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Article

Coefficient Bounds for Certain Subclasses of q-Starlike Functions

1
School of Mathematics and Computing Science, Hunan First Normal University, Changsha 410205, Hunan, China
2
Department of Mathematics, Riphah International University, Islamabad 44000, Pakistan
3
Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Abbottabad 22060, Pakistan
4
Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(10), 969; https://doi.org/10.3390/math7100969
Submission received: 7 September 2019 / Revised: 30 September 2019 / Accepted: 1 October 2019 / Published: 14 October 2019

Abstract

:
By making use of q-calculus, we define and investigate several new subclasses of bi-univalent mappings related to the q-Noor integral operator. The coefficient bounds | u 2 | , | u 3 | and the Fekete–Szegő problem u 3 μ u 2 2 for mappings belonging to these classes are derived.

1. Introduction and Preliminaries

Let χ : = v C : v < 1 , and A be the class of mappings l which are analytic in χ and normalized by the conditions l 0 = l 0 1 = 0 . Thus, every l from A has the following series representation:
l ( v ) = v + n = 2 u n v n .
A mapping l A is said to be in the class S if l is univalent in χ . It is well-known that every univalent mapping l A has an inverse l 1 which is defined by
l 1 ( l ( v ) ) = v v χ ,
and
l ( l 1 ( ϖ ) ) = ϖ ϖ < r 0 ( l ) ; r 0 ( l ) 1 4 .
Indeed, the inverse mapping l 1 is specified by
m ( ϖ ) = l 1 ( ϖ ) = ϖ u 2 ϖ 2 + ( 2 u 2 2 u 3 ) ϖ 3 ( 5 u 2 3 5 u 2 u 3 + u 4 ) ϖ 4 + .
A mapping l A is bi-univalent in χ if both l and l 1 are univalent in χ . Let Σ denote the class of bi-univalent mappings in χ specified by the Taylor–Maclaurin series expansion (1). Many researchers calculated the second coefficient and obtained the values of coefficients for different subclasses of bi-univalent mappings. Levin [1] proved that u 2 < 1.51 . Branan et al. [2] showed that u 2 2 , and Netanyahu [3] proved that max u 2 = 4 / 3 . Also, Srivastava et al. [4], Frasin and Aouf [5] estimated the coefficients | u 2 | and u 3 for certain subclasses of bi-univalent mapping. For more recent investigations on bi-univalent mappings, one can refer to [1,6,7,8,9,10,11,12,13].
Let l and m be analytic in χ . Then l is subordinate to m, if there exists a Schwarz mapping s, which is analytic in χ with s ( 0 ) = 0 and | s ( v ) | < 1 for v χ such that l ( v ) = m ( s ( v ) ) . Furthermore, if the mapping m is univalent in χ , then we obtain the following relationship (see [14,15]):
l ( v ) m ( v ) l ( 0 ) m ( 0 ) = 0 and l ( χ ) m ( χ ) .
For two analytic mappings
l ( v ) = v + n = 2 u n v n and m ( v ) = v + n = 2 b n v n ,
the convolution of l and m is defined by
l ( v ) m ( v ) = v + n = 2 u n b n v n .
We now recall several essential definitions involving q ¯ -calculus which will be utilized in the present paper. Throughout this paper, we will consider q ¯ to be a fixed number in ( 0 , 1 ) .
Definition 1
([16]). For ϱ N and q ¯ 0 , 1 , the number [ ϱ ] is defined by
[ ϱ ] = 1 q ¯ ϱ 1 q ¯ [ 0 ] = 0 , k = 0 n 1 q ¯ k = 1 + q ¯ + q ¯ 2 + + q ¯ n 1 .
Definition 2.
For ϱ N , the q ¯ -number shifted factorial is defined by
[ ϱ ] ! = [ 1 ] [ 2 ] [ 3 ] [ ϱ ] [ 0 ] ! = 1 .
We observe that lim q ¯ 1 [ ϱ ] ! = ϱ ! .
Definition 3
([17,18]). The q ¯ -derivative operator for l A is described as:
q ¯ l ( v ) = l ( q ¯ v ) l ( v ) ( q ¯ 1 ) v ( v χ ) .
For n N and v χ , we see that
q ¯ v n = [ n ] v n 1 , q ¯ n = 1 u n v n = n = 1 [ n ] u n v n 1 .
Definition 4.
The q ¯ -generalized Pochhammer symbol for ϱ R and n N is defined by
[ ϱ ] n = [ ϱ ] [ ϱ + 1 ] [ ϱ + 2 ] [ ϱ + ( n 1 ) ] ,
and for ϱ > 0 , q ¯ -gamma mapping is described as:
Γ q ¯ ( ϱ + 1 ) = [ ϱ ] Γ q ¯ ( ϱ ) and Γ q ¯ ( 1 ) = 1 .
Definition 5
([12]). Let l A and λ > 1 . The q ¯ -Noor integral operator is defined by
F q ¯ , λ + 1 ( v ) 1 F q ¯ , λ + 1 ( v ) = v q ¯ l ( v ) .
Now, we define the operator I q ¯ λ as follows:
I q ¯ λ l ( v ) = l ( v ) F q ¯ , λ + 1 ( v ) 1 ( v χ ; λ > 1 ) ,
where
F q ¯ , λ + 1 ( v ) 1 = v + n = 2 [ n ] ! Γ q ¯ ( 1 + λ ) Γ q ¯ ( n + λ ) v n .
Thus, we have
I q ¯ λ l ( v ) = v + n = 2 [ n ] ! Γ q ¯ ( 1 + λ ) Γ q ¯ ( n + λ ) u n v n = v + n = 2 n ! [ λ + 1 ] n 1 u n v n .
Clearly,
I q ¯ 0 l ( v ) = v q ¯ l ( v ) and I q ¯ 1 l ( v ) = l ( v ) .
Note that when q ¯ 1 , q ¯ -Noor integral operator reduces to Noor integral operator, and the following identity holds:
v q ¯ I q ¯ λ + 1 l ( v ) = 1 + [ λ ] q ¯ λ I q ¯ λ l ( v ) [ λ ] q ¯ λ I q ¯ λ + 1 l ( v ) .
If q ¯ 1 , the equality (5) implies that
v I λ + 1 l ( v ) = 1 + λ I λ l ( v ) λ I λ + 1 l ( v ) ,
which is the well-known recurrent formula for the Noor integral operator.
Furthermore, a mapping l A is called starlike of order α ( 0 α < 1 ) , if
v l v l v > α .
We use the notation S α for the class of starlike mappings of order α . In particular, when α = 0 , the class S : = S 0 denotes the familiar class of starlike mappings.
One way to generalize the class S α is to replace the derivative in (6) by the q ¯ -difference operator q ¯ and replace the right-half plane ϖ : ( ϖ ) > α with a suitable domain. The appropriate definition turns out to be the following:
Definition 6
([19]). A mapping l A is said to be in the class S q ¯ α if
v q ¯ l v l v α 1 α 1 1 q ¯ < 1 1 q ¯ ( 0 α < 1 ) .
Observe that as q ¯ 1 , the disk
ϖ 1 q ¯ 1 < 1 q ¯ 1
becomes the right-half plane and the class S q ¯ α reduces to S α . In particular, when α = 0 , the class S q ¯ α concides with the class S q ¯ : = S q ¯ 0 , which is proposed by Ismail et al. [19] (see also Srivastava et al. [20]), it was shown that the relationship in (7) is equivalent to
v q ¯ l v l v α 1 α 1 + v 1 q ¯ v .
Now, by using the q ¯ -Noor integral operator, we define the following function classes.
Definition 7.
For l Σ and 0 α < 1 , l belongs to the bi-univalent class N Σ ( α , q ¯ ) if and only if
v q ¯ l ( v ) l ( v ) α 1 α 1 + v 1 q ¯ v ,
and
ϖ q ¯ m ( ϖ ) m ( ϖ ) α 1 α 1 + ϖ 1 q ¯ ϖ ,
where m ( ϖ ) = l 1 ( ϖ ) is defined by (2).
Definition 8.
For l Σ , 0 α < 1 and λ > 1 , l belongs to the bi-univalent class N Σ ( α , λ , q ¯ ) if and only if
v q ¯ I q ¯ λ l ( v ) I q ¯ λ l ( v ) α 1 α 1 + v 1 q ¯ v ,
and
ϖ q ¯ I q ¯ λ m ( ϖ ) I q ¯ λ m ( ϖ ) α 1 α 1 + ϖ 1 q ¯ ϖ ,
where m ( ϖ ) = l 1 ( ϖ ) is defined by (2).
Example 1.
The mappings
v 1 v a n d log ( 1 v )
are examples of the above defined subclasses of bi-univalent mappings.
Lemma 1
([21]). If p ( v ) = 1 + p 1 v + p 2 v 2 + p 3 v 3 + is an analytic mapping in χ with positive real parts, then
p 2 p 1 2 2 2 p 1 2 2 .

2. Main Results

Firstly, we derive the following result.
Theorem 1.
For 0 α < 1 , if l N Σ ( α , q ¯ ) , then
u 2 ( 1 + q ¯ ) ( 1 α ) 2 2 q ¯ ( 1 + q ¯ ) ( 1 α ) [ 2 ] 1 q ¯ 2 ,
u 3 ( 1 α ) ( 1 + q ¯ ) [ 2 ] q ¯ 2 q ¯ + [ 2 ] ( 1 + q ¯ ) ( 1 α ) ,
and
u 3 2 u 2 2 ( 1 α ) ( 1 + q ¯ ) 2 [ 2 ] q ¯ 2 1 + q ¯ α .
Proof. 
For the mapping l N Σ ( α , q ¯ ) of the form (1), we know that
v q ¯ l ( v ) l ( v ) α 1 α = 1 + q ¯ 1 α u 2 v + q ¯ 1 α [ 2 ] u 3 u 2 2 v 2 + ,
and for its inverse map m = l 1 , we have
ϖ q ¯ m ( ϖ ) m ( ϖ ) α 1 α = 1 q ¯ 1 α u 2 ϖ + q ¯ 1 α [ 2 ] 2 u 2 2 u 3 u 2 2 ϖ 2 .
Since both mappings l and its inverse map m = l 1 are in N Σ ( α , q ¯ ) , by the definition of subordination, there exist two Schwarz mappings p ( v ) = n = 1 c n v n , and h ( ϖ ) = n = 1 d n ϖ n , where v, ϖ χ . It follows from (14) and (15) that
v q ¯ l ( v ) l ( v ) α 1 α = 1 + ( p ( v ) ) 1 q ¯ ( p ( v ) ) = 1 + ( 1 + q ¯ ) c 1 v + ( 1 + q ¯ ) c 2 + q ¯ c 1 2 v 2 + ,
and
ϖ q ¯ m ( ϖ ) m ( ϖ ) α 1 α = 1 + ( h ( ϖ ) ) 1 q ¯ ( h ( ϖ ) ) = 1 + ( 1 + q ¯ ) d 1 ϖ + ( 1 + q ¯ ) d 2 + q ¯ d 1 2 ϖ 2 + .
For the coefficients of the Schwarz mappings p ( v ) and h ( ϖ ) , we find from [22] that c n 1 1 and d n 1 1 . By equating the corresponding coefficients of (14) and (16), we get
q ¯ 1 α u 2 = ( 1 + q ¯ ) c 1 ,
and
q ¯ 1 α [ 2 ] u 3 u 2 2 = ( 1 + q ¯ ) c 2 + q ¯ c 1 2 .
Similarly, by equating the corresponding coefficients of (15) and (17), we have
q ¯ 1 α u 2 = ( 1 + q ¯ ) d 1 ,
and
q ¯ 1 α [ 2 ] 2 u 2 2 u 3 u 2 2 = ( 1 + q ¯ ) d 2 + q ¯ d 1 2 .
From (18) and (20), we obtain
c 1 = d 1 ,
and
2 q ¯ 2 ( 1 α ) 2 u 2 2 = ( 1 + q ¯ ) 2 c 1 2 + d 1 2 .
By means of (19) and (21), we know that
2 [ 2 ] q ¯ 1 α 2 q ¯ 1 α u 2 2 = ( 1 + q ¯ ) ( c 2 + d 2 ) + q ¯ ( 1 + q ¯ ) c 1 2 + d 1 2 .
By using (22) and (23) in (24), we have
u 2 ( 1 + q ¯ ) ( 1 α ) 2 2 q ¯ ( 1 + q ¯ ) ( 1 α ) [ 2 ] 1 q ¯ 2 .
Now, in view of (19), (21) and (22), we see that
2 2 q ¯ 1 α u 3 = ( 1 + q ¯ ) ( d 2 c 2 ) + 2 [ 2 ] q ¯ 1 α u 2 2 .
From (23) and (25), we know that
u 3 ( 1 α ) ( 1 + q ¯ ) [ 2 ] q ¯ 2 q ¯ + [ 2 ] ( 1 + q ¯ ) ( 1 α ) .
By means of (21), we get
q ¯ 1 α [ 2 ] 2 u 2 2 u 3 = ( 1 + q ¯ ) d 2 + q ¯ d 1 2 + q ¯ 1 α u 2 2 .
It follows from (20) and (26) that
2 u 2 2 u 3 = ( 1 α ) ( 1 + q ¯ ) [ 2 ] q ¯ 2 q ¯ d 2 + q ¯ 2 + ( 1 α ) ( 1 + q ¯ ) d 1 2 .
Thus, by virtue of c n 1 1 and d n 1 1 , we deduce that
u 3 2 u 2 2 ( 1 α ) ( 1 + q ¯ ) 2 [ 2 ] q ¯ 2 1 + q ¯ α .
Theorem 2.
For 0 α < 1 and λ > 1 , if l N Σ ( α , λ , q ¯ ) , then
u 2 [ λ + 1 ] 1 ( 1 + q ¯ ) ( 1 α ) 2 [ λ + 1 ] 2 T ,
where
T = q ¯ ( 1 + q ¯ ) ( 1 α ) 2 2 [ 3 ] ! [ λ + 1 ] 1 2 2 [ 2 ] ! 2 [ λ + 1 ] 2 Z
with
Z = 2 [ 2 ] ! 2 q ¯ 2 ,
u 3 ( 1 + q ¯ ) ( 1 α ) [ λ + 1 ] 2 2 [ 3 ] ! q ¯ 2 q ¯ + [ 2 ] [ 3 ] ! [ λ + 1 ] 1 2 ( 1 + q ¯ ) ( 1 α ) [ 2 ] ! 2 [ λ + 1 ] 2 ,
and
u 3 2 u 2 2 [ λ + 1 ] 2 ( 1 α ) ( 1 + q ¯ ) 2 [ 2 ] [ 3 ] ! q ¯ 2 1 + q ¯ α .
Proof. 
For the mapping l N Σ ( α , λ , q ¯ ) of the form (1), we have
v q ¯ I q ¯ λ l ( v ) I q ¯ λ l ( v ) α 1 α = 1 + 2 ! q ¯ 1 α [ λ + 1 ] 1 u 2 v + q ¯ 1 α [ 2 ] 3 ! [ λ + 1 ] 2 u 3 2 ! [ λ + 1 ] 1 2 u 2 2 v 2 + ,
and for its inverse map m = l 1 , we get
ϖ q ¯ I q ¯ λ m ( ϖ ) I q ¯ λ m ( ϖ ) α 1 α = 1 2 ! q ¯ 1 α [ λ + 1 ] 1 u 2 ϖ + q ¯ 1 α [ 2 ] 3 ! [ λ + 1 ] 2 2 u 2 2 u 3 2 ! [ λ + 1 ] 1 2 u 2 2 ϖ 2 + .
By noting that both mappings l and its inverse map m = l 1 are in the class N Σ ( α , λ , q ¯ ) , by the definition of subordination, there exist two Schwarz mappings p ( v ) = n = 1 c n v n and h ( ϖ ) = n = 1 d n ϖ n , where v, ϖ χ , it follows that
v q ¯ I q ¯ λ l ( v ) I q ¯ λ l ( v ) α 1 α = 1 + ( p ( v ) ) 1 q ¯ ( p ( v ) ) = 1 + ( 1 + q ¯ ) c 1 v + ( 1 + q ¯ ) c 2 + q ¯ c 1 2 v 2 + ,
and
ϖ q ¯ I q ¯ λ m ( ϖ ) I q ¯ λ m ( ϖ ) α 1 α = 1 + ( h ( ϖ ) ) 1 q ¯ ( h ( ϖ ) ) = 1 + ( 1 + q ¯ ) d 1 ϖ + ( 1 + q ¯ ) d 2 + q ¯ d 1 2 ϖ 2 + .
By observing that the coefficients of the Schwarz mappings p ( v ) and h ( ϖ ) satisfy the conditions c n 1 1 and d n 1 1 , and by equating the corresponding coefficients of (29) and (31), we see that
2 ! q ¯ 1 α [ λ + 1 ] 1 u 2 = ( 1 + q ¯ ) c 1 ,
and
q ¯ 1 α [ 2 ] 3 ! [ λ + 1 ] 2 u 3 2 ! [ λ + 1 ] 1 2 u 2 2 = ( 1 + q ¯ ) c 2 + q ¯ c 1 2 ,
Similarly, by comparing the corresponding coefficients of (30) and (32), we get
2 ! q ¯ 1 α [ λ + 1 ] 1 u 2 = ( 1 + q ¯ ) d 1 ,
and
q ¯ 1 α [ 2 ] 3 ! [ λ + 1 ] 2 2 u 2 2 u 3 2 ! [ λ + 1 ] 1 2 u 2 2 = ( 1 + q ¯ ) d 2 + q ¯ d 1 2 .
From (33) and (35), we obtain
c 1 = d 1 ,
and
2 2 ! q ¯ ( 1 α ) [ λ + 1 ] 1 2 u 2 2 = ( 1 + q ¯ ) 2 c 1 2 + d 1 2 .
It follows from (34) and (36) that
2 q ¯ 1 α 2 [ 3 ] ! [ λ + 1 ] 2 2 ! [ λ + 1 ] 1 2 u 2 2 = ( 1 + q ¯ ) ( c 2 + d 2 ) + q ¯ ( 1 + q ¯ ) c 1 2 + d 1 2 .
In view of (37), (38) and (39), we get the required estimate (27).
Now, by means of (34), (36) and (37), we obtain
2 q ¯ 1 α [ 2 ] 3 ! [ λ + 1 ] 2 u 3 = ( 1 + q ¯ ) ( d 2 c 2 ) + 2 [ 2 ] 3 ! q ¯ ( 1 α ) [ λ + 1 ] 2 u 2 2 .
By virtue of (38), (40), c n 1 1 and d n 1 1 , we know that
u 3 ( 1 + q ¯ ) ( 1 α ) [ λ + 1 ] 2 2 [ 3 ] ! q ¯ 2 q ¯ + [ 2 ] [ 3 ] ! [ λ + 1 ] 1 2 ( 1 + q ¯ ) ( 1 α ) [ 2 ] ! 2 [ λ + 1 ] 2 .
From (36), we get
q ¯ 1 α 2 [ 2 ] [ 3 ] ! [ λ + 1 ] 2 u 2 2 [ 2 ] [ 3 ] ! [ λ + 1 ] 2 u 3 = ( 1 + q ¯ ) d 2 + q ¯ d 1 2 + q ¯ 1 α [ 2 ] ! [ λ + 1 ] 1 2 u 2 2 .
By using (35) on right hand side of Equation (42), we see that
2 u 2 2 u 3 = ( 1 α ) ( 1 + q ¯ ) [ λ + 1 ] 2 [ 2 ] [ 3 ] ! q ¯ 2 q ¯ d 2 + q ¯ 2 + ( 1 α ) ( 1 + q ¯ ) d 1 2 .
By means of (43), c n 1 1 and d n 1 1 , we conclude that
u 3 2 u 2 2 [ λ + 1 ] 2 ( 1 α ) ( 1 + q ¯ ) 2 [ 2 ] [ 3 ] ! q ¯ 2 1 + q ¯ α .
Theorem 3.
For 0 α < 1 , if l N Σ ( α , q ¯ ) and μ R , then
u 3 μ u 2 2 ( 1 + q ¯ ) ( 1 α ) 2 q ¯ 0 h 1 μ 1 2 2 q ¯ , 4 ( 1 α ) ( 1 + q ¯ ) h 1 μ h 1 μ 1 2 2 q ¯ ,
where
h 1 μ = [ 2 ] 2 μ ( 1 + q ¯ ) ( 1 α ) q ¯ ( 1 + q ¯ ) ( 1 α ) [ 2 ] 2 2 [ 1 ] 2 q ¯ 2 .
Proof. 
Let the mapping l given by (1) be in the class N Σ ( α , q ¯ ) . From (23) and (24), we know that
u 2 2 = ( 1 + q ¯ ) 2 ( 1 α ) 2 ( c 2 + d 2 ) q ¯ ( 1 + q ¯ ) ( 1 α ) [ 2 ] 2 2 [ 1 ] 2 q ¯ 2 .
Also, by means of (25), we obtain
u 3 = ( 1 + q ¯ ) ( 1 α ) ( d 2 c 2 ) 2 2 q ¯ + [ 2 ] 2 u 2 2 .
It follows from (44) and (45) that
u 3 μ u 2 2 = ( 1 + q ¯ ) ( 1 α ) h 1 μ + 1 2 2 q ¯ d 2 + h 1 μ 1 2 2 q ¯ c 2 ,
where
h 1 μ = [ 2 ] 2 μ ( 1 + q ¯ ) ( 1 α ) q ¯ ( 1 + q ¯ ) ( 1 α ) [ 2 ] 2 2 [ 1 ] 2 q ¯ 2 .
Thus, in view of (12), we deduce that
u 3 μ u 2 2 ( 1 + q ¯ ) ( 1 α ) 2 q ¯ 0 h 1 μ 1 2 2 q ¯ , 4 ( 1 α ) ( 1 + q ¯ ) h 1 μ h 1 μ 1 2 2 q ¯ .
Theorem 4.
For 0 α < 1 and λ > 1 , if l N Σ ( α , λ , q ¯ ) and μ R , then
u 3 μ u 2 2 ( 1 + q ¯ ) ( 1 α ) [ λ + 2 ] 2 2 [ 3 ] ! q ¯ 0 h 2 μ [ λ + 2 ] 2 2 2 [ 3 ] ! q ¯ , 4 ( 1 α ) ( 1 + q ¯ ) h 2 μ h 2 μ [ λ + 2 ] 2 2 2 [ 3 ] ! q ¯ ,
where
h 2 μ = [ 2 ] 2 2 [ 3 ] ! q ¯ μ T 1
with
T 1 = [ λ + 2 ] 2 [ λ + 1 ] 1 2 ( 1 + q ¯ ) ( 1 α ) q ¯ ( 1 + q ¯ ) ( 1 α ) [ λ + 1 ] 1 2 [ 2 ] 2 [ 3 ] ! 2 [ 1 ] [ 2 ] ! 2 [ λ + 2 ] 2 2 q ¯ 2 .
Proof. 
Let the mapping l given by (1) be in the class N Σ ( α , λ , q ¯ ) . In view of (38) and (39), we know that
u 2 2 = [ λ + 2 ] 2 [ λ + 2 ] 1 2 ( 1 + q ¯ ) 2 ( 1 α ) 2 ( c 2 + d 2 ) q ¯ ( 1 + q ¯ ) ( 1 α ) [ λ + 2 ] 1 2 [ 2 ] 2 [ 3 ] ! 2 [ 1 ] [ 2 ] ! 2 [ λ + 2 ] 2 2 q ¯ 2 [ 2 ] ! 2 ,
and from (40), we have
u 3 = ( 1 + q ¯ ) ( 1 α ) [ λ + 2 ] 2 ( d 2 c 2 ) 2 2 [ 3 ] ! q ¯ + [ 2 ] 2 u 2 2 .
It follows from (48) and (49) that
u 3 μ u 2 2 = ( 1 + q ¯ ) ( 1 α ) h 2 μ + [ λ + 2 ] 2 2 2 [ 3 ] ! q ¯ d 2 + h 2 μ [ λ + 2 ] 2 2 2 [ 3 ] ! q ¯ c 2 ,
where
h 2 μ = [ 2 ] 2 μ T 1
with T 1 given by (47). Thus, by virtue of (12) and (50), we conclude that the desired inequality holds. □

3. Conclusions

The main purpose of this article is to achieve several attentiveness and constructive manipulations of q ¯ -calculus in Geometric Function Theory. By utilizing the q ¯ -calculus theory, we initiate and examine a new subclass of bi-univalent functions in open unit disk χ , and we find the second and the third Taylor–Maclaurin coefficients of mappings as well as find the Fekete–Szegő problem u 3 μ u 2 2 in this function class. These results are improvements on the estimates obtained in the recent studies. In particular, we discuss the applications of q ¯ -calculus by using the q ¯ -Noor integral operator.

Author Contributions

Formal analysis, Z.-G.W., S.K., S.H. and T.M.; Investigation, L.-L.F. and M.N. All authors worked jointly on the results, and they read and approved the final manuscript.

Funding

The present investigation was supported by the Natural Science Foundation of Hunan Province under Grant no. 2016JJ2036 of P. R. China.

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions, which was essential to improve the quality of this paper.

Conflicts of Interest

The authors declare no conflict of interest.

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MDPI and ACS Style

Fan, L.-L.; Wang, Z.-G.; Khan, S.; Hussain, S.; Naeem, M.; Mahmood, T. Coefficient Bounds for Certain Subclasses of q-Starlike Functions. Mathematics 2019, 7, 969. https://doi.org/10.3390/math7100969

AMA Style

Fan L-L, Wang Z-G, Khan S, Hussain S, Naeem M, Mahmood T. Coefficient Bounds for Certain Subclasses of q-Starlike Functions. Mathematics. 2019; 7(10):969. https://doi.org/10.3390/math7100969

Chicago/Turabian Style

Fan, Lin-Lin, Zhi-Gang Wang, Shahid Khan, Saqib Hussain, Muhammad Naeem, and Tahir Mahmood. 2019. "Coefficient Bounds for Certain Subclasses of q-Starlike Functions" Mathematics 7, no. 10: 969. https://doi.org/10.3390/math7100969

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