Next Article in Journal
On the Finite-Time Boundedness and Finite-Time Stability of Caputo-Type Fractional Order Neural Networks with Time Delay and Uncertain Terms
Next Article in Special Issue
Fractional Integral of the Confluent Hypergeometric Function Related to Fuzzy Differential Subordination Theory
Previous Article in Journal
On Sliding Mode Control for Singular Fractional-Order Systems with Matched External Disturbances
Previous Article in Special Issue
A Special Family of m-Fold Symmetric Bi-Univalent Functions Satisfying Subordination Condition
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Certain New Subclass of Multivalent Q-Starlike Functions Associated with Q-Symmetric Calculus

1
Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi Arabia
2
Department of Mathematics, Riphah International University, Islamabad 44000, Pakistan
*
Author to whom correspondence should be addressed.
Fractal Fract. 2022, 6(7), 367; https://doi.org/10.3390/fractalfract6070367
Submission received: 22 May 2022 / Revised: 26 June 2022 / Accepted: 28 June 2022 / Published: 30 June 2022
(This article belongs to the Special Issue New Trends in Geometric Function Theory)

Abstract

:
In our present investigation, we extend the idea of q-symmetric derivative operators to multivalent functions and then define a new subclass of multivalent q-starlike functions. For this newly defined function class, we discuss some useful properties of multivalent functions, such as the Hankel determinant, symmetric Toeplitz matrices, the Fekete–Szego problem, and upper bounds of the functional a p + 1 μ a p + 1 2 and investigate some new lemmas for our main results. In addition, we consider the q-Bernardi integral operator along with q-symmetric calculus and discuss some applications of our main results.

1. Introduction and Definitions

The class of all analytic and p-valent functions ζ ( p N = { 1 , 2 , } ) in the open unit disk
U = z : z C and z < 1 ,
denoted by A p and the series expansion of ζ A p is given as:
ζ ( z ) = z p + n = p + 1 a n z n .
Clearly, A p coincides with the set A of normalized analytic univalent (Schlicht) functions if p = 1 . In addition, let us select the symbol S which represents all analytic functions that are univalent in U and satisfy the condition
ζ ( 0 ) = 0 and ζ ( 0 ) = 1 .
A function ζ ( z ) A p is called a p-valently starlike S p and convex K p , which are defined as:
ζ ( z ) S p Re z ζ ( z ) ζ ( z ) > 0 , ( z U )
and
ζ ( z ) K p Re 1 + z ζ ( z ) ζ ( z ) > 0 , ( z U ) .
A function ζ ( z ) A p is called a p-valently starlike function of order α , which are defined as:
ζ ( z ) S p α Re z ζ ( z ) ζ ( z ) > α , ( z U ) .
Further, p-valently convex functions of order α for ζ ( z ) A p are defined as follows:
ζ ( z ) K p ( α ) Re 1 + z ζ ( z ) ζ ( z ) > α , ( z U ) ,
for some 0 α < p . For α = 0 , then
S p 0 = S p
and
K p ( 0 ) = K p .
The class of all analytic functions p which satisfy the condition Re p ( z ) > 0 in the open unit disk U denoted by P and the series expansion of p P is given as:
p ( z ) = 1 + n = 1 c n z n .
Basic (or q-) calculus is used in many different areas of mathematics and other sciences. Jackson [1,2] used the idea of quantum (or q-) calculus and defined the q-analogue of derivative ( D q ) and integral operators. In geometric function theory (GFT) of complex analysis, the usage of the operator ( D q ) is quite significant. Ismail et al. [3] were among the first researchers to work on q-calculus along with GFT and have defined the q- extensions of starlike functions. However in the context of GFT, it was Srivastava [4] who used the operator D q in GFT for the first time. Subsequently, a number of q-analogue operators have been defined. More in-depth information about operators in q-calculus, is available in [5,6,7,8,9]. In [10,11], Arif et al. defined and investigated some new subclasses of multivalent functions by implementing the concept of the q-derivative operator, while Zang et al. [12] used the basic concepts of q-calculus to define the generalized conic domain. For some recent investigations of q-function theory, we refer readers to [13,14,15,16,17,18,19].
The q-symmetric quantum calculus has proven to be useful in several fields, in particular in quantum mechanics [20]. As noted in [20], the q-symmetric derivative has important properties for the q-exponential function which turns out not to be true with the usual derivative. The application of q-symmetric calculus has been discussed in the field of mathematics and physics, particularly in quantum mechanics [21]. Recently, a number of researchers have started to study q-symmetric calculus in the field of GFT. Kanas et al. [22] used the basic concepts of q-symmetric calculus to define the symmetric operator of a q-derivative and studied its applications by defining some new subclasses of analytic univalent functions in open unit disc U. Khan et al. [23] made use of the q-symmetric calculus and have defined certain subclasses of analytic and starlike functions with the conic domains. Furthermore, in [24], using a q-symmetric operator, generalization of the conic domain was given, and, with the help of these conic domains, certain subclasses of starlike and convex functions were defined. Very recently, Khan et al. [25] used the q-symmetric operator with q-Chebyshev polynomials and defined certain subclasses of analytic and bi-univalent functions. Motivated by these recent studies, in this article, we define a new subclass of multivalent starlike functions and discuss some of their interesting properties.
Here, we provide some basic concepts and definitions of the q-symmetric calculus. We suppose 0 < q < 1 and N 0 = 0 , 1 , 2 , 3 , .
In 1984, Biedenharn [26] defined a q-symmetric number [ n ] q :
[ n ] q = q n q n q 1 q , n N
and for n = 0 , then the q-symmetric number will be zero. For any n Z + 0 , the q-symmetric number shift factorial is defined by:
[ n ] q ! = [ n ] q [ n 1 ] q [ n 2 ] q [ 2 ] q [ 1 ] q , n 1 ,
and
[ 0 ] q ! = 1 , lim q 1 [ n ] q ! = n ! .
Definition 1.
For ζ A p , Kamel and Yosr [27] defined the q-symmetric derivative (q-difference) operator as follows:
D ˜ q ζ ( z ) = 1 z p ζ ( q z ) ζ ( q 1 z ) q q 1 , z U , = p q z p 1 + n = p + 1 [ n ] q a n z n 1 , z 0 , q 1 ,
Note that
lim q 1 D ˜ q ζ ( z ) = ζ ( z ) .
By making use of the q-symmetric derivative operator ζ A p , we now define new subclasses of q-starlike and q-convex functions of order α .
Definition 2.
A function ζ ( z ) A p is in the class S ˜ p α , q iff
Re z D ˜ q ζ ( z ) ζ ( z ) > α , ( z U ) ,
for some 0 α < p .
Definition 3.
A function ζ ( z ) A p is in the class K ˜ p ( α , q ) iff
Re D ˜ q z D ˜ q ζ ( z ) D ˜ q ζ ( z ) > α , ( z U ) ,
for some 0 α < p .
Remark 1.
Let ζ ( z ) A p , then it follows that
ζ ( z ) K ˜ p ( α , q ) z D ˜ q ζ ( z ) p q S ˜ p α , q
and
ζ ( z ) S ˜ p α , q 0 z p q ζ ( θ ) θ d q θ K ˜ p ( α , q ) .
Special cases: We can see that
S ˜ p q = S ˜ p 0 , q , S ˜ α , q = S ˜ 1 α , q , K ˜ p q = K ˜ p ( 0 , q ) and K ˜ ( α , q ) = K ˜ 1 ( α , q ) .
Remark 2.
By taking q 1 , then S ˜ p α , q = S p α and K ˜ p ( α , q ) = K p ( α ) as introduced by Mishra and Gochhayat in [28].
In [29], Nonaan and Thomas introduced the jth Hankel determinant:
H j ( n ) = a n a n + 1 a n + j 1 a n + 1 a n + 2 a n + j 2 a n + j 1 a n + j 2 a n + 2 j 2
where n N 0 ,   a 1 = 1 , and j N . For j = 2 and n = 1 , then the Hankel determinant Equation (3) represents a Fekete–Szegö functional H 2 ( 1 ) = a 3 a 2 2 and for j = 2 and n = 1 then Equation (3) represents the functional a 2 a 4 a 3 2 . For complex number μ then this functional is further generalized as a 3 μ a 2 2 , (see [30]). The third Hankel determinant H 3 ( 1 ) was studied by Babalola [31].
The symmetric Toeplitz determinant T j ( n ) is defined as follows:
T j ( n ) = a n a n + 1 a n + j 1 a n + 1 a n + j 1 a n ,
so that
T 2 ( 2 ) = a 2 a 3 a 3 a 2 , T 2 ( 3 ) = a 3 a 4 a 4 a 3 , T 3 ( 2 ) = a 2 a 3 a 4 a 3 a 2 a 3 a 4 a 3 a 2
and so on. The problem of finding the best possible bounds for a n + 1 a n has a long history (see [32]). A number of authors have studied the symmetric Toeplitz determinant T j ( n ) from different points of view (for details see [18,33]).
By replacing n = n + p 1 , into Equation (4) then T j ( n ) can be written as:
T j ( n + p 1 ) = a n + p 1 a n + p a n + p + j 2 a n + p a n + p + j 2 a n + p 1 ,
so that
T 2 ( p + 1 ) = a p + 1 a p + 2 a p + 2 a p + 1 , T 2 ( p + 2 ) = a p + 2 a p + 3 a p + 3 a p + 2 ,
and
T 3 ( p + 1 ) = a p + 1 a p + 2 a p + 3 a p + 2 a p + 1 a p + 2 a p + 3 a p + 2 a p + 1 .

2. A Set of Lemmas

In this section, we provide some new and known lemmas to prove our main results.
Lemma 1
([32]). If a function u ( z ) = 1 + n = 1 c n z n P , then
c n 2 , n 1 .
The inequality is sharp for
ζ ( z ) = 1 + z 1 z .
Lemma 2.
If a function u ( z ) = 1 + n = 1 c n z n P which satisfies Re u ( z ) α , for some α , 0 α < p , then
c n 2 p q α , n 1 .
The inequality is sharp for
u ( z ) = 1 + p q 2 α z 1 z = 1 + n = 1 2 p q α z n .
Remark 3.
The Lemma 2 is the generalization of lemmas which was introduced in [32,34].
Lemma 3
([35]). If a function u ( z ) = 1 + n = 1 c n z n P and satisfies Re ( u ( z ) ) > α , for some α ( 0 α < 1 ) , then
2 c 2 = c 1 2 + x ( 4 c 1 2 )
and
4 c 3 = c 1 3 + 2 ( 4 c 1 2 ) c 1 x ( 4 c 1 2 ) c 1 x 2 + 2 ( 4 c 1 2 ) ( 1 x 2 ) z ,
for some x , z C , with z 1 and x 1 .
By virtue of Lemma 3, we have
Lemma 4.
If u ( z ) = [ p ] q + n = 1 c n z n P satisfy Re ( u ( z ) ) > α , for some α ( 0 α < p ) , then
2 p q α c 2 = c 1 2 + 4 p q α 2 c 1 2 x
and
4 p q α 2 c 3 = c 1 3 + 2 4 p q α 2 c 1 2 c 1 x 4 p q α 2 c 1 2 c 1 x 2 + 2 p q α 4 p q α 2 c 1 2 ( 1 x 2 ) z ,
for some x , z C , with z 1 , and x 1 .
Proof. 
Let l ( z ) = u ( z ) α p q α = 1 + n = 1 c n p q α z n P , replacing c 2 and c 3 by c 2 p q α and c 3 p q α in Lemma 3, respectively, we immediately have the relations of Lemma 4. □
Remark 4.
For q 1 ,then the above Lemma 4, reduces to the lemma which was introduced by Hayami et al. [34] and p = 1 , q 1 ,then it reduces to the lemma introduced by Singh in [35].
Lemma 5
([36]). If u ( z ) = 1 + n = 1 c n z n satisfy Re ( u ( z ) ) > 0 , also let μ C , then
c n μ c k c n k 2 max 1 , 2 μ 1 , 1 k n k .

3. Main Results

Theorem 1.
A function ζ ( z ) A p is in the class S ˜ p α , q , then
a p + 1 2 ( p q α ) Υ 1 , a p + 2 2 ( p q α ) Υ 2 1 + 2 ( p q α ) Υ 1 , a p + 3 2 ( p q α ) Υ 3 1 + 2 ( p q α ) Λ 2 ρ 3 + 2 ( p q α ) ,
where Λ 2 , Υ 1 , Υ 2 , Υ 3 are given by Equations (10) and (12)–(14).
Proof. 
Let ζ S ˜ p ( α , q ) , then u ( z ) = p q + n = 1 c n z n such that
Re ( u ( z ) ) > α
and
z D ˜ q ζ ( z ) ζ ( z ) = u ( z ) .
Implies that
z D ˜ q ζ ( z ) = u ( z ) ζ ( z ) .
Therefore, we get
n q p q a n = l = p n 1 a l c n l ,
where n p + 1 , a p = 1 , c 0 = p q . From Equation (5), we have
a p + 1 = c 1 Υ 1 ,
a p + 2 = 1 Υ 2 c 2 + c 1 2 Υ 1 ,
a p + 3 = 1 Υ 3 c 3 + Λ 1 c 1 c 2 + Λ 2 c 1 3 ,
where
Λ 1 = Λ 2 ρ 3 ,
Λ 2 = 1 Υ 1 Υ 2 ,
ρ 3 = [ p + 1 ] q + [ p + 2 ] q 2 p q ,
and
Υ 1 = [ p + 1 ] q p q ,
Υ 2 = [ p + 2 ] ˜ q p q ,
Υ 3 = p + 3 q p q .
Now using the Lemma 2, we obtain our required result of Theorem 1. □
Theorem 2.
A function ζ ( z ) A p is in the class S ˜ p α , q , then
T 3 ( ( p + 1 ) Λ 3 Ω 4 + 4 p q α 2 Ω 5 + Ω 7 + Ω 8 1 2 p q α Ω 6 Ω 8 ,
where
Λ 3 = 4 p q α 2 Ω 1 + Ω 2 1 + Ω 3 , Ω 1 = 2 p q α Υ 1 ,
Ω 2 = 2 p q α Υ 3 ,
Ω 3 = 2 p q α Λ 2 ρ 3 + 2 p q α ,
Ω 4 = 1 Υ 1 2 , Ω 5 = 2 Λ 2 Λ 2 Λ 2 ρ 4 , Ω 6 = 4 Λ 2 Υ 2 Λ 2 ρ 3 ρ 4 ,
Ω 7 = 2 Υ 2 2 ,
Ω 8 = ρ 4 = 1 Υ 1 Υ 3 .
Proof. 
The simple calculation of T 3 ( p + 1 ) is in the following order.
T 3 ( p + 1 ) = a p + 1 a p + 3 a p + 1 2 2 a p + 2 2 + a p + 1 a p + 3 ,
where a p + 1 , a p + 2 , and a p + 3 is given by Equations (6)–(8).
Now if ζ S ˜ ( α , q ) , then
a p + 1 a p + 3 a p + 1 + a p + 3 , Ω 1 + Ω 2 1 + Ω 3 ,
where Ω 1 , Ω 2 , Ω 3 , is given by Equations (16)–(18).
We need to maximize a p + 1 2 2 a p + 2 2 + a p + 1 a p + 3 for ζ S ˜ ( α , q ) , now we write a p + 1 , a p + 2 , a p + 3 in terms of c 1 , c 2 , c 3 . With the help of Equations (6)–(8), we get
a p + 1 2 2 a p + 2 2 + a p + 1 a p + 3 Ω 4 c 1 2 Ω 5 c 1 4 Ω 6 c 1 2 c 2 Ω 7 c 2 2 + Ω 8 c 1 c 3 , Ω 4 c 1 2 + Ω 5 c 1 4 + Ω 7 c 2 2 + Ω 8 c 1 c 3 Ω 6 c 1 c 2 Ω 8 .
Using the Lemmas 2 and 5, along with Equations (21) and (22), we obtain the required result. □
Taking q 1 , α = 0 , and p = 1 , then we have the known corollary.
Corollary 1
([37]). A function ζ ( z ) A is in the class S ˜ , then
T 3 ( 2 ) 84 .
Theorem 3.
A function ζ ( z ) A p is in the class S ˜ p α , q , then
a p + 1 a p + 3 a p + 2 2 4 ( p q α ) 2 ( Υ 2 ) 2 ,
where Υ 2 is given by Equation (13).
Proof. 
Combining Equations (6)–(8), we have
a p + 1 a p + 3 a p + 2 2 = ρ 4 c 1 c 3 + Λ 2 ρ 3 B ˜ c 1 2 c 2 D ˜ c 2 2 + Λ 2 ρ 4 Λ 2 Λ 2 c 1 4 ,
where
D ˜ = 1 Υ 2 2 , B ˜ = 2 Λ 2 Υ 2 .
By using Lemma 3 we take
Υ = 4 p q α 2 c 1 2
and Z = 1 x 2 z . We assume that
c = c 1 , 0 c 2 p q α .
Therefore
a p + 1 a p + 3 a p + 2 2 = λ 1 c 4 + λ 2 Υ c 2 x λ 3 Υ c 2 x 2 λ 4 Υ 2 x 2 + λ 5 Υ c Z ,
where
λ 1 = ρ 4 4 ( p q α ) 2 + Λ 2 ρ 3 B ˜ 2 ( p q α ) D ˜ 4 ( p q α ) 2 D ˜ Λ 2 ρ 4 Λ 2 Λ 2 4 ( p q α ) 2 , λ 2 = ρ 4 2 ( p q α ) 2 + Λ 2 ρ 3 B ˜ 2 ( p q α ) D ˜ 2 ( p q α ) 2 , λ 3 = ρ 4 4 ( p q α ) 2 , λ 4 = D ˜ 4 ( p q α ) 2 , λ 5 = ρ 4 2 ( p q α ) .
Using the triangle inequality on Equation (23), we obtain
a p + 1 a p + 3 a p + 2 2 λ 1 c 4 + λ 2 Υ c 2 x + λ 3 Υ c 2 x 2 + λ 4 Υ 2 x 2 + λ 5 1 x 2 c Υ = G ( c , x ) .
Now, trivially, we have
G ( c , x ) > 0
on [ 0 , 1 ] , in equality Equation (24) shows that G ( c , x ) is an increasing function in an interval [ 0 , 1 ] . So the maximum value obtained at x = 1 and
M a x G ( c , 1 ) = G ( c ) ,
and
G ( c ) = λ 1 c 4 + λ 2 Υ c 2 + λ 3 Υ c 2 + λ 4 Υ 2 .
Hence, by putting Υ = 4 c 1 2 , and after some simplification, we have
G ( c ) = λ 1 λ 2 λ 3 + λ 4 c 4 + 4 λ 2 + λ 3 2 λ 4 c 2 + 16 λ 4 .
We consider G ( c ) = 0 , for the optimum value of G ( c ) , which implies that c = 0 .
Therefore, G ( c ) has a maximum value at c = 0 and the maximum value of G ( c ) is given by
16 λ 4 .
Which occurs at c = 0 or
c 2 = 4 λ 2 + λ 3 2 λ 4 λ 1 λ 2 λ 3 + λ 4 .
Hence, by putting λ 4 = D ˜ 4 ( p q α ) 2 and D ˜ = 1 Υ 2 2 in Equation (25), and after simplification, we obtain the desired result. □
Taking q 1 , p = 1 and α = 0 , we have the following known corollary.
Corollary 2
([30]). A function ζ ( z ) A is in the class S ˜ p then
a 2 a 4 a 3 2 1 .

3.1. Fekete–Szegö Problem

Theorem 4.
A function ζ ( z ) A p is in the class S ˜ p α , q , 0 α < p , then
a p + 2 μ a p + 1 2 2 p q α Υ 2 ρ 1 ρ 2 μ , i f μ ρ 5 , 2 p q α Υ 2 , i f ρ 5 μ ρ 6 , 2 p q α Υ 1 2 Υ 2 ρ 2 μ ρ 1 , i f μ ρ 6 ,
where
ρ 1 = 2 p q α Υ 1 + Υ 1 2 , ρ 2 = 2 p q α Υ 2 ,
ρ 5 = Υ 1 2 p q α + Υ 1 1 2 p q α Υ 2 , ρ 6 = Υ 1 p q α + Υ 1 p q α Υ 2 .
where Υ 1 and Υ 2 is given by Equations (12) and (13).
Proof. 
Using Equations (6) and (7) and supposing that
c 1 = c 0 c 2 ( p q α ) .
We derive
a p + 2 μ a p + 1 2 = 1 ρ 7 ρ 1 ρ 2 μ c 2 + Υ 1 2 4 p q α 2 c 2 ρ = A ( ρ ) ,
where
ρ 7 = 2 p q α Υ 1 2 Υ 2 .
Applying the triangle inequality, we deduce
A ( ρ ) 1 ρ 7 ρ 1 ρ 2 μ c 2 + Υ 1 2 4 p q α 2 c 2 = 1 ρ 7 2 p q α ρ 11 ρ 12 μ c 2 + ρ 9 , i f μ ρ 8 , 1 ρ 7 2 p q α Υ 2 μ ρ 10 c 2 + ρ 9 , i f μ ρ 8 .
where
ρ 8 = 2 p q α Υ 1 + Υ 1 2 2 p q α Υ 2 , ρ 8 = 2 p q α Υ 1 + Υ 1 2 2 p q α Υ 2 ,
ρ 10 = Υ 1 p q α + Υ 1 , ρ 11 = Υ 1 , ρ 12 = Υ 2 , ρ 13 = 2 p q α Υ 1 2 Υ 2 .
Now we have
a p + 2 μ a p + 1 2 2 p q α Υ 2 ρ 1 ρ 2 μ , if μ ρ 5 , c = 2 p q α , 2 p q α Υ 2 , if ρ 5 μ ρ 8 , c = 0 , 2 p q α Υ 2 , if ρ 8 μ ρ 6 , c = 0 , ρ 13 ρ 2 μ 2 p q α ρ 11 + ρ 11 2 , if μ ρ 6 , c = 2 p q α .
 □
Corollary 3
([34]). A function ζ ( z ) A p is in the class S ˜ p α , q 1 = S ˜ p α , 0 α < p , then
a p + 2 μ a p + 1 2 p α 2 p α + 1 4 p α μ , i f μ 1 2 , p α , i f 1 2 μ p α + 1 2 ( p α ) , p α 4 p α μ 2 p α + 1 , i f μ p α + 1 2 ( p α ) .

3.2. Applications

Definition 4
([38]). Let ζ A p , let the q-analogue of the Benardi integral operator for multivalent functions defined by
B p , β q ζ ( z ) = p + β q z β 0 z t β 1 ζ ( t ) d q t , z U , β > p , = z p + n = 1 p + β q [ n + β + p ] q a n + p z n + p ,
= z p + n = 1 B ˜ n + p a n + p z n + p .
Remark 5.
For q 1 , then B ˜ p , β q = B ˜ p , β , studied in [19].
Remark 6.
For p = 1 , then B ˜ p , β q = B ˜ β q , introduced in [39].
Remark 7.
If q 1 and p = 1 , then B ˜ p , β q = B ˜ β , studied in [40].
Theorem 5.
A function ζ ( z ) A p is in the class S ˜ p α , q , as ζ ( z ) S ˜ p α , q , , 0 α < p , and
B ˜ p , β q ζ ( z ) = z p + n = 1 B ˜ n + p a n + p z n + p ,
and B ˜ p , β q is given by Equation (26), then
a p + 1 2 ( [ p ] ˜ q α ) Υ 1 B ˜ p + 1 , a p + 2 2 ( [ p ] q α ) Υ 2 B ˜ p + 2 1 + 2 ( [ p ] q α ) Υ 1 B ˜ p + 1 , a p + 3 2 ( [ p ] q α ) Υ 3 B ˜ p + 3 1 + 2 ( [ p ] q α ) ρ 14 ρ 15 ,
where
ρ 14 = Υ 1 B ˜ p + 1 + Υ 2 B ˜ p + 2 + 2 ( [ p ] q α ) , ρ 15 = Υ 1 Υ 2 B ˜ p + 1 B ˜ p + 2 ,
and Υ 1 , Υ 2 and Υ 3 are given by Equations (12)–(14).
Proof. 
The proof follows easily using Equation (27) and Theorem 1. □
Theorem 6.
Let an analytic function ζ given by Equation (1) be in the class S ˜ p ( α , q ) , in addition B ˜ p , β q is the integral operator defined by Equation (26) and is of the form Equation (27), then
T 3 ( ( p + 1 ) Υ 3 Ω 4 B ˜ p + 1 2 + 4 [ p ] q α 2 Ω 10 + Ω 7 B ˜ p + 2 2 + Ω 8 B ˜ p + 1 B ˜ p + 3 1 2 [ p ] q α B ˜ p + 3 B ˜ p + 1 Ω 11 Ω 8 ,
where
Υ 3 = 4 [ p ] q α 2 Ω 1 B ˜ p + 1 + Ω 2 B ˜ p + 3 1 + Ω 9 , Ω 9 = Λ p ρ 14 B ˜ p + 1 B ˜ p + 2 , Ω 10 = Λ 4 Λ 5 , Ω 11 = Λ 6 Λ 7 ,
Λ 4 = 2 Λ 2 Λ 2 B ˜ p + 1 2 B ˜ p + 2 2 , Λ 5 = Λ 2 ρ 4 B ˜ p + 1 2 B ˜ p + 2 B ˜ p + 3 , Λ 6 = 4 Λ 2 [ p + 2 ] q [ p ] q B ˜ p + 1 B ˜ p + 2 2 , Λ 7 = Λ 8 Λ 2 ρ 4 B ˜ p + 1 2 B ˜ p + 2 B ˜ p + 3 , Λ 8 = Υ 1 B ˜ p + 1 + Υ 2 B ˜ p + 2 .
Proof. 
The proof follows easily using Equation (27) and Theorem 2. □
Theorem 7.
A function ζ A p be in the class S ˜ p ( α , q ) , and B ˜ p , β q is defined by Equation (26) then
a p + 1 a p + 3 a p + 2 2 4 ( [ p ] q α ) 2 Υ 2 2 B ˜ p + 2 .
Theorem 8.
A function ζ A p be in the class S ˜ p ( α , q ) , and B ˜ p , β q is defined by Equation (26) then
a p + 2 μ a p + 1 2 2 [ p ] q α Υ 2 B ˜ p + 2 ρ 16 ρ 2 B ˜ p + 2 μ , i f μ ρ 17 , 2 [ p ] q α Υ 2 B ˜ p + 2 , i f ρ 17 μ ρ 18 , 2 Λ 2 [ p ] q α Υ 1 B ˜ p + 1 2 B ˜ p + 2 ρ 2 B ˜ p + 2 μ ρ 16 , i f μ ρ 18 ,
where
ρ 16 = 2 [ p ] q α Υ 1 B ˜ p + 1 + Υ 1 2 B ˜ p + 1 2 , ρ 17 = Υ 1 B ˜ p + 1 2 [ p ] q α + Υ 1 B ˜ p + 1 1 2 [ p ] q α Υ 2 B ˜ p + 2 , ρ 18 = Υ 1 B ˜ p + 1 [ p ] q α + Υ 1 B ˜ p + 1 [ p ] q α Υ 2 B ˜ p + 2 ,
and Λ 2 is given by Equation (10).

4. Conclusions

In this paper, we discussed the idea of a q-symmetric derivative operator to multivalent functions and then defined a new subclass of multivalent q-starlike functions. For this class, we investigated some useful properties of multivalent functions, with respect to the Hankel determinant, the symmetric Toeplitz matrices, and the Fekete–Szego problem. We used a q-Bernardi integral operator, along with a q-symmetric derivative operator for multivalent functions, and discussed some applications of our main results. Using the same approach, we applied a q-symmetric derivative operator to define a new subclass of meromorphic q-convex functions K ˜ p ( α , q ) of order α in U and investigated the results for K ˜ p ( α , q ) , respectively.

Author Contributions

M.F.K.: Conceptualization, formal analysis, resources, software, supervision, writing original draft, funding acquisition, preparation and writing review and editing; A.G.: Methodology, resources, software, validation and writing original draft preparation; S.K.: Conceptualization, formal analysis, supervision, visualization and project administration. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Not applicable.

Acknowledgments

We would like to express our sincere thanks to the editor and reviewers for their consideration of the paper, and for their careful reading and helpful remarks for improvement of the paper.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Jackson, F.H. On q-functions and a certain difference operator. Trans. R. Soc. Edinb. 1908, 46, 253–281. [Google Scholar] [CrossRef]
  2. Jackson, F.H. On q-definite integrals. Pure Appl. Math. Q. 1910, 41, 193–203. [Google Scholar]
  3. Ismail, M.E.H.; Merkes, E.; Styer, D. A generalization of starlike functions. Complex Var. Theory Appl. 1990, 14, 77–84. [Google Scholar] [CrossRef]
  4. Srivastava, H.M. Univalent functions, fractional calculus, and associated generalized hypergeometric functions. In Univalent Functions, Fractional Calculus, and Their Applications; Srivastava, H.M., Owa, S., Eds.; Halsted Press: Chichester, UK; John Wiley and Sons: New York, NY, USA, 1989; pp. 329–354. [Google Scholar]
  5. Arif, M.; Haq, M.U.; Lin, J.-L. A subfamily of univalent functions associated with q-analogue of Noor integral operator. J. Funct. Spaces 2018, 2018, 3818915. [Google Scholar] [CrossRef] [Green Version]
  6. Kanas, S.; Raducanu, D. Some class of analytic functions related to conic domains. Math. Slovaca 2014, 64, 1183–1196. [Google Scholar] [CrossRef]
  7. Rehman, M.S.; Ahmad, Q.Z.; Srivastava, H.M.; Khan, N.; Darus, M.; Khan, B. Applications of higher-order q-derivatives to the subclass of q-starlike functions associated with the Janowski functions. AIMS Math. 2020, 6, 1110–1125. [Google Scholar] [CrossRef]
  8. Shi, L.; Ahmad, B.; Khan, N.; Khan, M.G.; Araci, S.; Mashwani, W.K.; Khan, B. Coefficient estimates for a subclass of meromorphic multivalent q-close-to-convex functions. Symmetry 2021, 13, 1840. [Google Scholar] [CrossRef]
  9. Khan, B.; Srivastava, H.M.; Arjika, S.; Khan, S.; Khan, N.; Ahmad, Q.Z. A certain q-Ruscheweyh type derivative operator and its applications involving multivalent functions. Adv. Differ. 2021, 2021, 279. [Google Scholar] [CrossRef]
  10. Arif, M.; Barkub, O.; Srivastava, H.M.; Abdullah, S.; Khan, S.A. Some Janowski type harmonic q-starlike functions associated with symmetrical points. Mathematics 2020, 8, 629. [Google Scholar] [CrossRef]
  11. Arif, M.; Srivastava, H.M.; Uma, S. Some applications of a q-analogue of the Ruscheweyh type operator for multivalent functions. RACSAM 2019, 113, 1211–1221. [Google Scholar] [CrossRef]
  12. Zhang, X.; Khan, S.; Hussain, S.; Tang, H.; Shareef, Z. New subclass of q-starlike functions associated with generalized conic domain. AIMS Math. 2020, 5, 4830–4848. [Google Scholar] [CrossRef]
  13. Ahmad, Q.Z.; Khan, N.; Raza, M.; Tahir, M.; Khan, B. Certain q-difference operators and their applications to the subclass of meromorphic q-starlike functions. Filomat 2019, 33, 3385–3397. [Google Scholar] [CrossRef]
  14. Hu, Q.; Srivastava, H.M.; Ahmad, B.; Khan, N.; Khan, M.G.; Mashwani, W.K.; Khan, B. A subclass of multivalent Janowski type q-starlike functions and its consequences. Symmetry 2021, 13, 1275. [Google Scholar] [CrossRef]
  15. Mahmood, S.; Raza, M.; AbuJarad, E.S.; Srivastava, G.; Srivastava, H.M.; Malik, S.N. Geometric properties of certain classes of analytic functions associated with a q-integral operator. Symmetry 2019, 11, 719. [Google Scholar] [CrossRef] [Green Version]
  16. Rehman, M.S.; Ahmad, Q.Z.; Srivastava, H.M.; Khan, B.; Khan, N. Partial sums of generalized q-Mittag-Leffler functions. AIMS Math. 2019, 5, 408–420. [Google Scholar] [CrossRef]
  17. Srivastava, H.M. Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis. Iran. J. Sci. Technol. Trans. A Sci. 2020, 44, 327–344. [Google Scholar] [CrossRef]
  18. Srivastava, H.M.; Khan, N.; Darus, M.; Khan, S.; Ahmad, Q.Z.; Hussain, S. Fekete-Szegö type problems and their applications for a subclass of q-starlike functions with respect to symmetrical points. Mathematics 2020, 8, 842. [Google Scholar] [CrossRef]
  19. Wang, Z.G.; Raza, M.; Ayaz, M.; Arif, M. On certain multivalent functions involving the generalized Srivastava-Attiya operator. J. Nonlinear Sci. Appl. 2016, 9, 6067–6076. [Google Scholar] [CrossRef] [Green Version]
  20. Lavagno, A. Basic-deformed quantum mechanics. Rep. Math. Phys. 2009, 64, 79–88. [Google Scholar] [CrossRef] [Green Version]
  21. Cruz, A.M.D.; Martins, N. The q-symmetric variational calculus. Comput. Math. Appl. 2012, 64, 2241–2250. [Google Scholar] [CrossRef] [Green Version]
  22. Kanas, S.; Altinkaya, S.; Yalcin, S. Subclass of k uniformly starlike functions defined by symmetric q-derivative operator. Ukr. Math. 2019, 70, 1727–1740. [Google Scholar] [CrossRef] [Green Version]
  23. Khan, S.; Hussain, S.; Naeem, M.; Darus, M.; Rasheed, A. A subclass of q-starlike functions defined by using a symmetric q-derivative operator and related with generalized symmetric conic domains. Mathematics 2021, 9, 917. [Google Scholar] [CrossRef]
  24. Khan, S.; Khan, N.; Hussain, A.; Araci, S.; Khan, B.; Al-Sulami, H.H. Applications of symmetric conic domains to a subclass of q-starlike functions. Symmetry 2022, 14, 803. [Google Scholar] [CrossRef]
  25. Khan, B.; Liu, Z.G.; Shaba, T.G.; Araci, S.; Khan, N.; Khan, M.G. Applications of q-derivative operator to the subclass of bi-univalent functions involving q-chebyshev polynomials. J. Math. 2022, 2022, 8162182. [Google Scholar] [CrossRef]
  26. Biedenharn, L.C. The quantum group SUq(2) and a q-analogue of the boson operators. J. Phys. A 1984, 22, 873–878. [Google Scholar] [CrossRef]
  27. Kamel, B.; Yosr, S. On some symmetric q-special functions. Le Mat. 2013, 68, 107–122. [Google Scholar]
  28. Mishra, A.K.; Gochhayat, P. Invariance of some subclass of multivalent functions under a differintegral operator. Complex Var. Elliptic Equ. 2010, V55, 677–689. [Google Scholar] [CrossRef]
  29. Noonan, J.W.; Thomas, D.K. On the second Hankel derminant of areally mean p-valent functions. Trans. Am. Math. Soc. 1976, 233, 337–346. [Google Scholar]
  30. Janteng, A.; Halim, A.S.; Darus, M. Hankel determinant for starlike and convex functions. Int. J. Math. Anal. 2007, 2007, 619–625. [Google Scholar]
  31. Babalola, K.O. On H3(1) Hankel determinant for some classes of univalent functions. Inequal. Theory Appl. 2007, 6, 1–7. [Google Scholar]
  32. Duren, P.L. Univalent functions. Grundlehren der Mathematischen Wissenschaften (Band 259); Springer: New York, NY, USA, 1983. [Google Scholar]
  33. Hussain, S.; Khan, S.; Roqia, G.; Darus, M. Hankel Determinant for certain classes of analytic functions. J. Comput. Theor. Nanosci. 2016, 13, 9105–9110. [Google Scholar] [CrossRef]
  34. Hayami, T.; Owa, S. Hankel determinant for p-valently starlike and convex functions of order α. Gen. Math. 2009, 17, 29–44. [Google Scholar]
  35. Singh, G.; Singh, G. On the second Hankel determinant for a new subclass of analytic functions. J. Math. Sci. Appl. 2014, 2, 1–3. [Google Scholar]
  36. Efraimidis, I. A generalization of Livingstons coefficient inequalities for functions with positive real part. J. Math. Anal. Appl. 2016, 435, 369–379. [Google Scholar] [CrossRef]
  37. Ali, M.F.; Thomas, D.K.; Vasudevarao, A. Toeplitz determinants whose element are the coefficients of univalent functions. Bull. Aust. Math. Soc. 2018, 97, 253–264. [Google Scholar] [CrossRef]
  38. Tang, H.; Khan, S.; Hussain, S.; Khan, N. Hankel and Toeplitz determinant for a subclass of multivalent q-starlike functions of order. AIMS Math. 2021, 6, 5421–5439. [Google Scholar] [CrossRef]
  39. Noor, K.I.; Riaz, S.; Noor, M.A. On q-Bernardi integral operator. TWMS J. Pure Appl. Math. 2017, 8, 3–11. [Google Scholar]
  40. Bernardi, S.D. Convex and starlike univalent functions. Trans. Am. Math. Soc. 1969, 135, 429–446. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Share and Cite

MDPI and ACS Style

Khan, M.F.; Goswami, A.; Khan, S. Certain New Subclass of Multivalent Q-Starlike Functions Associated with Q-Symmetric Calculus. Fractal Fract. 2022, 6, 367. https://doi.org/10.3390/fractalfract6070367

AMA Style

Khan MF, Goswami A, Khan S. Certain New Subclass of Multivalent Q-Starlike Functions Associated with Q-Symmetric Calculus. Fractal and Fractional. 2022; 6(7):367. https://doi.org/10.3390/fractalfract6070367

Chicago/Turabian Style

Khan, Mohammad Faisal, Anjali Goswami, and Shahid Khan. 2022. "Certain New Subclass of Multivalent Q-Starlike Functions Associated with Q-Symmetric Calculus" Fractal and Fractional 6, no. 7: 367. https://doi.org/10.3390/fractalfract6070367

APA Style

Khan, M. F., Goswami, A., & Khan, S. (2022). Certain New Subclass of Multivalent Q-Starlike Functions Associated with Q-Symmetric Calculus. Fractal and Fractional, 6(7), 367. https://doi.org/10.3390/fractalfract6070367

Article Metrics

Back to TopTop