Citation for this paper:
Srivastava, H.M., Zahoor, Q., Khan, N., Khan, N. & Khan, B. (2019). Hankel and
Toeplitz Determinants for a Subclass of q-Starlike Functions Associated with a
General Conic Domain. Mathematics, 7(2), 181.
https://doi.org/10.3390/math7020181
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Hankel and Toeplitz Determinants for a Subclass of q-Starlike Functions Associated
with a General Conic Domain
Hari M. Srivastava, Qazi Zahoor Ahmad, Nasir Khan, Nazar Khan and Bilal Khan
February 2019
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open
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Attribution (CC BY) license (
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).
This article was originally published at:
http://dx.doi.org/10.3390/math7020181
Article
Hankel and Toeplitz Determinants for a Subclass of
q-Starlike Functions Associated with a General
Conic Domain
Hari M. Srivastava1,2,* , Qazi Zahoor Ahmad3, Nasir Khan4, Nazar Khan3and Bilal Khan3 1 Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada 2 Department of Medical Research, China Medical University Hospital, China Medical University,
Taichung 40402, Taiwan
3 Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan;
zahoorqazi5@gmail.com (Q.Z.A.); nazarmaths@gmail.com (N.K.); bilalmaths789@gmail.com (B.K.)
4 Department of Mathematics, FATA University, Akhorwal (Darra Adam Khel), FR Kohat 26000, Pakistan;
dr.nasirkhan@fu.edu.pk
* Correspondence: harimsri@math.uvic.ca
Received: 29 January 2019; Accepted: 12 February 2019; Published: 15 February 2019
Abstract: By using a certain general conic domain as well as the quantum (or q-) calculus, here we define and investigate a new subclass of normalized analytic and starlike functions in the open unit diskU. In particular, we find the Hankel determinant and the Toeplitz matrices for this newly-defined
class of analytic q-starlike functions. We also highlight some known consequences of our main results.
Keywords: analytic functions; starlike and q-starlike functions; q-derivative operator; q-hypergeometric functions; conic and generalized conic domains; Hankel determinant; Toeplitz matrices
MSC:Primary 05A30, 30C45; Secondary 11B65, 47B38
1. Introduction and Definitions
Let the class of functions, which are analytic in the open unit disk
U = {z : z∈ C and |z| <1},
be denoted byL (U). Also letAdenote the class of all functions f , which are analytic in the open unit diskUand normalized by
f(0) =0 and f0(0) =1.
Then, clearly, each f ∈ Ahas a Taylor–Maclaurin series representation as follows: f(z) =z+
∞
∑
n=2
anzn (z∈ U). (1)
Suppose thatS is the subclass of the analytic function classA, which consists of all functions which are also univalent inU.
A function f ∈ Ais said to be starlike inUif it satisfies the following inequality: < z f
0(z)
f(z)
>0 (z∈ U).
We denote byS∗the class of all such starlike functions inU.
For two functions f and g, analytic inU, we say that the function f is subordinate to the function
g and write this subordination as follows:
f ≺g or f(z) ≺g(z), if there exists a Schwarz function w which is analytic inU, with
w(0) =0 and |w(z)| <1, such that
f(z) =g w(z).
In the case when the function g is univalent inU, then we have the following equivalence (see, for
example, [1]; see also [2]):
f(z) ≺g(z) (z∈ U) ⇐⇒ f(0) =g(0) and f(U) ⊂g(U). Next, for a function f ∈ Agiven by (1) and another function g∈ Agiven by
g(z) =z+
∞
∑
n=2
bnzn (z∈ U),
the convolution (or the Hadamard product) of f and g is defined here by
(f∗g) (z):=z+
∞
∑
n=2
anbnzn=:(g∗f) (z). (2)
LetPdenote the well-known Carathéodory class of functions p, analytic in the open unit diskU,
which are normalized by
p(z) =1+ ∞
∑
n=1 cnzn, (3) such that < p(z) >0 (z∈ U).Following the works of Kanas et al. (see [3,4]; see also [5]), we introduce the conic domainΩk
(k=0)as follows: Ωk= u+iv : u>k q (u−1)2+v2 . (4)
In fact, subjected to the conic domainΩk(k=0), Kanas and Wi´sniowska (see [3,4]; see also [6])
studied the corresponding class k-S T of k-starlike functions inU(see Definition1below). For fixed
k,Ωkrepresents the conic region bounded successively by the imaginary axis(k=0), by a parabola
(k=1), by the right branch of a hyperbola(0<k<1), and by an ellipse(k>1). For these conic regions, the following functions play the role of extremal functions.
pk(z) = 1+z 1−z =1+2z+2z 2+ · · · (k=0) 1+ 2 π2 log 1+ √ z 1−√z 2 (k=1) 1+ 2 1−k2 sinh 2 2 π arccos k arctan h√z (05k<1) 1+ 1 k2−1 " 1+sin π 2K(κ) Z u√(z) κ 0 dt p (1−t2)(1−κ2t2) !# (k>1), (5) where u(z) = z− √ κ 1−√κz (z∈ U), and κ∈ (0, 1)is so chosen that
k=cosh πK0(κ) 4K(κ) .
Here K(κ)is Legendre’s complete elliptic integral of first kind and
K0(κ) =K
p 1−κ2
,
that is, K0(κ)is the complementary integral of K(κ)(see, for example, ([7], p. 326, Equation 9.4 (209))).
Indeed, from (5), we have
pk(z) =1+p1z+p2z2+p3z3+ · · ·. (6)
The class k-S T is defined as follows.
Definition 1. A function f ∈ Ais said to be in the class k-S T if and only if z f0(z)
f(z) ≺pk(z) (∀z∈ U; k=0).
We now recall some basic definitions and concept details of the q-calculus which will be used in this paper (see, for example, ([7], p. 346 et seq.)). Throughout the paper, unless otherwise mentioned, we suppose that 0<q<1 and
N = {1, 2, 3· · · } = N0\ {0} (N0:={0, 1, 2,· · · }).
Definition 2. Let q∈ (0, 1)and define the q-number[λ]qby
[λ]q = 1−qλ 1−q (λ∈ C) n−1 ∑ k=0 qk=1+q+q2+ · · · +qn−1 (λ=n∈ N).
Definition 3. Let q∈ (0, 1)and define the q-factorial[n]q! by
[n]q!= 1 (n=0) n ∏ k=1 [k]q (n∈ N).
Definition 4(see [8,9]). The q-derivative (or q-difference) operator Dq of a function f defined, in a given subset ofC, by Dqf(z) = f(z) − f(qz) (1−q)z (z6=0) f0(0) (z=0), (7)
provided that f0(0)exists.
From Definition4, we can observe that lim q→1− Dqf (z) = lim q→1− f(z) − f(qz) (1−q)z = f 0 (z)
for a differentiable function f in a given subset ofC. It is also known from (1) and (7) that
Dqf(z) =1+ ∞
∑
n=2
[n]qanzn−1. (8)
Definition 5. The q-Pochhammer symbol[ξ]n,q(ξ∈ C; n∈ N0)is defined as follows:
[ξ]n,q= q ξ; q n (1−q)n = 1 (n=0) [ξ]q[ξ+1]q[ξ+2]q· · · [ξ+n−1]q (n∈ N).
Moreover, the q-gamma function is defined by the following recurrence relation: Γq(z+1) = [z]qΓq(z) and Γq(1) =1.
Definition 6(see [10]). For f ∈ A, let the q-Ruscheweyh derivative operatorRλ
q be defined, in terms of the
Hadamard product(or convolution)given by (2), as follows:
Rλ qf(z) = f(z) ∗ Fq,λ+1(z) (z∈ U; λ> −1), where Fq,λ+1(z) =z+ ∞
∑
n=2 Γq(λ+n) [n−1]q!Γq(λ+1)z n =z+∑
∞ n=2 [λ+1]q,n−1 [n−1]q! z n.We next define a certain q-integral operator by using the same technique as that used by Noor [11].
Definition 7. For f ∈ A, let the q-integral operatorFq,λbe defined by
Fq,λ+1−1 (z) ∗ Fq,λ+1(z) =z Dqf(z). Then Iλ q f(z) = f(z) ∗ Fq,λ+1−1 (z) =z+ ∞
∑
n=2 ψn−1anzn (z∈ U; λ> −1), (9) where Fq,λ+1−1 (z) =z+ ∞∑
n=2 ψn−1znand ψn−1= [n]q!Γq(λ+1) Γq(λ+n) = [n]q! [λ+1]q,n−1. Clearly, we have I0 qf(z) =z Dqf(z) and Iq1f(z) = f(z).
We note also that, in the limit case when q→1−, the q-integral operatorFq,λgiven by Definition7
would reduce to the integral operator which was studied by Noor [11]. The following identity can be easily verified:
zDq Iλ+1 q f(z) = 1+[λ]q qλ ! Iλ q f(z) − [λ]q qλ I λ+1 q f(z). (10)
When q→1−, this last identity in (10) implies that
zIλ+1f(z)0= (1+λ) Iλf(z) −λIλ+1f(z),
which is the well-known recurrence relation for the above-mentioned integral operator which was studied by Noor [11].
In geometric function theory, several subclasses belonging to the class of normalized analytic functions classAhave already been investigated in different aspects. The above-defined q-calculus gives valuable tools that have been extensively used in order to investigate several subclasses of
A. Ismail et al. [12] were the first who used the q-derivative operator Dq to study the q-calculus
analogous of the classS∗of starlike functions in
U(see Definition8below). However, a firm footing
of the q-calculus in the context of geometric function theory was presented mainly and basic (or q-) hypergeometric functions were first used in geometric function theory in a book chapter by Srivastava (see, for details, ([13], p. 347 et seq.); see also [14]).
Definition 8(see [12]). A function f ∈ Ais said to belong to the classS∗ q if f(0) = f0(0) −1=0 (11) and z f(z) Dqf z− 1 1−q 5 1 1−q. (12)
It is readily observed that, as q→1−, the closed disk: w − 1 1−q 5 1 1−q
becomes the right-half plane and the classSq∗of q-starlike functions reduces to the familiar classS∗
of normalized starlike functions inUwith respect to the origin(z=0). Equivalently, by using the
principle of subordination between analytic functions, we can rewrite the conditions in (11) and (12) as follows (see [15]): z f(z) Dqf (z) ≺bp(z) b p(z) = 1+z 1−qz . (13)
The notationSq∗was used by Sahoo and Sharma [16].
Now, making use of the principle of subordination between analytic functions and the above-mentioned q-calculus, we present the following definition.
Definition 9. A function p is said to be in the class k-Pqif and only if
p(z) ≺ 2pk(z)
(1+q) + (1−q)pk(z)
, where pk(z)is defined by (5).
Geometrically, the function p(z) ∈k-Pqtakes on all values from the domainΩk,q (k=0)which
is defined as follows: Ωk,q = w :< (1+q)w (q−1)w+2 >k (1+q)w (q−1)w+2−1 . The domainΩk,qrepresents a generalized conic region.
It can be seen that
lim
q→1−Ωk,q=Ωk,
whereΩkis the conic domain considered by Kanas and Wi´sniowska [3]. Below, we give some basic
facts about the class k-Pq.
Remark 1. First of all, we see that
k-Pq ⊆ P 2k 2k+1+q , wherePh 2k 2k+1+q i
is the well-known class of functions with real part greater than 2k+1+q2k . Secondly, we have lim
q→1−k-Pq= P (pk),
whereP (pk)is the well-known function class introduced by Kanas and Wi´sniowska [3]. Thirdly, we have
lim
q→1−0-Pq = P,
wherePis the well-known class of analytic functions with positive real part.
Definition 10. A function f is said to be in the classS T (k, λ, q)if and only if
zDqIqλf (z) f(z) ∈k-Pq (k=0; λ=0), or, equivalently, < (1+q)z(DqI λ qf)(z) f (z) (q−1)z(Dqf (z)Iqλf)(z)+2 >k (1+q)z(DqI λ qf)(z) f (z) (q−1)z(Dqf (z)Iqλf)(z)+2 −1 .
Remark 2. First of all, it is easily seen that
S T (0, 1, q) = Sq∗,
whereSq∗is the function class introduced and studied by Ismail et al. [12]. Secondly, we have lim
where k-S T is a function class introduced and studied by Kanas and Wi´sniowska [4]. Finally, we have lim
q→1−S T (0, 1, q) = S ∗,
whereS∗is the well-known class of starlike functions in
Uwith respect to the origin(z=0).
Remark 3. Further studies of the new q-starlike function classS T (k, λ, q), as well as of its more consequences, can next be determined and investigated in future papers.
Let n∈ N0and j∈ N. The following jth Hankel determinant was considered by Noonan and
Thomas [17]: Hj(n) = an an+1 . . . an+j−1 an+1 . . . . . . . . . . . an+j−1 . . . . an+2(j−1) ,
where a1=1. In fact, this determinant has been studied by several authors, and sharp upper bounds on
H2(2)were obtained by several authors (see [18–20]) for various classes of functions. It is well-known
that the Fekete–Szegö functional a3−a22
can be represented in terms of the Hankel determinant asH2(1). This functional has been further generalized as a 3−µa22 for some real or complex µ. Fekete and Szegö gave sharp estimates of
a3−µa22 for µ real and f ∈ S, the class of normalized univalent functions inU. It is also known that the functional a2a4−a23
is equivalent to H2(2) (see [18]). Babalola [21] studied the Hankel determinantH3(1)for some subclasses of normalized
analytic functions inU. The symmetric Toeplitz determinantTj(n)is defined by
Tj(n) = an an+1 . . . an+j−1 an+1 . . . . . . . . . . . an+j−1 . . . . an , so that T2(2) = a2 a3 a3 a2 , T2(3) = a3 a4 a4 a3 , T3(2) = a2 a3 a4 a3 a2 a3 a4 a3 a2 , and so on.
For f ∈ S, the problem of finding the best possible bounds for||an+1| − |an||has a long history
(see, for details, [22]). It is a known fact from [22] that
|an+1| − |an| <c
for a constant c. However, the problem of finding exact values of the constant c forS and its various subclasses has proved to be difficult. In a very recent investigation, Thomas and Abdul-Halim [23] succeeded in obtaining some sharp estimates forTj(n)for the first few values of n and j involving
symmetric Toeplitz determinants whose entries are the coefficients an of starlike and
In the present investigation, our focus is on the Hankel determinant and the Toeplitz matrices for the function classS T (k, λ, q)given by Definition10.
2. A Set of Lemmas
In order to prove our main results in this paper, we need each of the following lemmas.
Lemma 1(see [20]). If the function p(z)given by (3) is in the Carathéodory classPof analytic functions with positive real part inU, then
2c2=c21+x 4−c21 and 4c3=c31+2 4−c21c1x−c1 4−c21x2+24−c21 1− x 2 z for some x, z∈ Cwith|x| 51 and|z| 51.
Lemma 2(see [24]). Let the function p(z)given by (3) be in the Carathéodory classPof analytic functions with positive real part inU. Also let µ∈ C. Then
|cn−µckcn−k| 52 max(1,|2µ−1|) (15k5n−1).
Lemma 3(see [22]). Let the function p(z)given by (3) be in the Carathéodory classPof analytic functions with positive real part inU. Then
|cn| 52 (n∈ N).
This last inequality is sharp.
3. Main Results
Throughout this section, unless otherwise mentioned, we suppose that q∈ (0, 1), λ> −1 and k∈ [0, 1].
Theorem 1. If the function f(z)given by (1) belongs to the classS T (k, λ, q), where k∈ [0, 1], then
|a2| 5 (1+q)p1 2qψ1 , a35 1 2qψ2 p1+ p2−p1+ q2+1 p2 1 2q ! and a45 (1+q) 4(q+q2+q3)ψ 3 2p1+4 p2−p1+ 2+q2 p2 1 4q + 2p3+2p1−4p2− 2 1+q2 −q p2 1 q + 4q2−3q+2 q p1p2 + q 2+2q−1 2q2 p 3 1 ! , (14)
where pj(j=1, 2, 3)are positive and are the coefficients of the functions pk(z)defined by (6). Each of the above
results is sharp for the function g(z)given by
g(z) = 2pk(z)
(1+q) + (1−q)pk(z)
Proof. Let f(z) ∈ S T (k, λ, q). Then, we have z Dqf(z) f(z) =q(z) ≺Sk(z), (15) where Sk(z) = 2pk(z) (1+q) + (1−q)pk(z),
and the functions pk(z)are defined by (6).
We now define the function p(z)with p(0) =1 and with a positive real part inUas follows:
p(z) =1+S −1 k q(z) 1−S−1k q(z) =1+c1z+c2z 2+ · · ·. (16)
After some simple computation involving (16), we get q(z) =Sk
p(z) +1 p(z) −1
. We thus find that
Sk p(z) +1 p(z) −1 =1+ q+1 2 " p1c1 2 z+ ( p1c2 2 + p2 4 − p1 4 + (q−1)p2 1 8 !! c21 ) z2 + ( p1c3 2 + p2 2 − p1 2 + (q−1)p21 4 !! c1c2 + p1 8 − p2 4 − (q−1)p21 8 + p3 8 − (q−1)p1p2 8 + (q−1)2p31 32 ! c31 ) z3 # + · · ·. (17)
Now, upon expanding the left-hand side of (15), we have zDqIqλf (z) f(z) =1+qψ1a2z+ n q+q2ψ2a3−qψ12a22 o z2 +nq+q2+q3ψ3a4− 2q+q2ψ1ψ2a2a3+qψ13a32 o z3+ · · ·. (18)
Finally, by comparing the corresponding coefficients in (17) and (18) along with Lemma3, we obtain the result asserted by Theorem1.
Theorem 2. If the function f(z)given by (1) belongs to the classS T (k, λ, q), then
T3(2)5 1 +q 2qψ1 p21+ 1+q 4(q+q2+q3)ψ 3 [Ω1+Ω2] · " 4 (1+q) 2 16q2ψ2 1 ! p21+16|Ω3| + p21 4q2ψ2 2 +2Ω5p21 2− Ω4 Ω5p21 # ,
where Ω1=2p1+4 p2−p1+ 2+q2 4q p 2 1 , Ω2= 2p3+2p1−4p2− 21+q2−qp21 + 4q 2−3q+2 q p1p2+ q2+q+1 2q2 p 3 1 , Ω3= 1 2q2ψ2 2 p2 4 − p1 4 + q2+1 p2 1 8q !2 −Ω5· " p3 4 + p1 4 − p2 2 −2 1+q 2 −q p2 1 8q + 4q2−3q+2 8q p1p2+ q2+2q−1 16q2 p31 # , Ω4= p1 2q2ψ2 2 p2 4 − p1 4 + q2+1 p2 1 8q ! −Ω5p1 p2−p1+ 2+q2 p2 1 4q ! , Ω5= (1+q)2 16q2(1+q+q2)ψ1ψ3
and pj(j=1, 2)are positive and are the coefficients of the functions pk(z)defined by (6).
Proof. Upon comparing the corresponding coefficients in (17) and (18), we find that
a2= (1+q)p1c1 4qψ1 , (19) a3= 1 2qψ2 " p1c2 2 + p2 4 − p1 4 + q2+1 p2 1 8q ! c21 # , (20) a4= (1+q) 4(q+q2+q3)ψ 3 " p1c3+ p2−p1+ 2+q2 p2 1 4q ! c1c2 + p3 4 + p1 4 − p2 2 − 2 1+q2 −q p2 1 8q + 4q2−3q+2 8q p1p2 + q 2+2q−1 16q2 p 3 1 ! c31 # . (21)
By a simple computation,T3(2)can be written as follows:
T3(2) = (a2−a4)
a22−2a23+a2a4
. Now, if f ∈ S T (k, λ, q), then it is clearly seen that
|a2−a4| 5 |a2| + |a4| 5 1+q 2qψ1 p21+ 1+q 4(q+q2+q3)ψ 3 (Ω1+Ω2).
We need to maximize a22−2a23+a2a4
for a function f ∈ S T (k, λ, q). So, by writing a2, a3, and a4in terms of c1, c2, and c3, with the help of (19)–(21), we get
a 2 2−2a23+a2a4 = (1+q)2 16q2ψ2 1 ! p21c21−Ω3c41−Ω4c21c2− p21 8q2ψ2 2 c22+Ω5p21c1c3 . (22)
Finally, by applying the trigonometric inequalities, Lemmas2and3along with(22), we obtain the result asserted by Theorem2.
As an application of Theorem2, we first set ψn−1=1 and k=0 and then let q→1−. We thus
arrive at the following known result.
Corollary 1(see [25]). If the function f(z)given by(1)belongs to the classS∗, then
T3(2)584.
Theorem 3. If the function f(z)given by (1) belongs to the classS T (k, λ, q), then a2a4−a 2 3 5 1 4q2ψ2 2 p21, (23)
where k∈ [0, 1]and pj(j=1, 2, 3)are positive and are the coefficients of the functions pk(z)defined by (6).
Proof. Making use of (19)–(21), we find that
a2a4−a23= A(q) 16q2ψ 1ψ3 p21c1c3+ A (q)ψ22−ψ1ψ3 16q2ψ 1ψ22ψ3 p1p2− A (q)ψ22−ψ1ψ3 16q2ψ 1ψ22ψ3 p21 +A(q) 2+q 2 ψ22−2 1+q2ψ1ψ3 64q2ψ 1ψ3 p31 ! c21c2+ 1 16q2ψ2 2 p21c22 + " A(q) 64q2ψ 1ψ3 p1p3+ A(q)ψ22−ψ1ψ3 64q2ψ 1ψ22ψ3 ! p21+ ψ1ψ3−A(q)ψ 2 2 32q2ψ 1ψ22ψ3 ! p1p2 + 2 1+q 2 ψ1ψ3− 2 1+q2−q A(q)ψ22 128q3ψ1ψ2 2ψ3 ! p31 + A(q) 4q 2−3q+2 ψ22−2 1+q2ψ1ψ3 128q3ψ 1ψ22ψ3 ! p21p2 + A(q) q 2+2q−1 ψ22− 1+q22ψ1ψ3 256q4ψ1ψ2 2ψ3 ! p41− 1 64q2ψ2 2 p22 # c41, (24) where A(q) = (1+q) 2 1+q+q2.
We substitute the values of c2and c3from the above Lemma and, for simplicity, take Y=4−c21
and Z= (1− |x|2)z. Without loss of generality, we assume that c=c
1(05c52), so that a2a4−a23= " q(1−q)A(q)ψ22 128q2ψ 1ψ3 p31+ A(q) 64q2ψ 1ψ3 p1p3 + A(q) 4q 2−3q+2 ψ22−2 1+q2ψ1ψ3 128q3ψ 1ψ22ψ3 ! p21p2 + A(q) q 2+2q−1 ψ22− 1+q22ψ1ψ3 256q4ψ 1ψ22ψ3 ! p41− 1 64q2ψ2 2 p22 # c4 + " A(q)ψ22−ψ1ψ3 32q2ψ 1ψ22ψ3 p1p2 + A(q) 2+q 2 ψ22−2 1+q2ψ1ψ3 128q2ψ 1ψ3 p31 # c2xY · " − A(q) 64q2ψ 1ψ3 p 2 1c2Yx2− 1 64q2ψ2 2 p21x2Y2+ A(q) 32q2ψ 1ψ3 p 2 1cYZ # . (25)
Upon setting Z= (1− |x|2)z and taking the moduli in (25) and using trigonometric inequality,
we find that a2a4−a 2 3 5 |λ1|c 4+| λ2| |x|Yc2+ A(q) 64q2ψ 1ψ3 p21Y|x|2c2 + 1 64q2ψ2 2 p21|x|2Y2+ A(q) 32q2ψ 1ψ3 p 2 1c2Y 1− |x|2 =Λ(c,|x|), (26) where λ1= q (1−q)A(q)ψ22 128q2ψ 1ψ3 p31+ A(q) 64q2ψ 1ψ3 p1p3 + A(q) 4q 2−3q+2 ψ22−2 1+q2ψ1ψ3 128q3ψ1ψ2 2ψ3 ! p21p2 + A(q) q 2+2q−1 ψ22− 1+q22ψ1ψ3 256q4ψ 1ψ22ψ3 ! p41− 1 64q2ψ2 2 p22 λ2= A(q)ψ22−ψ1ψ3 32q2ψ 1ψ22ψ3 ; p1p2 + A(q) 2+q 2 ψ22−2 1+q2ψ1ψ3 128q2ψ 1ψ3 p 3 1.
Now, trivially, we have
Λ0(|
x|) >0 on[0, 1], and so
Λ(|x|) 5Λ(1).
Hence, by puting Y=4−c21and after some simplification, we have a2a4−a 2 3 = |λ1| − |λ2| + ψ1ψ3−A(q)ψ22 64q2ψ 1ψ3 p21 ! c4 + 4|λ2| + A (q)ψ22−ψ1ψ3 16q2ψ 1ψ3 p21 !! c2+ 1 4q2ψ2 2 p21 =G(c). (27)
For optimum value of G(c), we consider G0(c) = 0, which implies that c = 0. So G(c)has a maximum value at c=0. We therefore conclude that the maximum value of G(c)is given by
1 4q2ψ2 2 p21, which occurs at c=0 or c2= − 128|λ2|q 2ψ 1ψ3+4A(q)ψ22−2ψ1ψ3p21 64q2(|λ1| − |λ2|)ψ1ψ3+ψ1ψ3−A(q)ψ2 2p21 .
This completes the proof of Theorem3.
If we put ψn−1=1 and let q→1−in Theorem3, we have the following known result.
Corollary 2(see [26]). If the function f(z)given by (1) belongs to the class k-S T, where k∈ [0, 1], then
a2a4−a 2 3 5 p21 4 . If we put p1=2 and ψn−1=1,
by letting q→1−in Theorem3, we have the following known result.
Corollary 3(see [18]). If f ∈ S∗, then a2a4−a 2 3 51. By letting k=1, ψn−1=1, q→1−and p1= 8 π2, p2= 16 3π2 and p3= 184 45π2
in Theorem3, we have the following known result.
Corollary 4(see [27]). If the function f(z)given by (1) belong to the classS P, then a2a4−a 2 3 5 16 π4.
4. Concluding Remarks and Observations
Motivated significantly by a number of recent works, we have made use of a certain general conic domain and the quantum (or q-) calculus in order to define and investigate a new subclass of normalized analytic functions in the open unit diskU, which we have referred to as q-starlike functions.
For this q-starlike function class, we have successfully derived several properties and characteristics. In particular, we have found the Hankel determinant and the Toeplitz matrices for this newly-defined class of q-starlike functions. We also highlight some known consequences of our main results which are stated and proved as theorems and corollaries.
Author Contributions: conceptualization, Q.Z.A. and N.K. (Nazar Khan); methodology, N.K. (Nasir Khan); software, B.K.; validation, H.M.S.; formal analysis, H.M.S.; writing—original draft preparation, H.M.S.; writing—review and editing, H.M.S.; supervision, H.M.S.
Funding:This research received no external funding.
References
1. Miller, S.S.; Mocanu, P.T. Differential subordination and univalent functions. Mich. Math. J. 1981, 28, 157–171. [CrossRef]
2. Miller, S.S.; Mocanu, P.T. Differential Subordination: Theory and Applications; Series on Monographs and Textbooks in Pure and Applied Mathematics, No. 225; Marcel Dekker Incorporated: New York, NY, USA; Basel, Switzerland, 2000.
3. Kanas, S.; Wi´sniowska, A. Conic regions and k-uniform Convexity. J. Comput. Appl. Math. 1999, 105, 327–336. [CrossRef]
4. Kanas, S.; Wi´sniowska, A. Conic domains and starlike Functions. Rev. Roum. Math. Pures Appl. 2000, 45, 647–657.
5. Kanas, S.; Srivastava, H.M. Linear operators associated with k-uniformly convex functions. Integral Transform. Spec. Funct. 2000, 9, 121–132. [CrossRef]
6. Mahmood, S.; Jabeen, M.; Malik, S.N.; Srivastava, H.M.; Manzoor, R.; Riaz, S.M.J. Some coefficient inequalities of q-starlike functions associated with conic domain defined by q-derivative. J. Funct. Spaces
2018, 2018, 8492072. [CrossRef]
7. Srivastava, H.M.; Karlsson, P.W. Multiple Gaussian Hypergeometric Series; Ellis Horwood Limited: Chichester, UK, 1985.
8. Jackson, F.H. On q-definite integrals. Quart. J. Pure Appl. Math. 1910, 41, 193–203. 9. Jackson, F.H. q-difference equations. Am. J. Math. 1910, 32, 305–314. [CrossRef]
10. Kanas, S.; R˘aducanu, D. Some class of analytic functions related to conic domains. Math. Slov. 2014, 64, 1183–1196. [CrossRef]
11. Noor, K.I. On new classes of integral operators. J. Nat. Geom. 1999, 16, 71–80.
12. Ismail, M.E.H.; Merkes, E.; Styer, D. A generalization of starlike functions. Complex Var. Theory Appl. 1990, 14, 77–84. [CrossRef]
13. 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.; Ellis Horwood Limited: Chichester, UK, 1989; pp. 329–354.
14. Srivastava, H.M.; Bansal, D. Close-to-convexity of a certain family of q-Mittag-Leffer functions. J. Nonlinear Var. Anal. 2017, 1, 61–69.
15. Uçar, H.E.Ö. Coefficient inequality for q-starlike Functions. Appl. Math. Comput. 2016, 76, 122–126.
16. Sahoo, S.K.; Sharma, N.L. On a generalization of close-to-convex functions. Ann. Polon. Math. 2015, 113, 93–108. [CrossRef]
17. Noonan, J.W.; Thomas, D.K. On the second Hankel derminant of areally mean p-valent functions. Trans. Am. Math. Soc. 1976, 223, 337–346.
18. Janteng, A.; Abdul-Halim, S.; Darus, M. Hankel determinant for starlike and convex functions. Int. J. Math. Anal. 2007, 1, 619–625.
19. Mishra, A.K.; Gochhayat, P. Second Hankel determinant for a class of analytic functions defined by fractional derivative. Internat. J. Math. Math. Sci. 2008, 2008, 153280. [CrossRef]
20. Singh, G.; Singh, G. On the second Hankel determinant for a new subclass of analytic functions. J. Math. Sci. Appl. 2014, 2, 1–3.
21. Babalola, K.O. On H3(1)Hankel determinant for some classes of univalent functions. Inequal. Theory Appl.
2007, 6, 1–7.
22. Duren, P.L. Univalent Functions (Grundlehren der Mathematischen Wissenschaften 259); Springer: New York, NY, USA; Berlin/Heidelberg, Germany; Tokyo, Japan, 1983.
23. Thomas, D.K.; Abdul-Halim, S. Toeplitz matrices whose elements are the coefficients of starlike and close-to-convex functions. Bull. Malays. Math. Sci. Soc. 2017, 40, 1781–1790. [CrossRef]
24. Efraimidis, I. A generalization of Livingston’s coefficient inequalities for functions with positive real part. J. Math. Anal. Appl. 2016, 435, 369–379. [CrossRef]
25. 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. [CrossRef]
26. Ramachandran, C.; Annamalai, S. On Hankel and Toeplitz determinants for some special class of analytic functions involving conical domains defined by subordination. Internat. J. Engrg. Res. Technol. 2016, 5, 553–561.
27. Lee, S.K.; Ravichandran, V.; Supramaniam, S. Bounds for the second Hankel determinant of certain univalent functions. J. Inequal. Appl. 2013, 2013, 281–297. [CrossRef]
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