) − 1 ℬ q , p − m − λ f ( z ) z p − 1 = q γ [ p + j + γ ] q [ p + γ ] q − 1 T 2 ( γ , λ , q ) a p + j 2 z 2 j ( 1 + T ( γ , λ , q ) a p + j z j ) [ p + γ ] q < q γ [ p + j + γ ] q [ p + γ ] q − 1 T 2 ( γ , λ , q ) ∣ a p + j ∣ 2 ( 1 − T ( γ , λ , q ) ∣ a p + j ∣ ) [ p + γ ] q , z ∈ E .

Consider five boundary points

z 1 = e − i arg ( a p + j ) j , z 2 = e i π − 6 arg ( a p + j ) 6 j , z 3 = e i π − 4 arg ( a p + j ) 4 j z 4 = e i π − 3 arg ( a p + j ) 3 j , z 5 = e i π − 2 arg ( a p + j ) 2 j .

We know about these five boundary points

ℬ q , p − m − λ f ( z 1 ) z 1 p = 1 + T ( γ , λ , q ) a p + j e − i arg ( a p + j ) = 1 + T ( γ , λ , q ) ∣ a p + j ∣ ,

ℬ q , p − m − λ f ( z 2 ) z 2 p = 1 + T ( γ , λ , q ) a p + j e i π 6 − arg ( a p + j ) = 1 + 3 + i 2 T ( γ , λ , q ) ∣ a p + j ∣ ,

ℬ q , p − m − λ f ( z 3 ) z 3 p = 1 + T ( γ , λ , q ) a p + j e i π 4 − arg ( a p + j ) = 1 + 2 ( 1 + i ) 2 T ( γ , λ , q ) ∣ a p + j ∣ ,

ℬ q , p − m − λ f ( z 4 ) z 4 p = 1 + T ( γ , λ , q ) a p + j e i π 3 − arg ( a p + j ) = 1 + ( 1 + 3 i ) 2 T ( γ , λ , q ) ∣ a p + j ∣ ,

and

ℬ q , p − m − λ f ( z 5 ) z 5 p = 1 + T ( γ , λ , q ) a p + j e i π 2 − arg ( a p + j ) = 1 + i T ( γ , λ , q ) ∣ a p + j ∣ .

Thus, from (19), we obtain α 5 and is given by

α 5 = 1 5 ℬ q , p − m − λ f ( z 1 ) z 1 p + ℬ q , p − m − λ f ( z 2 ) z 2 p + ℬ q , p − m − λ f ( z 3 ) z 3 p + ℬ q , p − m − λ f ( z 4 ) z 4 p + ℬ q , p − m − λ f ( z 5 ) z 5 p

= 1 + ( 1 + i ) ( 3 + 2 + 3 ) 10 T ( γ , λ , q ) ∣ a p + j ∣ .

This gives us that

∣ 1 − e − i β α 5 ∣ = 2 ( 3 + 2 + 3 ) 10 T ( γ , λ , q ) ∣ a p + j ∣ .

For such α 5 and β , we take ρ > 1 with

q γ [ p + j + γ ] q [ p + γ ] q − 1 T 2 ( γ , λ , q ) ∣ a p + j ∣ 2 [ p + γ ] q ( 1 − T ( γ , λ , q ) ∣ a p + j ∣ ) ≤ q γ ∣ e i β − α 5 ∣ 2 j ρ 2 [ p + γ ] q ( 1 + ∣ e i β − α 5 ∣ ρ ) .

It follows from the above that

ρ ≥ 10 2 ( 3 + 2 + 3 ) T ( γ , λ , q ) a p + j > 1 .

For α 5 and ρ > 1 , f ( z ) satisfies

ℬ q , p − m − λ f ( z ) z p − 1 < ρ ∣ e i β − α 5 ∣ , z ∈ E .

Theorem 3

If f ( z ) ∈ ℋ ( p ) satisfies

ℬ q , p − m − λ + l f ( z ) z p − 1 < ρ 1 + j [ p + γ ] q ∣ e i β − α s ∣ , z ∈ E .

For some α s given by (19), with α s ≠ 1 , ( l = 1 , 2 , 3 , … , s ), and for some real ρ > 1 , then

ℬ q , p − m − λ + l − 1 f ( z ) z p − 1 < ρ ∣ e i β − α s ∣ , z ∈ E .

Proof

Define the function u ( z ) by

u ( z ) = e i β ( ℬ q , p − m − λ + l − 1 f ( z ) ) z p − α s e i β − α s − 1 = e i β e i β − α s ∑ k = p + j ∞ [ p + γ ] q [ k + γ ] q m − l + 1 Γ q ( k + γ ) Γ q ( p + γ + λ ) Γ q ( k + γ + λ ) Γ q ( p + γ ) a k z k − p .

Then, for u ( z ) given in (1) and

ℬ q , p − m − λ + l − 1 f ( z ) = z p + ( 1 − e − i β α s ) z p u ( z ) .

By the definition of ℬ q , p − m − λ + l f ( z ) , we know that

ℬ q , p − m − λ + l f ( z ) = z 1 − γ [ p + γ ] q ∂ q ( z γ ℬ q , p − m − λ + l − 1 f ( z ) ) = z p 1 + ( 1 − e − i β α s ) u ( z ) 1 + z ∂ q u ( z ) [ p + γ ] q u ( z ) .

Our condition implies that

(33) ℬ q , p − m − λ + l f ( z ) z p − 1 = ( 1 − e − i β α s ) u ( z ) 1 + z ∂ q u ( z ) [ p + γ ] q u ( z ) < ρ ∣ e i β − α s ∣ 1 + j [ p + γ ] q

for all z ∈ E . Let there exist a point z 0 , ( 0 < ∣ z 0 ∣ < 1 ) such that

max { ∣ u ( z ) ∣ ; ∣ z ∣ ≤ ∣ z 0 ∣ } = ∣ u ( z 0 ) ∣ = ρ > 1 .

Then, by Lemma 1, we can write that u ( z 0 ) = ρ e i θ , ( 0 ≤ θ ≤ 2 π ) , and z 0 ∂ q u ( z 0 ) = k u ( z 0 ) , k ≥ j . Therefore, we have

ℬ q , p − m − λ + l f ( z 0 ) z 0 p − 1 = ( 1 − e − i β α s ) u ( z 0 ) 1 + z 0 ∂ q u ( z 0 ) [ p + γ ] q u ( z 0 ) ≥ ρ ∣ e i β − α s ∣ 1 + j [ p + γ ] q ,

which contradicts inequality (33). This means that there is no z 0 ∈ E , such that

∣ u ( z 0 ) ∣ = ρ > 1 .

As we know that

∣ u ( z ) ∣ = e i β ( ℬ q , p − m − λ + l − 1 f ( z ) ) z p − α s e i β − α s − 1 = e i β e i β − α s ℬ q , p − m − λ + l − 1 f ( z ) z p − 1 < ρ .

Thus, our theorem is completed.□

Theorem 4

If f ( z ) ∈ ℋ ( p ) satisfies

(34) ℬ q , p − m − λ + l f ( z ) z p − 1 < ρ 1 + j [ p + γ ] q l ∣ e i β − α s ∣ , z ∈ E ,

for some α s defined by, (19) with α s ≠ 1 , ( l = 1 , 2 , 3 , … , s ), and for ρ > 1 , then

ℬ q , p − m − λ f ( z ) z p − 1 < ρ ∣ e i β − α s ∣ , z ∈ E ,

or, equivalently, f ( z ) ∈ T p , j ( q , α s , β , ρ , λ , m ) .

Proof

Theorem 3 demonstrates that if f ( z ) satisfy inequality (34), then

ℬ q , p − m − λ + l − 1 f ( z ) z p − 1 < ρ ∣ e i β − α s ∣ 1 + j [ p + γ ] q l − 1 , z ∈ E .

Similarly, we have

ℬ q , p − m − λ + l − 2 f ( z ) z p − 1 < ρ ∣ e i β − α s ∣ 1 + j [ p + γ ] q l − 2 , z ∈ E .

Considering this further, we obtain that

ℬ q , p − m − λ f ( z ) z p − 1 < ρ ∣ e i β − α s ∣ , z ∈ E .

This completes the theorem.□

Example 3

For f ( z ) ∈ ℋ ( p ) provided by

f ( z ) = z p + a p + j z p + j , z ∈ E ,

which satisfies

(35) ℬ q , p − m − λ + l f ( z ) = z p + [ p + γ ] q [ p + j + γ ] q m − l Γ q ( p + j + γ ) Γ q ( p + γ + λ ) Γ q ( p + j + γ + λ ) Γ q ( p + γ ) a p + j z p + j .

It follows from (35) that

ℬ q , p − m − λ + l f ( z ) z p − 1 = [ p + γ ] q [ p + j + γ ] q m − l Γ q ( p + j + γ ) Γ q ( p + γ + λ ) Γ q ( p + j + γ + λ ) Γ q ( p + γ ) ∣ a p + j ∣ ∣ z ∣ j < T ( γ , λ , q ) [ p + j + γ ] q [ p + γ ] q l ∣ a p + j ∣ , z ∈ E ,

where T ( γ , λ , q ) given by (31). Now, we consider the five boundary points given in Example 2. Then, we see

(36) ∣ e i β − α 5 ∣ = 2 ( 3 + 2 + 3 ) 10 T ( γ , λ , q ) ∣ a p + j ∣ .

With relation given in (36), we take ρ > 1 such that

T ( γ , λ , q ) [ p + j + γ ] q [ p + γ ] q l ∣ a p + j ∣ ≤ ρ ∣ e i β − α 5 ∣ .

It follows from the above that

ρ ≥ 10 2 ( 3 + 2 + 3 ) [ p + j + γ ] q [ p + γ ] q l > 1 .

Thus, we have that

ℬ q , p − m − λ f ( z ) z p − 1 ≤ T ( γ , λ , q ) ∣ a p + j ∣ ≤ ρ [ p + γ ] q [ p + j + γ ] q l ∣ e i β − α 5 ∣ < ρ ∣ e i β − α 5 ∣ , z ∈ E .

Remark 7

These results correlate with uses of the q -Libera integral operator if we assume that γ = 1 in the results of this section. Let us assume that the q -Libera integral operator is written as

(37) ℬ q , p − m − λ f ( z ) = ℒ q , p − m − λ f ( z ) = z p + ∑ k = p + j ∞ [ p + 1 ] q [ k + 1 ] q m Γ q ( p + 1 + λ ) [ k ] q ! Γ q ( k + 1 + λ ) [ p ] q ! a k z k .

Taking γ = 1 in Theorems 1, 3, and 4, then we achieve the findings listed below, which are applications of the q -Libera integral operator presented in (37).

Theorem 5

If f ( z ) ∈ ℋ ( p ) satisfies

ℒ q , p − m − λ + 1 f ( z ) ℒ q , p − m − λ f ( z ) − 1 < q γ ∣ e i β − α s ∣ j ρ [ p + 1 ] q ( 1 + ∣ e i β − α s ∣ ρ ) , z ∈ E ,

for some α s given by (19) with α s ≠ 1 such that z l ∈ ∂ E ( l = 1 , 2 , 3 , … , s ), and for some real ρ > 1 , then

ℒ q , p − m − λ f ( z ) z p − 1 < ρ ∣ e i β − α s ∣ , z ∈ E

that is f ( z ) ∈ T p , j ( q , α s , β , ρ , λ , m ) .

Theorem 6

If f ( z ) ∈ ℋ ( p ) satisfies

( ℒ q , p − m − λ + l f ( z ) ) z p − 1 < ρ 1 + j [ p + 1 ] q ∣ e i β − α s ∣ , z ∈ E ,

for some α s given by (19) with α s ≠ 1 , ( l = 1 , 2 , 3 , … , s ), and for some real ρ > 1 , then

ℒ q , p − m − λ + l − 1 f ( z ) z p − 1 < ρ ∣ e i β − α s ∣ , z ∈ E .

Theorem 7

If f ( z ) ∈ ℋ ( p ) satisfies

( ℒ q , p − m − λ + l f ( z ) ) z p − 1 < ρ 1 + j [ p + 1 ] q l ∣ e i β − α s ∣ , z ∈ E ,

for some α s given by (19) with α s ≠ 1 , ( l = 1 , 2 , 3 , … , s ) , and for some real ρ > 1 , then

ℒ q , p − m − λ f ( z ) z p − 1 < ρ ∣ e i β − α s ∣ , z ∈ E .

Or, equivalently, f ( z ) ∈ T p , j ( q , α s , β , ρ , λ , m ) .

3 Application of the Carathéodory lemma

This section utilizes the Carathéodory lemma to analyze the coefficients of functions f ( z ) within the class T p , j ( q , α s , β , ρ , λ , m ) .

Lemma 2

[36]. Let us have the function

g ( z ) = 1 + ∑ k = 1 ∞ c k z k

be analytic in E and Re g ( z ) > 0 , z ∈ E . Then,

(38) ∣ c k ∣ ≤ 2 , ( k ≥ 1 ) .

Inequality (38) is sharp.

We obtain the following theorem by using the aforementioned lemma.

Theorem 8

If f ( z ) ∈ ℋ ( p ) is in T p , j ( q , α s , β , ρ , λ , m ) , then

(39) ∣ a k ∣ ≤ 2 ∣ e i β − α s ∣ ρ T 1 ( γ , λ , q ) , ( k = p + j , p + j + 1 , … ) ,

where

(40) T 1 ( γ , λ , q ) = [ p + γ ] q [ p + γ ] q m Γ q ( k + γ ) Γ q ( p + γ + λ ) Γ q ( k + γ + λ ) Γ q ( p + γ ) .

The result is sharp for f ( z ) defined by

ℬ q , p − m − λ f ( z ) = z p e i θ − ( 1 + 2 ∣ e i β − α s ∣ ρ ) z e i θ − z .

Proof

Let f ( z ) ∈ T p , j ( q , α s , β , ρ , λ , m ) , we see that

ℬ q , p − m − λ f ( z ) z p − 1 < ∣ e i β − α s ∣ ρ , z ∈ E .

If we define a function g ( z ) with f ( z ) ∈ T p , j ( q , α s , β , ρ , λ , m ) by

g ( z ) = ℬ q , p − m − λ f ( z ) z p − ( 1 − ∣ e i β − α s ∣ ρ ) ∣ e i β − α s ∣ ρ , z ∈ E ,

then g ( z ) is analytic in E along with Re g ( z ) > 0 , and g ( 0 ) = 1 , z ∈ E . The following power series expansion is also present for g ( z ) :

(41) g ( z ) = 1 + ∑ k = p + j ∞ T 1 ( γ , λ , q ) ∣ e i β − α s ∣ ρ a k z k − p .

Thus, by using Lemma 2 on (41), we obtain

T 1 ( γ , λ , q ) ∣ e i β − α s ∣ ρ ∣ a k ∣ ≤ 2 ,

where

k = p + j , p + j + 1 , … .

This demonstrates the coefficient inequalities (39). Consider that

g ( z ) = e i θ + z e i θ − z = 1 + ∑ k = 1 ∞ 2 e i θ z k = 1 + ∑ k = 1 ∞ c k z k

and g ( z ) analytic in E , along with g ( 0 ) = 1 , Re ( g ( z ) ) > 0 and ∣ c k ∣ = 2 , ( k = 1 , 2 , 3 … ) . Thus, assuming f ( z ) in a way that

g ( z ) = ℬ q , p − m − λ f ( z ) z p − ( 1 − ∣ e i β − α s ∣ ρ ) ∣ e i β − α s ∣ ρ = e i θ + z e i θ − z ,

we have

ℬ q , p − m − λ f ( z ) z p = ( e i θ + z ) ∣ e i β − α s ∣ ρ + ( 1 − ∣ e i β − α s ∣ ρ ) ( e i θ − z ) e i θ − z ℬ q , p − m − λ f ( z ) = z p e i θ − ( 1 + 2 ∣ e i β − α s ∣ ρ ) z e i θ − z .

Theorem is thus completed.□

We obtain the following result for the q -Libera integral operator by assuming that γ = 1 in Theorem 8.

Theorem 9

If f ( z ) ∈ ℋ ( p ) satisfies

ℒ q , p − m − λ f ( z ) z p − 1 < ∣ e i β − α s ∣ ρ , z ∈ E ,

then

∣ a k ∣ ≤ 2 ∣ e i β − α s ∣ ρ T 0 ( γ , λ , q ) , ( k = p + j , p + j + 1 , … ) ,

where

T 0 ( γ , λ , q ) = [ p + 1 ] q [ p + 1 ] q m [ k ] q ! Γ q ( p + 1 + λ ) Γ q ( k + 1 + λ ) [ p ] q ! .

The result is sharp for f ( z ) as given by

ℒ q , p − m − λ f ( z ) = z p e i θ − ( 1 + 2 ∣ e i β − α s ∣ ρ ) z e i θ − z .

Theorem 10

If f ( z ) ∈ ℋ ( p ) satisfies

(42) ∑ k = p + j ∞ T 1 ( γ , λ , q ) ∣ a k ∣ ≤ ∣ e i β − α s ∣ ρ .

Then, f ( z ) ∈ T p , j ( q , α s , β , ρ , λ , m ) , where, p , j ∈ N and T 1 ( γ , λ , q ) is given by (40).

Proof

For f ( z ) ∈ ℋ ( p ) , we let

ℬ q , p − m − λ f ( z ) z p − 1 = ∑ k = p + j ∞ T 1 ( γ , λ , q ) a k z k < ∑ k = p + j ∞ T 1 ( γ , λ , q ) ∣ a k ∣ ≤ ∣ e i β − α s ∣ ρ , z ∈ E , j ∈ N .

Therefore, if f ( z ) ∈ ℋ ( p ) satisfies (42), then we know f ( z ) ∈ T p , j ( q , α s , β , ρ , λ , m ) . □

Remark 8

If we take γ = 1 in Theorem 10, then we obtain result related to the q -Libera integral operator ℒ q , p − m − λ f ( z ) .

4 Conclusions

Currently, the application of q -calculus in GFT is a highly trending and active area of research, surpassing traditional classical techniques. While classical calculus techniques have been widely used in GFT, this article takes a novel approach by applying q -calculus techniques to explore new possibilities in the field of GFT. This article leverages the q -calculus operator theory to introduce the q -Bernardi and q -Jung-Kim-Srivastava integral operators for multivalent functions. Building on these operators, we develop a generalized fractional q -integral operator, ℬ q , p − m − λ f ( z ) of order λ , which encompasses various known operators for specific parameter values. The applications of these operators are investigated by defining a subclass T p , j ( q , α s , β , ρ , λ , m ) of multivalent functions via the generalized fractional q -integral operator, which enables us to study their characteristics and potential uses. Section 2 presents new results and examples for functions belonging to this new class, while Section 3 explores applications of the Carathéodory lemma, yielding additional new results.

Furthermore, this concept can be extended to meromorphic and bi-univalent functions, leveraging the operators introduced in this article. By applying these operators, we can establish numerous new subclasses of analytic and meromorphic functions, which can be explored for various useful properties, such as coefficient inequalities, Fekete-Szegő problems, partial sums, sufficient conditions, convex combinations, closure theorems, growth and distortion bounds, radii of close-to-starlikeness, and starlikeness.

Acknowledgements

The authors acknowledge Maha Alammari for her valuable input and contribution to the revised version of this manuscript.

Funding information: This research did not receive external funding.
Author contributions: Conceptualization, SK, AK, and IAS; data curation, SK, AS, and NMA; formal analysis, MA, SK, IAS, and WH; funding acquisition, AS; investigation, AS, AK, SK, and NMA; methodology, SK, AK, WH, and AS; project administration, AS, SK, and NMA; resources, IAS, AS, and SK; software, SK, AK, WH, and NMA; supervision, MA, SK, and IAS; validation, IAS and AS; visualization, AK and SK; writing–original draft, MA, AK, SK; writing–review and editing, SK, AS, and IAS. All authors have read and approved the final manuscript.
Conflict of interest: The authors declare that they have no conflict of interest.
Data availability statement: No data is used in this work.

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Received: 2024-03-28

Revised: 2024-07-28

Accepted: 2025-04-07

Published Online: 2025-07-29

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

Artikel in diesem Heft

https://doi.org/10.1515/dema-2025-0128

Schlagwörter für diesen Artikel

analytic functions; q-Bernardi integral operator; multivalent analytic functions; q-difference operator; fractional q-integral operator; q-Jack’s lemma

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