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Article

On a Certain Subclass of p-Valent Analytic Functions Involving q-Difference Operator

by
Abdel Moneim Y. Lashin
1,2,
Abeer O. Badghaish
1 and
Badriah Maeed Algethami
1,*
1
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2
Department of Mathematics, Faculty of science, Mansoura University, Mansoura 35516, Egypt
*
Author to whom correspondence should be addressed.
Symmetry 2023, 15(1), 93; https://doi.org/10.3390/sym15010093
Submission received: 7 December 2022 / Revised: 25 December 2022 / Accepted: 27 December 2022 / Published: 29 December 2022
(This article belongs to the Special Issue Symmetry in Quantum Calculus)

Abstract

:
This paper introduces and studies a new class of analytic p-valent functions in the open symmetric unit disc involving the Sălăgean-type q-difference operator. Furthermore, we present several interesting subordination results, coefficient inequalities, fractional q-calculus applications, and distortion theorems.

1. Introduction

As a result of Euler and Heine’s pioneering work, Frank Hilton Jackson developed q-calculus in a systematic manner at the beginning of the previous century. In his work, Jackson systematically developed the concepts of the q-derivative (Jackson [1]), as well as the q-integral (Jackson [2]). Calculus without limits is called q-calculus. Due to its applications in mathematics, mechanics, and physics, symmetric q-calculus is experiencing rapid growth. Ismail et al. [3] were the first to apply q-calculus to geometric function theory (GFT) by generalizing the set of starlike functions into q-analogs, called q-starlike functions. Several authors have extensively investigated the q-difference operator in GFT based on the same idea. Some recent works related to this operator on analytic functions include [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. Several properties of certain analytic multivalent functions are considered in this paper using the q-analog of the Sălăgean differential operator. Let A p j denote the class of functions that have the form
f ( z ) = z p + l = p + j a l z l , ( p , j N : = { 1 , 2 , } ) ,
that are analytic in the open unit disc E = z C : z < 1 . Let A = A 1 1 . In [1,2], the q-derivative operator q of a function f was defined by Jackson as follows:
q f ( z ) = f ( q z ) f ( z ) ( q 1 ) z ( z 0 ) , f ( 0 ) ( z = 0 ) .
For a function f ( z ) A p j , we deduce that
q f ( z ) = p q z p 1 + l = 2 l q a l z l 1 ,
where
l q = 1 q l 1 q .
As q 1 , l q l . Jackson [1] introduced the q-integral
0 z f ( t ) d q t = z ( 1 q ) l = 0 q l f ( z q l ) ,
as long as the series converges. For a function f ( z ) = z l , one can observe that
0 z f ( t ) d q t = 0 z t l d q t = 1 [ l + 1 ] q z l + 1 ( l 1 ) .
For a function f ( z ) A p j , El-Qadeem and Mamon [23] defined the p-valent q-Sălăgean operator by
D p , q 0 f ( z ) = f ( z ) , D p , q 1 f ( z ) = D p , q f ( z ) = z q f ( z ) p q = z p + l = p + j a l l q p q z l , D p , q 2 f ( z ) = D p , q D p , q f ( z ) = z p + l = p + j a l l q p q 2 z l ,
therefore,
D p , q n f ( z ) = D p , q D p , q n 1 f ( z ) = z p + l = p + j a l l q p q n z l .
When p = 1 , the q-Sălăgean operator was introduced by Govindaraj and Sivasubramanian [24]. The q-shifted factorial, see [25], is defined for a C by
( a ; q ) n =   1 if n = 0 , ( 1 a ) ( 1 a q ) ( 1 a q 2 ) ( 1 a q n 1 ) , if n N = { 1 , 2 , } ,
let ( a ; q ) = n = 0 ( 1 a q n ) . Recalling the q-analog definitions given by Gasper and Rahman [26], the q-Gamma function is given by
Γ q ( z ) = q , q q z , q 1 q 1 z 0 < q < 1 ,
and the q-binomial expansion is given by
( x y ) ν = x ν y x , q ν = x ν n = 0 1 y x q n 1 y x q n + ν .
For functions f and g analytic in E , one can say that f is subordinate to g, written as f g or f ( z ) g ( z ) ( z E ) , if there exists a Schwarz function ω , that is analytic in E with ω 0 = 0 , ω z < 1 and f ( z ) = g ( ω z )   ( z E ) . In addition, if the function g is univalent in E , then the following equivalence will occur
f ( z ) g ( z ) f ( 0 ) g ( 0 ) ,
and
f ( E ) g ( E ) .
For functions f j ( z ) = l = 0 a l , j z l j = 1 , 2 analytic in E , the Hadamard product (or convolution) of f 1 ( z ) and f 2 ( z ) is defined by
f 1 f 2 z = l = 0 a l , 1 a l , 2 z l = f 2 f 1 z z E .
A function f A is convex, if and only if f ( E ) is a convex domain. We denote this subclass of A by K. Analytically, a function f A belongs to the class K if and only if
1 + z f z f z > 0 z E .
The proof can be found in [27]. A similar characterization can be made for the class S * of functions starlike in E . A function f A belongs to the class S * if and only if
z f z f z > 0 z E .
More details on the classes of starlike and convex functions can be found in [28,29]. A univalent function f : E C ^ = C { } is said to be concave if the complement C ^ \ f is convex (functions mapping on the exterior of a convex curve). An analytic, univalent function f A is said to be in the class C o ( α ) , if it is concave, satisfies f ( 1 ) = with an opening angle of f ( E ) at less than or equal to α π with α ( 1 , 2 ] . Due to the similarity with convex functions, sometimes the inequality
1 + z f z f z < 0 z E ,
is also used as a definition for concave functions (see e.g., [30]) see also, Avkhadiev et al. [31], Cruz and Pommerenke [32], and the references within. Recently, Nishiwaki and Owa [33] defined and studied the subclasses M β and N β of A as follows: for some β β > 1 , let M β be the subclass of A consisting of functions f ( z ) , which satisfy
z f z f z < β z E ,
let N β be the subclass of A consisting of functions f ( z ) , which satisfy
1 + z f z f z < β z E ,
(see [33,34,35,36]). With the use of the differential operator D p , q n , we introduce class A p , q ( n , j , β ) , which generalizes the above-mentioned classes M β and N β .
Definition 1.
We say that a function f ( z ) A p j belongs to the class A p , q ( n , j , β ) , if it satisfies the condition
z q ( D p , q n f ( z ) ) D p , q n f ( z ) < β z E ,
where n N 0 = N { 0 } , p N , β > p q , and 0 < q < 1 .
As f ( z ) = z p belongs to the class A p , q ( n , j , β ) , it is not empty. A p , q ( n , j , β ) generalizes the classes M β and N β as follows
Remark 1.
1.   lim q 1 A 1 , q ( 0 , 1 , β ) = M β ;
2. 
lim q 1 A 1 , q ( 1 , 1 , β ) = N β .
In this paper, we derive some interesting subordination results, coefficient inequalities, and various distortion theorems involving fractional q-calculus operators for functions in the class A p , q ( n , j , β ) . Moreover, some special cases are also indicated.

2. Coefficient Estimates

Theorem 1.
If f ( z ) A p j satisfies the condition
l = p + j l q p q n l q p q + l q + p q 2 β a l 2 β p q ,
for some β β > p q ; n N 0 , then f ( z ) A p , q ( n , j , β ) .
Proof. 
Let condition (6) be true. Then, we have
z q ( D p , q n f ( z ) ) D p , q n f ( z ) p q z q ( D p , q n f ( z ) ) D p , q n f ( z ) 2 β p q = l = p + j a l l q p q n l q p q z l 2 β p q z p + l = p + j a l l q p q n l q + p q 2 β z l z p + j l = p + j a l l q p q n ( l q p q ) 2 β p q z p + j l = p + j a l l q p q n l q + p q 2 β l = p + j a l l q p q n l q p q 2 β p q l = p + j a l l q p q n l q + p q 2 β
the last expression is bounded above by 1 if
l = p + j l q p q n l q p q + l q + p q 2 β a l 2 β p q .
This completes the proof of Theorem 1. □
Remark 2.
Letting p = 1 , q 1 , and n = 1 in Theorem 1, we obtain the result obtained by Nishwaki and Owa [33].
Corollary 1.
If f ( z ) A p j satisfies the condition
l = p + j l q p q n l q β a l β p q ,
for some β p q β < p + j q + p q 2 ; n N 0 , then f ( z ) A p , q ( n , j , β ) .
Proof. 
Since ( l q + p q 2 β ) is an increasing function of l ( l p + j ) , we have
l q + p q 2 β 0 i f p + j q + p q 2 β 0 ,
or
β p + j q + p q 2 .

3. Subordination Results

Definition 2
([37]). A sequence b l l = 1 of complex numbers is said to be subordinating factor sequence if, whenever f ( z ) = z + l = 2 a l z l , a 1 = 1 is analytic, univalent, and convex in E , we have
l = 1 b l a l z l f ( z ) z E .
Lemma 1
([37]). The sequence b l l = 1 is subordinating factor sequence if and only if
1 + 2 l = 1 b l z l > 0 z E .
Let A p , q * ( n , j , β ) denoted the class of functions f ( z ) A p ( j ) whose coefficients satisfy the condition (6).
Theorem 2.
Let f ( z ) A p , q * ( n , j , β ) , g ( z ) K , and
ε = p + j q p q n q p j q + p + j q + p q 2 β 2 p + j q p q n q p j q + p + j q + p q 2 β + β p q ,
then
ε z 1 p f ( z ) g ( z ) g ( z ) ( z E ) ,
and
f ( z ) z p 1 > 1 2 ε .
The constant
p + j q p q n q p j q + p + j q + p q 2 β 2 p + j q p q n q p j q + p + j q + p q 2 β + β p q ,
is the best estimate.
Proof. 
Let f ( z ) A p , q * ( n , j , β ) , and let g ( z ) = z + l = 2 c l z l belong to the subclass K . Then
ε z 1 p f ( z ) g ( z ) = l = 1 b l c l z l ( z E ) ,
where
b l = ε ( l = 1 ) , 0 ( 2 l j ) , ε a p + l 1 ( l j + 1 ) .
Hence, by using Definition 2, the subordination result (7) will be true, if b l l = 1 is the subordinating factor sequence. Since
Ψ ( l ) = l q p q n l q p q + l q + p q 2 β ,
is an increasing function of l ( l j + 1 ) , we have
1 + 2 l = 1 b l z l = 1 + 2 ε z + 2 l = j + 1 b l z l = 1 + p + j q p q n q p j q + p + j q + p q 2 β p + j q p q n q p j q + p + j q + p q 2 β + β p q z + 1 p + j q p q n q p j q + p + j q + p q 2 β + β p q × l = j + p p + j q p q n q p j q + p + j q + p q 2 β a l z l + 1 p 1 p + j q p q n q p j q + p + j q + p q 2 β p + j q p q n q p j q + p + j q + p q 2 β + β p q r l = j + p l q p q n l q p q + l q + p q 2 β p + j q p q n q p j q + p + j q + p q 2 β + β p q a l r j + 1
Thus, by using Theorem 1, and Lemma 1 we deduce that
1 + 2 l = 1 b l z k 1 p + j q p q n q p j q + p + j q + p q 2 β r p + j q p q n q p j q + p + j q + p q 2 β + β p q β p q p + j q p q n q p j q + p + j q + p q 2 β + β p q r > 0 .
This proves the subordination result (7). Letting g ( z ) = z 1 z = l = 1 z l ( z E ) in (7), we easily get the result (8). □
Theorem 3.
Let f ( z ) be in the class A p , q * ( n , j , β ) , defined by (1). Then for z = r < 1 , we have
p q ! p m q ! 2 β p q p + j q ! p + j m q ! r j p + j q p q n p + j q p q + p + j q + p q 2 β r p m q m f z p q ! p m q ! + 2 β p q p + j q ! p + j m q ! r j p + j q p q n p + j q p q + p + j q + p q 2 β r p m .
The result is sharp for the function f ( z ) given by
f ( z ) = z p + 2 β p q p + j q ! p + j m q ! p + j q p q n p + j q p q + p + j q + p q 2 β z p + j .
Proof. 
Since Ψ ( l ) given by (9) is an increasing function of l ( l j + 1 ) , Theorem 1 gives
p + j q p q n p + j q p q + p + j q + p q 2 β l = p + j a l l = p + j l q p q n l q p q + l q + p q 2 β a l 2 β p q .
That is
l = p + j a l 2 β p q p + j q p q n p + j q p q + p + j q + p q 2 β .
The m t h q-derivative of the functions f ( z ) A p ( j ) is given by
q m f z = p q ! p m q ! z p m + l = p + j l q ! l m q ! a l z l m ,
then we have
q m f z p q ! p m q ! z p m l = p + j l q ! l m q ! a l z l m p q ! p m q ! r p m r p + j m p + j q ! p + j m q ! l = p + j a l
p q ! p m q ! 2 β p q p + j q ! p + j m q ! r j p + j q p q n p + j q p q + p + j q + p q 2 β r p m ,
and
q m f z p q ! p m q ! z p m + l = p + j l q ! l m q ! a l z l m p q ! p m q ! r p m + p + j q ! p + j m q ! r p + j m l = p + j a l
p q ! p m q ! + 2 β p q p + j q ! p + j m q ! r j p + j q p q n p + j q p q + p + j q + p q 2 β r p m .
Putting m = 0 in Theorem 3 we have the following corollary
Corollary 2.
Let f ( z ) defined by (1) be in the class A p , q * ( n , j , β ) .Then for z = r < 1 , we have
f z 1 2 β p q r j p + j q p q n p + j q p q + p + j q + p q 2 β r p ,
and
f z 1 + 2 β p q r j p + j q p q n p + j q p q + p + j q + p q 2 β r p .
This result is sharp.

4. Application of q -Fractional Calculus Operators

Let the function f ( z ) be defined by (1). Then the q-Bernardi integral operator J c , p q is given by
J c , p q f ( z ) = c + p q z c 0 z t c 1 f ( z ) d q t = z p + l = p + j c + p q c + l q a l z l c > p ,
this operator introduced by El-Qadeem and Mamon [23] (see also [17,38]). For f ( z ) A p j , we define the following q-fractional calculus operators given by Purohit and Raina [39,40].
Definition 3.
The fractional q-integral operator of order m ( m > 0 ) is defined, for a function f , by
Ω q , z m f ( z ) = 1 Γ q ( m ) 0 z z q t m 1 f ( t ) d q t ,
where f is analytic in a simply-connected region of the z-plane containing the origin and the function z q t m is single-valued when arg ( t q m z ) < π , t q m z ) < 1 and arg z < π .
Definition 4.
The fractional q-derivative operator of order m is defined, for a function f , by
Ω q , z m ( f ( z ) ) = 1 Γ q ( 1 m ) q 0 z z q t m f ( t ) d q t ( 1 > m 0 ) ,
where f suitably constrained and removing the multiplicity of z q t m as in Definition 3 above.
Remark 3.
From Definitions 3 and 4, we see that
Ω q , z m z γ = Γ q ( γ + 1 ) Γ q ( γ m + 1 ) z γ m ( m 0 , γ > 1 ) , Ω q , z m z γ = Γ q ( γ + 1 ) Γ q ( γ + m + 1 ) z γ + m ( m > 0 , γ > 1 ) .
This gives that, for f ( z ) A p j ,
Ω q , z m f ( z ) = Γ q p + 1 Γ q p + 1 + m z p + m + l = p + j Γ q l + 1 Γ q l + 1 + m a l z l + m ,
and
Ω q , z m f ( z ) ) = Γ q p + 1 Γ q p + 1 m z p m + l = p + j Γ q l + 1 Γ q l + 1 m a l z l m .
Using the formulas (13), (14), and (12), we have
Ω q , z m ( J c , p q f ( z ) ) = Γ q p + 1 Γ q p + 1 + m z p + m + l = p + j c + p q c + l q Γ q l + 1 Γ q l + 1 + m a l z l + m ,
Ω q , z m ( J c , p q f ( z ) ) = Γ q p + 1 Γ q p + 1 m z p m + l = p + j c + p q c + l q Γ q l + 1 Γ q l + 1 m a l z l m ,
J c , p q ( Ω q , z m f ( z ) ) = c + p q c + p + m q Γ q p + 1 Γ q p + 1 + m z p + m + l = p + j c + p q c + l + m q Γ q l + 1 Γ q l + 1 + m a l z l + m ,
and
J c , p q ( Ω q , z m f ( z ) ) = c + p q c + p m q Γ q p + 1 Γ q p + 1 m z p m + l = p + j c + p q c + l m q Γ q l + 1 Γ q l + 1 m a l z l m .
Here, we investigate the distortion properties of functions in the class A p , q * ( n , j , β ) involving the operators J q , z q , Ω q , z m , and Ω q , z m .
Theorem 4.
Let f ( z ) be in the class A p , q ( n , j , β ) , defined by (1). Then for z = r < 1 , we have
Ω q , z m ( J c , p q f ( z ) ) Γ q p + 1 Γ q p + 1 + m Υ p , q c , j , m n , β z j z p + m ,
Ω q , z m ( J c , p q f ( z ) ) Γ q p + 1 Γ q p + 1 + m + Υ p , q c , j , m n , β z j z p + m ,
where
Υ p , q c , j , m n , β = c + p q c + p + j q 2 Γ q p + j + 1 ( β p q ) Γ q p + j + 1 + m p + j q p q n p + j q p q + p + j q + p q 2 β ,
( 0 m < 1 , c > p , p N ) . Each of the assertions are sharp for f given by (10).
Proof. 
By using (11) and (15), we have
Ω q , z m ( J c , p q f ( z ) ) Γ q p + 1 Γ q p + 1 + m z p + m l = p + j c + p q c + l q Γ q k + 1 Γ q l + 1 + m a l z l + m Γ q p + 1 Γ q p + 1 + m z p + m c + p q c + p + j q Γ q p + j + 1 Γ q p + j + 1 + m z p + j + m l = p + j a l Γ q p + 1 Γ q p + 1 + m Υ p , q c , j , m n , β z j z p + m ,
where
Υ p , q c , j , m n , β = c + p q c + p + j q 2 Γ q p + j + 1 ( β p q ) Γ q p + j + 1 + m p + j q p q n p + j q p q + p + j q + p q 2 β .
Similarly, using (15) and (11) we have
Ω q , z m ( J c , p q f ( z ) ) Γ q p + 1 Γ q p + 1 + m + Υ p , q c , j , m n , β z j z p + m .
Thus, the proof of the theorem is completed. □
Theorem 5.
Let f ( z ) be in the class A p , q ( n , j , β ) , defined by (1). Then for z = r < 1 , we have
Ω q , z m ( J c , p q f ( z ) ) Γ q p + 1 Γ q p + 1 m Θ p , q c , j , m n , β z j z p m ,
Ω q , z m ( J c , p q f ( z ) ) Γ q p + 1 Γ q p + 1 m + Θ p , q c , j , m n , β z j z p m ,
where
Θ p , q c , j , m n , β = c + p q c + p + j q 2 Γ q p + j + 1 ( β p q ) Γ q p + j + 1 m p + j q p q n p + j q p q + p + j q + p q 2 β .
According to (10), each assertion is sharp.
Proof. 
Using (11) and (16), the assertions (21) and (22) of Theorem 5 can now be proved similarly to Theorem 4. □
Theorem 6.
Let f ( z ) be in the class A p , q ( n , j , β ) , defined by(1). Then for z = r < 1 , we have
J c , p q ( Ω q , z m f ( z ) ) c + p q c + p + m q Γ q p + 1 Γ q p + 1 + m Λ p , q c , j , m n , β z j z p + m
J c , p q ( Ω q , z m f ( z ) ) c + p q c + p + m q Γ q p + 1 Γ q p + 1 + m + Λ p , q c , j , m n , β z j z p + m ,
where
Λ p , q c , j , m n , β = c + p q c + p + j q 2 Γ q p + j + 1 ( β p q ) Γ q p + j + 1 + m p + j q p q n p + j q p q + p + j q + p q 2 β ,
( m > 0 , c > p , p N ) . The result is sharp for the function f given by (10).
Proof. 
We only prove the first inequality. The argument for the second inequality is similar and hence omitted. Using (17) and (11), we have
J c , p q ( Ω q , z m f ( z ) ) c + p q c + p + m q Γ q p + 1 Γ q p + 1 + m z p + m c + p q c + p + j + m q Γ q p + j + 1 Γ q p + j + 1 + m z p + j + m l = p + j a l c + p q c + p + m q Γ q p + 1 Γ q p + 1 + m z p + m c + p q c + p + j + m q 2 Γ q p + j + 1 z p + j + m ( β p q ) Γ q p + j + 1 + m p + j q p q n p + j q p q + p + j q + p q 2 β = c + p q c + p + m q Γ q p + 1 Γ q p + 1 + m Λ p , q c , j , m n , β z j z p + m ,
where
Λ p , q c , j , m n , β = c + p q c + p + j q 2 Γ q p + j + 1 ( β p q ) Γ q p + j + 1 + m p + j q p q n p + j q p q + p + j q + p q 2 β .
Theorem 7.
Let f ( z ) be in the class A p , q ( n , j , β ) , defined by (1). Then for z = r < 1 , we have
J c , p q ( Ω q , z m f ( z ) ) c + p q c + p m q Γ q p + 1 Γ q p + 1 m Φ p , q c , j , m n , β z j z p m ,
J c , p q ( Ω q , z m f ( z ) ) c + p q c + p m q Γ q p + 1 Γ q p + 1 m + Φ p , q c , j , m n , β z j z p m ,
where
Φ p , q c , j , m n , β = c + p q c + p + j q 2 Γ q p + j + 1 ( β p q ) Γ q p + j + 1 + m p + j q p q n p + j q p q + p + j q + p q 2 β ,
( 0 m < 1 , c > p , p N ) . Each of the assertions are sharp for f given by (10).
Proof. 
As the same manner in proving Theorem 6, we can easily deduce the proof of this theorem. □

5. Conclusions

Quantum calculus is classical calculus without limits. The field of q-calculus has recently attracted researchers’ attention. Its application in various branches of mathematics and physics is responsible for this extraordinary interest. Jackson [1,2] was one of the first to define the q-analog to derivative the integral operators and provide some applications for them. Numerous subclasses of normalized analytic functions in the open symmetric unit disc associated with q-derivatives have already been investigated in geometric function theory. Using the Sălăgean q-difference operator, we introduce a new class of analytic p-valent functions in the open symmetric unit disc. Several subordination results, coefficient inequalities, fractional q-calculus applications, and distortion theorems are also presented. The paper also generalizes some known results. For future work, we can study some new classes of analytic p-valent functions in the open symmetric unit disc in the same way.

Author Contributions

Conceptualization, A.M.Y.L. and A.O.B.; Methodology, A.M.Y.L., A.O.B. and B.M.A.; Investigation, A.M.Y.L.; supervision, A.M.Y.L. Writing—original draft, A.O.B. and B.M.A.; Writing—review and editing, A.M.Y.L., A.O.B. and B.M.A.; Project administration, A.M.Y.L.; Funding acquisition, A.M.Y.L., A.O.B. and B.M.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by King Abdulaziz University under grant no. (IFPDP-206-22).

Data Availability Statement

Data sharing not applicable.

Acknowledgments

This research work was funded by the Institutional Fund Projects under grant no. (IFPDP-206-22). Therefore, the authors gratefully acknowledge technical and financial support from the Ministry of Education and Deanship of Scientific Research (DSR), King Abdulaziz University (KAU), Jeddah, Saudi Arabia.

Conflicts of Interest

The authors declare no conflict of interest.

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MDPI and ACS Style

Lashin, A.M.Y.; Badghaish, A.O.; Algethami, B.M. On a Certain Subclass of p-Valent Analytic Functions Involving q-Difference Operator. Symmetry 2023, 15, 93. https://doi.org/10.3390/sym15010093

AMA Style

Lashin AMY, Badghaish AO, Algethami BM. On a Certain Subclass of p-Valent Analytic Functions Involving q-Difference Operator. Symmetry. 2023; 15(1):93. https://doi.org/10.3390/sym15010093

Chicago/Turabian Style

Lashin, Abdel Moneim Y., Abeer O. Badghaish, and Badriah Maeed Algethami. 2023. "On a Certain Subclass of p-Valent Analytic Functions Involving q-Difference Operator" Symmetry 15, no. 1: 93. https://doi.org/10.3390/sym15010093

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