J. Nig. Soc. Phys. Sci. 4 (2022) 961

Journal of the
Nigerian Society

of Physical
Sciences

On Lemniscate of Bernoulli of q-Janowski type

Afis Saliua,b,∗, Semiu Oladipupo Oladejob

aDepartment of Mathematics, University of the Gambia, MDI Road, Kanifing P.O. Box 3530, Serrekunda, The Gambia
bDepartment of Mathematics, Gombe State University P.M.B 127, Tudun Wada, Gombe, Gombe State, Nigeria

Abstract

In this article, we introduce the q-analogue of functions characterized by the lemniscate of Bernoulli in the right-half plane and define the class
L∗q(A, B). Furthermore, we study the geometric properties of this class, which include coefficient inequalities, subordination factor sequence
property, radii results and Fekete-Szegö problems. Some deductions of our results show relevant connections between this present work and the
existing ones in many literature. It is worthy of note that some of our results are sharp.

DOI:10.46481/jnsps.2022.961

Keywords: Univalent functions, Schwarz functions, lemniscate of Bernoulli, Subordination, Janowski functions.

Article History :
Received: 28 July 2022
Received in revised form: 12 September 2022
Accepted for publication: 13 September 2022
Published: 08 October 2022

c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license

(https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published articleâs title, journal citation, and DOI.

Communicated by: P. Thakur

1. Introduction and Preliminaries

The theory of univalent functions had its history from the
Riemann mapping theorem [1]. This theorem aroused the in-
terest of many researchers to start working in this field. For
example, Bieberbach [1] proved that for every univalent func-
tion

f (z) = z +
∞∑

n=2

anz
n (z ∈ U := {z ∈ C : |z| < 1}) , (1)

|a2| ≤ 2

and conjectured that, in general,

|an| ≤ n , n ≥ 2.

∗Corresponding author tel. no: +2348066221155, +22941154
Email address: asaliu@utg.edu.gm, saliugsu@gmail.com (Afis

Saliu )

This conjecture stood for more than fifty years until it was fi-
nally settled by De Branges [2] in 1985. Geometric Function
Theory (GFT) has witnessed many developements and intro-
duction of new subfamilies of univalent functions within these
intervening years. In particular, Ma and Minda [3] gave a com-
prehensive classification of classes S ∗ and C of starlike and
convex functions [1], respectively. For this purpose, they con-
sidered the class A of normalized analytic functions (1) and a
univalent function ϕ (which maps U onto domains symmetric
with the real axis and starlike with respect to 1) normalized such
that ϕ(0) = 1 and ϕ′(0) > 0. Thus, the classes are respectively
defined as follows:

S ∗(ϕ) =
{

f ∈ A :
z f ′(z)
f (z)

≺ ϕ

}
and

C(ϕ) =
{

f ∈ A : z f ′ ∈ S ∗(ϕ)
}
.

In particular, if we chose ϕ = (1 + z)/(1 − z), S ∗ and C respec-
tively become the usual classes S ∗ and C of starlike and convex

1



Saliu & Oladejo / J. Nig. Soc. Phys. Sci. 4 (2022) 961 2

functions. Sokół and Stankiewicz [4] introduced the class L∗
by setting ϕ(z) =

√
1 + z. This class consists of those functions

that map U onto the lemniscate of Bernoulli. They obtained
radius of convexity, growth and distortion results for this class.
Also, early coefficients of functions in L∗ were derived in [5].
Furthermore, Ali et al. [6] and Sokół [7] determined various
radii results associated with this class. Moreover, Raza and Ma-
lik [8] proved the third Hankel determinant associated with L∗.
For other related works, see [9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20] for details.

One of the most recent approaches in GFT is the concept
of calculus without the notion of limits known as q-calculus,
or quantum calculus [21, 22]. This idea was first discussed in
the theory of univalent functions by Ismail et al. [23]. They
introduced the class S ∗q of q-starlike functions, which have led
to many new results and techniques in quantum calculus associ-
ated with GFT. Recently, Khan et al. [24] gave a q-analogue of
√

1 + z and introduced the class L∗q. Moreover, they proved few
coefficient bounds and obtained the third Hankel determinant
for the class.

Motivated by these works and in particular, [24, 8] we in-
troduce a new class L∗q(A, B) and establish many coefficient
inequalities, subordination factor sequence, radii results and
Fekete-Szegö bounds for this class.

2. Definitions and Lemmas

Let W be the class of analytic functions

w(z) =
∞∑

n=1

wnz
n, z ∈ U (2)

such that w(0) = 0 and |w(z)| < 1. These functions are known as
Schwarz functions. If f (z) and g(z) are analytic functions in U,
then f (z) is subordinate to g(z) (written as f (z) ≺ g(z)) if there
exists w(z) ∈ W such that f (z) = g(w(z)), z ∈ U. In addition,
if g is univalent in U, then f (0) = g(0) and f (U) ⊂ g(U). An
analytic function

p(z) = 1 +
∞∑

n=1

cnz
n (3)

is a function with positive real part. The class of all such func-
tions is denoted by P. More generally, for −1 ≤ B < A ≤ 1, the
class P(A, B) consists of functions p(z) of the form (3) satisfy-
ing the subordination condition

p(z) ≺
1 + Az
1 + Bz

, z ∈ U.

On the other hand, p ∈ P(A, B) is called a Janowski function
[25] if and only if

p(z) =
(1 + A)h(z) + (1 − A)
(1 + B)h(z) + (1 − B)

, −1 ≤ B < A ≤ 1,

where h ∈ P.

Definition 2.1 (Subordinating Factor Sequence). [26] A se-
quence {bn}∞n=1 of complex numbers is called a subordinating
factor sequence if, whenever f (z) of the form (1) is analytic,
univalent and convex in U, we have the subornation given by

∞∑
n=1

anbnz
n
≺ f (z) (z ∈ U, a1 := 1).

Definition 2.2. [27] Let q ∈ (0, 1). Then the q-number [n]q is
given as

[n]q =




1−qn

1−q , n ∈ C,

n−1∑
ι=0

qι = 1 + q + q2 + · · · + qn−1, n ∈ N,

n, as q → 1−.

(4)

and the q-derivative of a complex valued function f (z) in U is
given by

Dq f (z) =




f (qz)− f (z)
(q−1)z , z , 0

f ′(0), z = 0,

f ′(z), as q → 1−.

(5)

From the above explanation, it is easy to see that for f (z) given
by (1),

Dq f (z) = 1 +
∞∑

n=2

[n]qanz
n−1. (6)

Using the idea of q-calculus, Khan et al. [24] extended the
work of Raza and Malik [8] by presenting the following class:

Definition 2.3. [24] A function of the form (1) belongs to the
class L∗q if and only if∣∣∣∣∣

(
zDq f (z)

f (z)

)2
−

1
1 − q

∣∣∣∣∣ < 11 − q, (7)
or equivalently,(

zDq f (z)
f (z)

)2
≺

2(1 + z)
2 + (1 − q)z

, z ∈ U, q ∈ (0, 1). (8)

Inspired by the work of Khan et al. [24] and Raza and Malik
[8], and in view of the Janowski functions, we introduce the
following class.

Definition 2.4. A function of the form (1) belongs to the class
L∗q(A, B) if and only if∣∣∣∣∣∣∣

(B − 1)
(

zDq f (z)
f (z)

)2
− (A − 1)

(B + 1)
(

zDq f (z)
f (z)

)2
− (A + 1)

−
1

1 − q

∣∣∣∣∣∣∣ <
1

1 − q
, (9)

or equivalently,(
zDq f (z)

f (z)

)2
≺

(AO1 + O3) z + 4
(BO1 + O3) z + 4

, zU, q ∈ (0, 1), (10)

where

O1 = 1 + q, O3 = 3 − q, −1 ≤ B < A ≤ 1. (11)
2



Saliu & Oladejo / J. Nig. Soc. Phys. Sci. 4 (2022) 961 3

Remark 2.5. For A = 1 and B = −1, we are back to the class
in Definition 2.3. In addition to that, if q → 1−, Definition 2.4
reduces to the one given in [4, 7].

To prove our main findings, we need the following results

Lemma 2.6. [28] If w ∈ W is of the form (2), then for a real
number σ,

|w2 −σw
2
1| ≤



−σ, for σ ≤−1,

1, for − 1 ≤ σ ≤ 1,

σ for σ ≥ 1.

When σ < −1 or σ > 1, equality holds if and only if w(z) = z
or one of its rotations. If −1 < σ < 1, then equality holds if
and only if w(z) = z2 or one of its rotations. Equality holds
for σ = −1 if and only if w(z) = z(x+z)1+xz (0 ≤ x ≤ 1) or one
of its rotations while for σ = 1, equality holds if and only if
w(z) = −z(x+z)1+xz (0 ≤ x ≤ 1) or one of its rotations.

Also, the sharp upper bound above can be improved as fol-
lows when −1 < σ < 1:

|w2 −σw
2
1| + (1 + σ)|w1|

2
≤ 1 (−1 < σ ≤ 0)

and
|w2 −σw

2
1| + (1 −σ)|w1|

2
≤ 1 (0 < σ < 1).

Lemma 2.7. [26] The sequence {bn}∞n=1 is a subordinating fac-
tor sequence if and only if

Re

(
1 + 2

∞∑
n=1

bnz
n

)
> 0, (zU).

In the next section, we present our main findings.

3. Main Results

Theorem 3.1. Let f ∈ A and suppose

∞∑
n=3

{ n−1∑
j=1

[
2
(
[ j]q[n − j]q − 1

)
(12)

+

∣∣∣∣(B + 1)[ j]q[n − j]q − (A + 1)
∣∣∣∣]∣∣∣a jan− j∣∣∣

}
<
∣∣∣B − A∣∣∣.

(13)

Then f ∈ L∗q(A, B)

Proof. Suppose condition (12) is satisfied. Then it suffices to
prove (9). Now,

∣∣∣∣∣∣∣
(B − 1)

(
zDq f (z)

f (z)

)2
− (A − 1)

(B + 1)
(

zDq f (z)
f (z)

)2
− (A + 1)

−
1

1 − q

∣∣∣∣∣∣∣
≤

∣∣∣∣∣∣∣
(B − 1)

(
zDq f (z)

f (z)

)2
− (A − 1)

(B + 1)
(

zDq f (z)
f (z)

)2
− (A + 1)

− 1

∣∣∣∣∣∣∣ +
q

1 − q

=2

∣∣∣∣∣
(
zDq f (z)

)2
− ( f (z))2

(B + 1)
(
zDq f (z)

)2
− (A + 1)( f (z))2

∣∣∣∣∣ + q1 + q
=2

∣∣∣∣∣∣∣∣∣∣

∞∑
n=2

(
n−1∑
j=1

a jan− j[ j]q[n − j]q

)
zn −

∞∑
n=2

(
n−1∑
j=1

a jan− j

)
zn

(1 + B)
∞∑

n=2

(
n−1∑
j=1

a jan− j[ j]q[n − j]q

)
zn − (A + 1)

∞∑
n=2

(
n−1∑
j=1

a jan− j

)
zn

∣∣∣∣∣∣∣∣∣∣
+

q
1 − q

=2

∣∣∣∣∣∣∣∣∣∣

∞∑
n=3

[
n−1∑
j=1

a jan− j
(
[ j]q[n − j]q − 1

)]
zn

(B − A)z2 +
∞∑

n=3

{
n−1∑
j=1

a jan− j
[
(B + 1)[ j]q[n − j]q − (A + 1)

]}
zn

∣∣∣∣∣∣∣∣∣∣
≤

2
∞∑

n=3

[
n−1∑
j=1

∣∣a jan− j∣∣([ j]q[n − j]q − 1)
]

|B − A| +
∞∑

n=3

[
n−1∑
j=1

∣∣a jan− j∣∣ ∣∣∣∣(B + 1)[ j]q[n − j]q − (A + 1)
∣∣∣∣
] + q

1 − q
.

3



Saliu & Oladejo / J. Nig. Soc. Phys. Sci. 4 (2022) 961 4

This last expression is bounded by 11−q provided (12) is sat-
isfied. Hence, we have the required result. We observe that
Theorem 3.1 implies the following result.

Corollary 3.2. Let f ∈ A. Then f ∈ L∗q(A, B) if
∞∑

n=2

[
2q[n−1]q+

∣∣(B + 1)[n]q − (A + 1)∣∣] |an| < |B − A| .(14)
Proof. Let

Φ j,n =

[
2
(
[ j]q[n − j]q − 1

)
+

∣∣∣∣(B + 1)[ j]q[n − j]q
− (A + 1)

∣∣∣∣
]∣∣∣a jan− j∣∣∣ < |B − A|

in (12). Then

∞∑
n=3

(
Φ1,n + Φ2,n + Φ3,n + · · · + Φn−1,n

)
< |B − A| ,

which implies that

∞∑
n=3

Φn−1,n < |B − A| .

That is
∞∑

n=3

[
2
(

[n−1]q−1
)

+
∣∣(B + 1)[n − 1]q − (A + 1)∣∣] |an−1| < |B − A|

or
∞∑

n=2

[
2q[n − 1]q +

∣∣(B + 1)[n]q − (A + 1)∣∣] |an| < |B − A|

For A = 1, B = −1 in Corollary ??corollary1, we are led to
the following assertion.

Corollary 3.3. Let f ∈ A. Then f ∈ L∗q(1,−1) if

∞∑
n=2

[n]q |an| < 1.

As q → 1− in Corollary ??corollary2, we have

Corollary 3.4. Let f ∈ A. Then f ∈ L∗ if
∞∑

n=2

n |an| < 1.

By Corollary 3.2, the class L∗q(A, B) is assumed to be a sub-
class of L∗q(A, B) consisting of f (z) of the form (14) and satisfy-
ing the condition (14). Therefore, by the coefficient inequality
for the class L∗q(A, B), we have the following results.

Theorem 3.5. Let −1 ≤ B1 ≤ B2 < A1 ≤ A2 ≤ 1. If f ∈ A,
then
L∗q(A2, B2) ⊂ L

∗
q(A1, B1).

Proof. By the hypothesis of the theorem, we have that

∞∑
n=2

[
2q[n − 1]q +

∣∣(B1 + 1)[n − 1]q − (A1 + 1)∣∣] |an|
≤

∞∑
n=2

[
2q[n − 1]q +

∣∣(B2 + 1)[n − 1]q − (A2 + 1)∣∣] |an|
≤ |B2 − A2|

≤ |B1 − A1| .

Thus, f ∈ L∗q(A1, B1).

Theorem 3.6. Let f j ∈ L∗q(A, B) and be of the form

f j(z) =
∞∑

n=1

an, jz
n, j = 1, 2, . . . m, a1, j = 1.

Then G(z) ∈ L∗q(A, B), where

G(z) =
m∑

j=1

|c j| f j(z) with
m∑

j=1

|c j| = 1.

Proof. In view of (14), we have

∞∑
n=2

[
2q[n − 1]q +

∣∣(B + 1)[n]q − (A + 1)∣∣
|B − A|

]∣∣an, j∣∣ < 1.
Therefore,

G(z) =
m∑

j=1

|c j| f j(z)

=

m∑
j=1

|c j|

(
z +

∞∑
n=2

an, jz
n

)

=

m∑
j=1

∞∑
n=1

|c j|an, jz
n

=

∞∑
n=1

( m∑
j=1

|c j|an, j

)
zn.

But

∞∑
n=2

[
2q[n − 1]q +

∣∣(B + 1)[n]q − (A + 1)∣∣
|B − A|

]∣∣∣∣ m∑
j=1

|c j|an, j

∣∣∣∣
≤

m∑
j=1

{ ∞∑
n=2

[
2q[n − 1]q +

∣∣(B + 1)[n]q − (A + 1)∣∣
|B − A|

]∣∣an, j∣∣}|c j|
<

m∑
j=1

|c j|

= 1.

Hence, G(z) ∈ L∗q(A, B).

4



Saliu & Oladejo / J. Nig. Soc. Phys. Sci. 4 (2022) 961 5

Theorem 3.7. Let g(z) = z +
∞∑

n=2
bnzn. If f, g ∈ L∗q(A, B), then

their weighted mean

F j(z) =
(1 − j) f (z) + (1 + j)g(z)

2
(15)

also belongs to L∗q(A, B).

Proof. From (15), we have

F j(z) = z +
∞∑

n=2

[
(1 − j)an + (1 + j)bn

2

]
zn.

To show that F j(z) ∈ L∗q(A, B), we need to show that

∞∑
n=2

[
2q[n − 1]q +

∣∣∣(B + 1)[n]q
− (A + 1)

∣∣∣]∣∣∣∣(1 − j)an + (1 + j)bn2
∣∣∣∣ < |B − A| .

For this, we have

∞∑
n=2

[
2q[n − 1]q +

∣∣∣(B + 1)[n]q
− (A + 1)

∣∣∣]∣∣∣∣(1 − j)an + (1 + j)bn2
∣∣∣∣

≤

(
1 − j

2

) ∞∑
n=2

[
2q[n − 1]q +

∣∣(B + 1)[n]q − (A + 1)∣∣] |an|
+

(
1 + j

2

) ∞∑
n=2

[
2q[n − 1]q +

∣∣(B + 1)[n]q − (A + 1)∣∣] |bn|
< |B − A|.

Theorem 3.8. Suppose f j(z) ∈ L∗q(A, B). Then the arithmetic
mean F(z) of f j(z) given by

F(z) =
1
n

n∑
j=1

f j(z)

is also in L∗q(A, B).

Proof. We have

F(z) =
1
n

n∑
j=1

f j(z)

=
1
n

n∑
j=1

(
z +

∞∑
k=2

ak, jz
k

)

= z +
∞∑

k=2


1

n

n∑
j=1

ak, j


 zk.

Since f j(z) ∈ L∗q(A, B), then

∞∑
k=2

[
2q[k − 1]q +

∣∣(B + 1)[k]q − (A + 1)∣∣]
∣∣∣∣∣∣1n

n∑
j=1

ak, j

∣∣∣∣∣∣
≤

1
n

n∑
j=1

{
∞∑

k=2

[
2q[k − 1]q +

∣∣(B + 1)[k]q − (A + 1)∣∣]
}∣∣ak, j∣∣

<
1
n

n∑
j=1

|B − A|

= |B − A| .

Thus, F ∈ L∗q(A, B).

In the next theorem, we provide a sharp subordination result
involving class L∗q(A, B).

Theorem 3.9. Let f ∈ L∗q(A, B). Then for every convex function
g(z) in U, we have

2q + |(1 + B)q + B − A|
2
[
2q + |B − A| + |(1 + B)q + B − A|

] ( f ∗ g) (z) ≺ g(z)(16)
and

Re f (z) > −

[
2q + |B − A| + |(1 + B)q + B − A|

]
2q + |(1 + B)q + B − A|

. (17)

The constant

2q + |(1 + B)q + B − A|
2
[
2q + |B − A| + |(1 + B)q + B − A|

]
cannot be replaced by any larger one.

Proof. Let f ∈ L∗q(A, B) and g(z) = z +
∞∑

n=2
cnzn be convex in U.

Then

2q + |(1 + B)q + B − A|
2
[
2q + |B − A| + |(1 + B)q + B − A|

] ( f ∗ g) (z)
=

2q + |(1 + B)q + B − A|
2
[
2q + |B − A| + |(1 + B)q + B − A|

] (z + ∞∑
n=2

anbnz
n

)
.

Therefore, by Definition 2.1, the assertion of our theorem holds
if the sequence{

2q + |(1 + B)q + B − A|
2
[
2q + |B − A| + |(1 + B)q + B − A|

]an}∞
n=1

is a subordinating factor sequence, with a1 = 1. In view of
Lemma 2.7, this will hold true if and only if

Re

[
1 +

∞∑
n=1

2q + |(1 + B)q + B − A|[
2q + |B − A| + |(1 + B)q + B − A|

]anzn
]
> 0 (z ∈ U).

Since
2q[n − 1]q +

∣∣(B + 1)[n]q − (A + 1)∣∣
is an increasing function of n (n ≥ 2), we have

5



Saliu & Oladejo / J. Nig. Soc. Phys. Sci. 4 (2022) 961 6

Re

[
1 + 2

∞∑
n=1

2q + |(1 + B)q + B − A|
2
[
2q + |B − A| + |(1 + B)q + B − A|

]anzn
]

= Re
[

1 +
2q + |(1 + B)q + B − A|[

2q + |B − A| + |(1 + B)q + B − A|
]z

+

∞∑
n=2

2q + |(1 + B)q + B − A|[
2q + |B − A| + |(1 + B)q + B − A|

]anzn]
≥ 1 −

2q + |(1 + B)q + B − A|[
2q + |B − A| + |(1 + B)q + B − A|

]r
−

∞∑
n=2

2q + |(1 + B)q + B − A|[
2q + |B − A| + |(1 + B)q + B − A|

] |an|rn

> 1 −
2q + |(1 + B)q + B − A|[

2q + |B − A| + |(1 + B)q + B − A|
]

−

∞∑
n=2

2q + |(1 + B)q + B − A|[
2q + |B − A| + |(1 + B)q + B − A|

] |an|
≥ 1 −

2q + |(1 + B)q + B − A|[
2q + |B − A| + |(1 + B)q + B − A|

]
−

1[
2q + |B − A| + |(1 + B)q + B − A|

] ∞∑
n=2

[
2q[n − 1]q

+
∣∣(B + 1)[n]q − (A + 1)∣∣] |an|

> 1 −
2q + |(1 + B)q + B − A|[

2q + |B − A| + |(1 + B)q + B − A|
]

−
|B − A|[

2q + |B − A| + |(1 + B)q + B − A|
]

= 0,

where we have used condition (12). This proves (16). Further-
more, condition (17) is achieved by setting

g0 (z) =
z

1 − z
= z +

∞∑
n=2

zn

To prove the sharpness of the constant 2q+|B−A|4(q+|B−A|) , we consider
the function given by

f (z) = z −
|B − A|

2q + |(1 + B)q + B − A|
z2.

Evidently, f ∈ L∗q(A, B) and from (16), we have

2q + |(1 + B)q + B − A|
2
[
2q + |B − A| + |(1 + B)q + B − A|

] f (z) ≺ z
1 − z

(zU).

It is observed that

min
{

Re
(

2q + |(1 + B)q + B − A|
2
[
2q + |B − A| + |(1 + B)q + B − A|

] f (z))} = −1
2
.

This completes the proof.

For A = 1, B = −1 in Theorem 3.9, we have the following
result.

Corollary 3.10. Let f ∈ L∗q ⊂ L
∗
q. then for every convex func-

tion g(z) in U, we have

1 + q
2(q + 2)

( f ∗ g)(z) ≺ g(z)

and
Re f (z) > −

2 + q
1 + q

.

Moreover, the constant 1+q2(2+q) cannot be a larger one.

As q → 1− in Corollary 3.10, we obtain the following as-
sertion.

Corollary 3.11. Let f ∈ L∗ ⊂ L∗. then for every convex func-
tion g(z) in U, we have

1
3

( f ∗ g)(z) ≺ g(z)

and
Re f (z) > −

3
2
.

The constant factor 13 cannot be replaced by a larger one.

In the following theorems, we present some radii results as-
sociated with the class L∗q(A, B).

Theorem 3.12. L∗q(A, B) ⊂ L
∗
q in the disc |z| < Rq(A, B),

−1 ≤ B < A ≤ 1, where

Rq(A, B) = min
{

4
2(A − B) +

[
2(1 + B) + (1 − q)(1 − A)

] , 1}.(18)
Proof. To determine L∗q radius, we need to find R such that 0 <
R < 1 and

H(zR) :=
(AO1 + O3) z + 4
(BO1 + O3) z + 4

≺
2(1 + z)

2 + (1 − q)z
:= Q(z) (z ∈ U),

where O1 and O3 are given by (11). This is equivalent to

|Q−1(H(Rz))| ≤ 1.

It is easy to see that

|Q−1(H(Rz))| =

∣∣∣∣∣ 2(A − B)Rz4 + [(2B − (1 − q)A) + O3] Rz
∣∣∣∣∣

≤
2(A − B)R

4 −
[
2(1 + B) + (1 − q)(1 − A)

]
R
.

The last expression is bounded by 1 if

2(A − B)R ≤ 4 −
[
2(1 + B) + (1 − q)(1 − A)

]
R.

This completes the proof.

Theorem 3.13. Let f ∈ S ∗q. Then f ∈ L
∗
q in the disc |z| < Rq,

where

Rq =
1

3 +
√

11 − 2q + |2q − 1|
. (19)

6



Saliu & Oladejo / J. Nig. Soc. Phys. Sci. 4 (2022) 961 7

Proof. Since f ∈ S ∗q, then

zDq f (z)
f (z)

≺
1 + z

1 − qz
(z ∈ U, see [29, 30, 31]) ,

and by subordination property, we have(
zDq f (z)

f (z)

)2
≺

(
1 + z
1 − qz

)2
:= S (z) (z ∈ U).

We need to determine the smallest positive radius R such that

S (Rz) ≺ Q(z) (z ∈ U),

which is equivalent to showing that∣∣Q−1(S (Rz))∣∣ ≤ 1 (z ∈ U).
Now, we can obviously see that

∣∣Q−1(S (Rz))∣∣ =
∣∣∣∣∣ 2
(
2R + (1 − q)R2z2

)
1 − 2Rz − (2q − 1)R2z2

∣∣∣∣∣
≤

2
(
2(1 − q) + R2

)
1 − 2R − |2q − 1|R2

.

This expression is bounded by 1 if

(2(1 − q) + |2q − 1|) R2 + 6R − 1 ≤ 0.

Let T (R) = (2(1 − q) + |2q − 1|) R2 +6R−1. Then T (0)T (1) < 0.
Therefore, there exists R ∈ (0, 1) such that T (R) = 0. Hence,
we have the result.

As q → 1− in Theorem 3.13, we get the following result.

Corollary 3.14. Let f ∈ S ∗. Then f ∈ L∗ in the disc |z| <
1

3+
√

10
.

In the next theorem, using Lemma 2.6, we present the Fekete-
Szegö inequality for the class L∗q(A, B).

Theorem 3.15. Let f ∈ L∗q. Then for a real number µ,

|a3−µa
2
2| ≤

(
A − B

8q

)



−
qV (q)+2µ(A−B)(1+q)2

16q ,

for µ ≤− q(16+V (q))2(A−B)(1+q)2 := ρ1,

1, for ρ1 ≤ µ ≤
q(16−V (q))

2(A−B)(1+q)2 := ρ2,

qV (q)+2µ(A−B)(1+q)2

16q , for µ ≥ ρ2, .

It is asserted also that

|a3 −µa
2
2| +

[
µ +

q(16 + V (q))
2(A − B)(1 + q)2

]
|a2|

2
≤

A − B
8q

,

−
q(16 + V (q))

2(A − B)(1 + q)2
< µ ≤−

qV (q)
2(A − B)(1 + q)2

and

|a3 −µa
2
2| −

[
µ−

q(16 − V (q))
2(A − B)(1 + q)2

]
|a2|

2
≤

A − B
8q

,

−
qV (q)

2(A − B)(1 + q)2
< µ ≤

q(16 − V (q))
2(A − B)(1 + q)2

where

V (q) = (A + 3B − 4)q2 − (A − 5B − 12)q − 2(A − B).(20)

These bounds are sharp.

Proof. Let f ∈ L∗q(A, B). Then

zDq f (z)
f (z)

=

√
1 + A1w(z)
1 + B1w(z)

,

where

w(z) =
∞∑

n=1

wnz
n
∈W, A1 =

AO1 + O3
4

, and B1 =
BO1 + O3

4
.

Therefore,

1 + qa2z + q
(
[2]qa3 − a

2
2

)
z2 + . . .

= 1 +
w1O1(A − B)

8
z −

O1(A − B)
8

[
1

16

(
AO1 + 4O3

+ 3BO1
)

w21 − w2

]
z2 + . . . .

On comparing coefficients, we have

a2 =
(A − B)O1w1

8q
and a3 =

A − B
8q

(
w2 −

V (q)
16

w21

)
,

so that
a3 −µa

2
2 =

A − B
8q

(w2 −σw
2
1),

where

σ =
qV (q) + 2µ(A − B)O21

16q
.

Then we have the required result by applying Lemma 2.6. These
inequalities are sharp for the functions



λ f1(zλ), for µ ∈ (−∞,ρ1) ∪ (ρ2,∞),

λ f2(zλ), for ρ1 ≤ µ ≤ ρ2,

λ f3(zλ), for µ = ρ1,

λ f4(zλ), for µ = ρ2,

where |λ| = 1 and

zDq f1(z)
f1(z)

=

√
1 + A1z
1 + B1z

zDq f2(z)
f2(z)

=

√
1 + A1z2

1 + B1z2

zDq f3(z)
f3(z)

=

√
1 + A1wx(z)
1 + B1wx(z)

zDq f4(z)
f4(z)

=

√
1 − A1wx(z)
1 − B1wx(z)

with
wx(z) =

z(x + z)
1 + xz

, 0 ≤ x ≤ 1.

7



Saliu & Oladejo / J. Nig. Soc. Phys. Sci. 4 (2022) 961 8

Remark 3.16. For A = 1, B = −1, Theorem 3.15 reduces to the
Fekete-Szegö inequalities presented by Khan [24]. Further, as
q → 1− we are led to the result of Raza and Malik [8].

4. Conclusion

In this work, we introduced the class L∗q(A, B) and obtained
Coefficient inequalities, subordination factor sequence property
radii results and Fekete-Szegö inequality for it. Overall, we pre-
sented many consequences of our investigation. These results
are fascinated essentially by their particular cases and conse-
quences. Also, to have more new hypotheses under present as-
sessments, new extension and applications can be investigated
with some positive and novel results in different fields of sci-
ence, particularly, in GFT. For more details about the applica-
tions, one may go through [32, 33].

Acknowledgments

We thank the referees for the positive enlightening com-
ments and suggestions, which have greatly helped us in making
improvements to this paper.

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