EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 2516-2537
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some New Applications of the Quantum Calculus for
New Families of Sigmoid Activation Bi-Univalent
Functions Connected to Horadam Polynomials

Nidhish Kumar Mishra1, Mohammad Faisal Khan1,∗, Showkat Ahmad Lone1

1 Department of Basic Sciences, College of Science and Theoretical Studies,
Saudi Electronic University, Riyadh11673, Kingdom of Saudi Arabia

Abstract. The study of q-calculus is becoming increasingly prominent in the field of geometric
function theory, reflecting a growing interest in its applications. In this research work, we first
develop a new type of modified Sigmoid-Salagean q-differential operator in the open unit disk D,
utilizing the concepts of quantum calculus and the Sigmoid activation function. Using this newly
defined q-analogous differential operator and Horadam polynomials, we introduce new subclasses
of bi-univalent functions in D. We determine upper bounds on initial coefficients, as well as the
Fekete-Szegö problems, for functions belonging to these special families. Additionally, we discuss
several interesting consequences related to the findings presented in this study.

2020 Mathematics Subject Classifications: 30C45, 30C50

Key Words and Phrases: Holomorphic function, Bi-univalent function, Horadam polynomials,
Modified Sigmoid function, q-Calculus, the q-Difference operator, Fekete-Szegö problem

1. Introduction

Let A be the set of normalized analytic functions that have the form

g(z) = z + d2z
2 + d3z

3 + ... = z +

∞∑
j=2

djz
j , (1)

in D = {z ∈ C : |z| < 1}. Suppose that, an analytic function g, which is a function of a
single-value in some domain ∆ ⊂ C. If g does not take the same value twice in ∆, we say
that it is a univalent function, that is, if g(z1) ̸= g(z2) for z1 ̸= z2, (see ([10], page 26).
The class of all univalent functions represented by S. The theorem of Koebe one-quarter

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5345

Email addresses: n.kumar@seu.edu.sa (N. K. Mishra),
f.khan@seu.edu.sa (M. F. Khan), s.lone@seu.edu.sa (S. A. Lone)

https://www.ejpam.com 2516 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2517

([10], page 31) prove that the range of each function g ∈ S contains the open disk of radius
1
4 . Every function g ∈ S has an inverse g−1 satisfies

z = g−1(g(z)), z ∈ D

and
ω = g(g−1(ω)), |ω| < r0(g), r0(g) ≥ 1/4.

The inverse function f = g−1 for each g ∈ S has Taylor series expansion as follows (see
([3], page 185)):

g−1(ω) = f(ω) = ω +
∞∑
j=2

1

j
K−j

j−1 (d2, d3, d4, ....dn)ω
j

= ω − d2ω
2 + (2d22 − d3)ω

3 − (5d32 − 5d2d3 + d4)ω
4 + · · · , (2)

where the coefficients of j parametric function Kp
j (d2, d3, d4, ....dn) are given by

Kp
1 = pd2,

Kp
2 =

p(p− 1)

2
d22 + pd3,

Kp
3 = p(p− 1)d2d3 + pd4 +

p(p− 1)(p− 2)

3!
d32.

An analytic function g ∈ A will be the bi-univalent in D, if both g and g−1 are univalent
in D and the family of bi-univalent functions of the form (1) is represented by the symbol
Σ. The subject has gained renewed attention in the last ten years, with several studies
published on the subject since 2011, for instance [16, 30]. There were intriguing findings
about the estimation of coefficients for certain types of univalent functions, (see [32, 40, 45–
47]).

Lemma 1. (Schwarz lemma ([10], page 3)). Let ψ(z) is analytic in D with ψ(0) = 0 and
|ψ(z)| < 1, z ∈ D, then we have

|ψ(z)| < |z| and |ψ′
(0)| < 1 in D.

Definition 1. The analytic function ζ1(z) is subordinate to the analytic function ζ2(z),
(written as ζ1(z) ≺ ζ2(z), z ∈ D), if there exist a Schwarz function ψ(z) in D such that

ζ1(z) = ζ2(ψ(z)), z ∈ D.

To be specific, if ζ2 is univalent in D, then (see also [10]):

ζ1(z) ≺ ζ2(z) ⇔ ζ1(0) = ζ2(0) and ζ1(D) ⊂ ζ2(D).



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2518

The study of coefficients for functions in particular classes has been a cornerstone of
univalent function research from its earliest beginnings. The Gronwall Area Theorem,
established in 1914, is a significant discovery in the theory of univalent functions, used
to determine bounds on the coefficients of the class of meromorphic functions. Different
approaches in the geometric theory of functions of a complex variable have been inspired
by Bieberbach’s famous hypothesis, presented in 1916 but only verified in 1984, which he
used to solve similar problems for the class S. When examining bi-univalent functions, as
in the classes investigated by Gronwall and Bieberbach, it is common practice to provide
estimates for the first two Taylor-Maclaurin coefficients. Obtaining comparable estimates
for different types of functions is known as the Fekete-Szegö problem. In 1933, it was
shown by Fekete and Szegö [14] that

∣∣d3 − δd22
∣∣ ≤


3− 4δ if δ < 0,

1 + 2 exp
(

2δ
δ−1

)
if 0 ≤ δ < 1,

4δ − 3 if δ ≥ 1,

is sharp and valid for every normalized univalent function. The Fekete-Szegö problem is
the one where the objective is to maximize the absolute value of the functional

∣∣d3 − δd22
∣∣ .

According to many writers, Fekete-Szegö inequalities have been shown for several types of
functions (see to references [9, 11]).

Geometric function theory have built and studied new classes of analytic functions
using the q-calculus and the fractional q-calculus. To construct a class of q-starlike func-
tions in D, Ismail et al. [23] first used the q-calculus (∂q) operator, which was created by
Jackson [24] in 1909. References such as [6, 26, 31, 35, 36] provide more information on
q-calculus.

Definition 2. Jackson [24, 25] developed the following q-derivative operator ∂q for an
analytic function g as follows:

∂qg(z) =
g(z)− g(qz)

(1− q) z
, 0 < q < 1; z ̸= 0.

It is evident that there is a limit relationship:

lim
q→1−

∂qg(z) = g
′
(z)

and
∂qg(0) = g

′
(0).

For the function g ∈ A, defined by (1), we deduce the following series

∂qg(z) = 1 +
∞∑
j=2

[j]q dnz
j−1,

where [j]q, called the q-analogue of j ∈ N is given by

[j]q =
1− qj

1− q
, j ∈ N.



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2519

As q → 1−, we have [j]q → j and [0]q → 0.

Fadipe-Joseph et al. [12] recently (2013) defined a modified Sigmoid function, Ψ(s) =
2

1+e−s , s ≥ 0, and showed that it has a positive real part and belongs to the class P of
Caratheodory functions.

Definition 3. Let AΨ denote the family of functions of the form

gΨ(z) = z +
∞∑
j=2

2

1 + e−s
djz

j = z +
∞∑
j=2

Ψ(s)djz
j , (3)

where Ψ(s) = 2
1+e−s , s ≥ 0, is a modified Sigmoid function. Clearly Ψ(0) = 1 and hence

A1 = A (see also [13]).

Now we use the definitions of Modified Sigmoid function gΨ(z) and q-difference oper-
ator ∂q, we define a Modified Sigmoid Sălăgean q-differential operator Dk

q : AΨ → AΨ as
follows:

Definition 4. For gΨ ∈ AΨ, k ∈ N∪{0}, the Sălăgean q-differential operator Dk
q : AΨ →

AΨ, is defined by

D0
qgΨ(z) = gΨ(z), D

1
qgΨ(z) = z∂qgΨ(z), ..., D

k
q gΨ(z) = ∂q(D

k−1
q gΨ(z)), z ∈ D.

For gΨ ∈ AΨ, defined by (3), we deduce the following series:

Dk
q gΨ(z) = z +

∞∑
j=2

[j]kq Ψ(s)djz
j . (4)

Remark 1. When s = 0 then Ψ(s) = 1, then we have the Sălăgean q-differential operator
[19].

Remark 2. When q → 1−, and Ψ(s) = 1, then we have the Sălăgean differential operator
[33].

The Horadam polynomials Υj(y) were used in a comparable setting by Srivastava et al.
[41]. The well-known Horadam polynomials Υj(y), as defined in Definition 5 in the field
of Geometric Function Theory of Complex Analysis, were recently examined by Hörçum
and Koçer [22], see also [21].

Definition 5. ([21, 22]). The following recurrence relation gives the Horadam polynomials
Υj(y, a, b; p, t) (or briefly Υj(y)):

Υj(y) = pyΥj−1(y) + tΥj−2(y) (5)

with
Υ1(y) = a, Υ2(y) = by,



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2520

where j ∈ N = {1, 2, ...}, y ∈ R, a, b, p and t are real constants. From (5) we have

Υ3(y) = pby2 + ta.

In addition, the characteristic equation for the recurrence relation (5) is provided by

s2 − pys− t = 0,

where

β1 =
py +

√
p2y2 + 4t

2
and β2 =

py −
√
p2y2 + 4t

2

are two real roots. The 𭟋(y, z) is the generating function of Υj(y) is given below (see
[22]):

𭟋(y, z) :=

∞∑
j=1

Υj(y)z
j−1 =

a+ (b− ap)yz

1− pyz − tz2
, (6)

where y ∈ R is independent of the argument z ∈ C, that is y ̸= ℜ(z).

Remark 3. By selecting the parameters a, b, p, and t properly, we present here a few
unique instances of Υj(y, a, b; p, t).
(i): Υj(y, 2, 2; 2, 1) = Qj(y), the Pell-Lucas polynomials.
(ii): Υj(y, 1, 1; 2,−1) = Tj(y), the first kind Chebyshev polynomials.
(iii): Υj(y, 1, 2; 2,−1) = Uj(y), the second kind Chebyshev polynomials.
(iv): Υj(y, 1, 1; 1, 1) = Fj(y), the Fibonacci polynomials.
(v): Υj(y, 2, 1; 1, 1) = Lj(y), the Lucas polynomials.
(vi): Υj(y, 1, 2; 2, 1) = Pj(y), the Pell polynomials.

Applications: The Horadam polynomial is a mathematical series used in texture
analysis and picture processing. The Horadam polynomial, a distinct kind of polynomial
sequence, is used for tasks such as filtering and resampling. Scale-space representations
of pictures can be generated and adjusted in image processing and computer vision by
using the Horadam polynomial. This polynomial is applicable for edge detection, texture
examination, and multi-scale picture analysis. The Horadam polynomial has been used for
texture analysis to extract features, segment images, and denoise pictures. This method
may be used to examine the statistical characteristics of textures, such as the distribution
of gray levels and the geographical arrangement of textures.

Abirami et al. [1] examined the initial coefficient estimates of Taylor-Maclaurin series
for bi-Mocanu-convex and bi-µ-starlike functions related to Horadam polynomials. Ad-
ditionally, Abirami et al. [2] discussed coefficient estimates for λ-bi-pseudo-starlike and
bi-Bazilevic functions. Alamoush [4, 5] introduced subclasses of bi-starlike and bi-convex
functions, utilizing the Poisson distribution series and Horadam polynomials, and also
explored a class of bi-univalent functions associated with Horadam polynomials. These
studies yielded initial coefficient estimates for the respective subclasses. Recent studies
[7, 8, 29, 37–39] have explored various classes of bi-univalent functions associated with
Horadam polynomials, Chebyshev polynomials, and other special functions. These works



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2521

have established initial coefficient estimates, Fekete-Szegö bounds, and coefficient esti-
mates for different classes of bi-univalent functions. We also taken notice of the fact that
unique polynomials like the ones mentioned above might play a significant role in the
fields of engineering, mathematics, statistics, and physical science. References [15, 21, 26]
provide more information on these polynomials. In the works of [44] and [43], you can find
further information on the Fekete-Szegö problem as it relates to Haradam polynomials.
See references [17, 18, 20, 27, 28, 34] for a discussion of the many uses and applica-
tions of orthogonal polynomial families as well as other special functions and specialized
polynomials. Illustrating from present trends in bi-univalent functions associated with
various polynomials, we establish the following unique families of the class Σ using the
Horadam polynomials Υj(y) linked to the Modified Sigmoid function (3) and its Sălăgean
q-differential operator.

Here we give the value of all parameters, which will be used in this article

µ ≥ 0, q ∈ (0, 1) , µ ≥ γ, 0 ≤ γ ≤ 1, k ∈ N ∪ {0}, ξ ≥ 1, τ ≥ 1

and

Ψ(s) =
2

1 + e−s
, s ≥ 0,

also
fΨ(ω) = g−1

Ψ (ω)

which is an extension of g−1 to D given by (2), a, b, p and t are as in (5) and 𭟋 is as in
(6).

Definition 6. A function g(z) in Σ is expressed as (1), then it is belong to the family

Sµ,k
Σ,y,γ(q,Ψ(s)), if

z∂q(D
k
q gΨ(z)) + µz2∂2q (D

k
q gΨ(z))

(1− γ)Dk
q gΨ(z) + γz∂q(Dk

q gΨ(z))
≺ 𭟋(y, z) + 1− α, z ∈ D

and
ω∂q(D

k
q fΨ(ω)) + µω2∂2q (D

k
q fΨ(ω))

(1− γ)DkfΨ(ω) + γω∂q(Dk
q fΨ(ω))

≺ 𭟋(y, ω) + 1− α, ω ∈ D.

Remark 4. For the special values of γ and µ, the family Sµ,k
Σ,y,γ(q,Ψ(s)) reduces to the

following new subfamilies.
(i): For γ = µ = 1

2 , we have Sµ,k
Σ,y,γ(q,Ψ(s)) = JΣ(y, k, q,Ψ(s)), a new family of bi-

univalent functions connected with Sigmoid activation functions and Horadam polynomi-
als.
(ii): For γ = 0, and µ = 1

2 , we obtain a new family Sµ,k
Σ,y,γ(q,Ψ(s)) = KΣ(y, k, q,Ψ(s)) of

bi-univalent functions connected with Sigmoid activation functions and Horadam polyno-
mials.
(iii): For γ = 1

2 , and µ = 1, we obtain a new family Sµ,k
Σ,y,γ(q,Ψ(s)) = LΣ(y, k, q,Ψ(s)) of



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2522

bi-univalent functions connected with Sigmoid activation functions and Horadam polyno-
mials.
(iv): For γ = 0, we obtain a new family Sµ,k

Σ,y,γ(q,Ψ(s)) = MΣ(y, µ, k, q,Ψ(s)) of bi-
univalent functions connected with Sigmoid activation functions and Horadam polynomials.

The Class LΣ(y, γ, µ, k, q,Ψ(s))

Definition 7. A function g(z) in Σ is given in (1), then it is belong to the family
LΣ(y, γ, µ, k, q,Ψ(s)), and Ψ(s) = 2

1+e−s , s ≥ 0, if

z∂q(D
kgΨ(z)) + µz2∂2q (D

k
q gΨ(z))

(1− γ) z + γz∂q(Dk
q gΨ(z))

≺ 𭟋(y, z) + 1− α, z ∈ D

and
ω∂q(D

k
q fΨ(ω)) + µω2∂2q (D

k
q fΨ(ω))

(1− γ)ω + γω∂q(Dk
q fΨ(ω))

≺ 𭟋(y, ω) + 1− α, ω ∈ D.

Remark 5. It is easy to observe that the special values of γ lead the family NΣ(y, γ, µ, k, q,Ψ(s))
to the following various subfamilies:
(i): For γ = 0, we obtain a new family LΣ(y, γ, µ, k, q,Ψ(s)) = NΣ(y, µ, k, q,Ψ(s)) of
bi-univalent functions connected with Sigmoid activation functions and Horadam polyno-
mials.
(ii): For γ = 1, we obtain a new family LΣ(y, γ, µ, k, q,Ψ(s)) = OΣ(y, µ, k, q,Ψ(s)) of
bi-univalent functions connected with Sigmoid activation functions and Horadam polyno-
mials.

The Class BΣ(y, ξ, τ, k, q,Ψ(s))

Definition 8. A function g(z) in Σ is expressed as (1), then it is belong to the family
BΣ(y, ξ, τ, k, q,Ψ(s)), and Ψ(s) = 2

1+e−s , s ≥ 0, if

(1− ξ) + ξ
[
∂q(z∂q

(
Dk

q gΨ(z)
)
)
]τ

∂q(Dk
q gΨ(z))

≺ 𭟋(y, z) + 1− α, z ∈ D,

and
(1− ξ) + ξ

[
∂q(ω∂q

(
Dk

q fΨ(ω)
)
)
]τ

∂q(Dk
q fΨ(ω))

≺ 𭟋(y, ω) + 1− α, ω ∈ D.

Remark 6. It is easy to observe that the special values of γ lead the family BΣ(y, ξ, τ, k, q,Ψ(s))
to the following various subfamilies:
(i): For τ = 1, we obtain a new family BΣ(y, ξ, τ, k, q,Ψ(s)) = MΣ(y, ξ, k, q,Ψ(s)) of
bi-univalent functions connected with Sigmoid activation functions and Horadam polyno-
mials.
(ii): For ξ = 1, we obtain a new family BΣ(y, ξ, τ, k, q,Ψ(s)) = NΣ(y, τ, k, q,Ψ(s)) of bi-
univalent functions connected with Sigmoid activation functions and Horadam polynomials.



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2523

2. Main Results

2.1. Coefficient estimates and Fekete-Szegö problem for the class Sµ,k
Σ,y,γ(q,Ψ(s))

Theorem 1. Let g(z) is of the form (1) belong to Sµ,k
Σ,y,γ(q,Ψ(s)). Then

|d2| ≤
|by|

√
|by|√∣∣∣{(Υ2 (y))

2 [2]q [3]
k
q Ψ(s)(q − γ + [3]q µ)−Q (Υ (y) , γ, q, µ)

}∣∣∣ , (7)

|d3| ≤
(by)2

[2]2kq Ψ2(s)(q − γ + [2]q µ)
2
+

|by|
[2]q [3]

k
q Ψ(s)(q − γ + [3]q µ)

, (8)

where

Q (Υ (y) , γ, q, µ) = [2]2kq Ψ2(s)(q − γ + [2]q µ)
{
(Υ2 (y))

2 (1 + γ) + Υ3(y)(q − γ + [2]q µ)
}
.

For δ ∈ R

∣∣d3 − δd22
∣∣ ≤


|by|

[2]q [3]
k
qΨ(s)(q−γ+[3]qµ)

, |1− δ| ≤ J,

|by|3|1−δ|
|{(Υ2(y))

2[2]q [3]
k
qΨ(s)(q−γ+[3]qµ)−Q(Υ(y),γ,q,µ)}| , |1− δ| ≥ J,

(9)

where

J =

∣∣∣{[2]q [3]kq Ψ(s)(q − γ + [3]q µ)− (Υ2(y))
2Q (Υ (y) , γ, q, µ)

}∣∣∣
[2]q [3]

k
q Ψ(s)(q − γ + [3]q µ)

.

Proof. Let g(z) ∈ Sµ,k
Σ,y,γ(q,Ψ(s)). Then, for the analytic functions m(z) and n(z) such

that
m(0) = n(0) = 0

and
|m(z)| < 1 and |n(ω)| < 1, z, ω ∈ D.

By Definition 6, we can write

z∂q(D
k
q gΨ(z)) + µz2∂2q (D

k
q gΨ(z))

(1− γ)Dk
q gΨ(z) + γz∂q(Dk

q gΨ(z))
= 𭟋(y,m (z)) + 1− α

and
ω∂q(D

k
q fΨ(ω)) + µω2∂2q (D

k
q fΨ(ω))

(1− γ)DkfΨ(ω) + γω∂q(Dk
q fΨ(ω))

= 𭟋(y, n (ω)) + 1− α.

Or

z∂q(D
k
q gΨ(z)) + µz2∂2q (D

k
q gΨ(z))

(1− γ)Dk
q gΨ(z) + γz∂q(Dk

q gΨ(z))



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2524

= 1 + Υ1 (y)− a+Υ2 (y)m(z) + Υ3 (y) (m(z))2 + . . . (10)

and

ω∂q(D
k
q fΨ(ω)) + µω2∂2q (D

k
q fΨ(ω))

(1− γ)DkfΨ(ω) + γω∂q(Dk
q fΨ(ω))

= 1 + Υ1 (y)− a+Υ2 (y)n(ω) + scΥ3 (y) (n(ω))
2 + . . . . (11)

Based on (10) and (11), in view of (5), we may deduce

z∂q(D
k
q gΨ(z)) + µz2∂2q (D

k
q gΨ(z))

(1− γ)Dk
q gΨ(z) + γz∂q(Dk

q gΨ(z))

= 1 + Υ2 (y)m1z +
[
Υ2 (y)m2 +Υ3 (y)m

2
1

]
z2 + · · · (12)

and

ω∂q(D
k
q fΨ(ω)) + µω2∂2q (D

k
q fΨ(ω))

(1− γ)DkfΨ(ω) + γω∂q(Dk
q fΨ(ω))

= 1 + Υ2 (y)n1ω +
[
Υ2 (y)n2 +Υ3 (y)n

2
1

]
ω2 + · · · . (13)

It is well known that if

|m(z)| = |m1z +m2z
2 +m3z

3 + ...| < 1, z ∈ D

and
|n(ω)| = |n1ω + n2ω

2 + n3ω
3 + ...| < 1, ω ∈ D,

then
|mi| ≤ 1 and |ni| ≤ 1, for (i ∈ N). (14)

Comparing the coefficients of (12) and (13), we have

[2]kq Ψ(s)(q − γ + [2]q µ)d2 = Υ2(y)m1, (15)

{
[2]q [3]

k
q Ψ(s))(q − γ + [3]q µ)d3

− [2]2kq Ψ2(s)(1 + γ)(q − γ + [2]q µ)d2

}
= Υ2(y)m

2 +Υ3(y)m
2
1, (16)

− [2]kq Ψ(s)(q − γ + [2]q µ)d2 = Υ2(y)n1 (17)

and

− [2]q [3]
k
q Ψ(s))(q − γ + [3]q µ)d3 +

{
2 [2]q [3]

k
q Ψ(s)(q − γ + [3]q µ)

− [2]2kq Ψ2(s)(1 + γ)(q − γ + [2]q µ)
}
d22



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2525

= Υ2(y)n2 +Υ3(y)n
2
1. (18)

From (15) and (17), we can see that

m1 = −n1 (19)

and also
2 [2]2kq Ψ2(s)(q − γ + [2]q µ)

2d2 =
(
m2

1 + n21
)
(Υ2 (y))

2 . (20)

Adding (16) and (18), then we obtain{
2 [2]q [3]

k
q Ψ(s)(q − γ + [3]q µ) −

[2]2kq Ψ2(s)(1 + γ)(q − γ + [2]q µ)
}
d22

= Υ2(y)(m2 + n2) + Υ3(y)(m
2
1 + n21). (21)

Putting the value of m2
1 + n21 from (20) in (21), we get

d22 =
(Υ2(y))

3(m2 + n2)

2
{
(Υ2 (y))

2 [2]q [3]
k
q Ψ(s)(q − γ + [3]q µ)−Q (Υ (y) , γ, q, µ)

} , (22)

where

Q (Υ (y) , γ, q, µ)

= [2]2kq Ψ2(s)(q − γ + [2]q µ)
{
(Υ2 (y))

2 (1 + γ)

+ Υ3(y)(q − γ + [2]q

}
µ),

which yields (7) on using (14). Using (19) in the subtraction of (18) from (16), we obtain

d3 = d22 +
Υ2(y)(m2 − n1)

2 [2]q [3]
k
q Ψ(s)(q − γ + [3]q µ)

. (23)

Then in view of (20), and (23), we get

d3 =
(Υ2(y))

2(m2
1 + n21)

2 [2]2kq Ψ2(s)(q − γ + [2]q µ)
2
+

Υ2(y)(m2 − n2)

2 [2]q [3]
k
q Ψ(s)(q − γ + [3]q µ)

,

which yields (8) on using (14). From (22) and (23), for δ ∈ R, we get

∣∣d3 − δd22
∣∣ = |Υ2(y)|

∣∣∣∣∣
(
T (δ, q, y) +

1

2 [2]q [3]
k
q Ψ(s)(q − γ + [3]q µ)

)
m2 +(

T (δ, q, y)− 1

2 [2]q [3]
k
q Ψ(s)(q − γ + [3]q µ)

)
n2

∣∣∣∣∣ ,



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2526

where

T (δ, q, y) =
(1− δ)

2
{
[2]q [3]

k
q Ψ(s)(q − γ + [3]q µ)− (Υ2 (y))

2Q (Υ (y) , γ, q, µ)
} .

In view of (5), we conclude that

∣∣d3 − δd22
∣∣ ≤


|Υ2(y)|

[2]q [3]
k
qΨ(s)(q−γ+[3]qµ)

; 0 ≤ |T (δ, q, y)| ≤ 1
2[2]q [3]

k
qΨ(s)(q−γ+[3]qµ)

,

2 |Υ2(y)| |T (δ, q, y)| ; |T (δ, q, y)| ≥ 1
2[2]q [3]

k
qΨ(s)(q−γ+[3]qµ)

,

which yields (9). Evidently, this concludes Theorem 1.

Remark 7. For µ = 0, γ = 0, k = 0 and Ψ(s) = 1, in Theorem 1 we obtain Corollary 1
and Corollary 3 proved in [26].

2.1.1. Coefficient estimates and Fekete-Szegö problem for the class LΣ(y, γ, µ, k, q,Ψ(s))

Theorem 2. Let g(z) of the form (1) belong to LΣ(y, γ, µ, k, q,Ψ(s)). Then

|d2| ≤
|b(y)|

√
|b(y)|√∣∣∣(Υ2 (y))

2 [3]k+1
q Ψ(s)(1− γ + [2]q µ)−R1 (Υ, y, γ, µ)

∣∣∣ (24)

and

|d3| ≤
∣∣b2y2∣∣

[2]2k+2
q Ψ2(s)(1− γ + µ)2

+
|b(y)|

2 [3]k+1
q Ψ(s)(1− γ + [2]q µ)

, (25)

where

R1 (Υ, y, γ, µ) = [2]2k+2
q Ψ2(s)(1− γ + µ)

{
(Υ2 (y))

2 γ +Υ3(y)(1− γ + µ)
}
.

For δ ∈ R

∣∣d3 − δd22
∣∣ ≤


|by|

[3]k+1
q Ψ(s)(1−γ+[2]qµ)

, |1− δ| ≤M,

|by|3|1−δ|
|2(by)2[3]k+1

q Ψ(s)(1−γ+[2]qµ)−R2(Υ,y,γ,µ)| , |1− δ| ≥M,
(26)

where

R2 (Υ, y, γ, µ) = [2]2k+2
q Ψ2(s)(1− γ + µ)

{
(by)2 γ +

(
pby2 + ra

)
(1− γ + µ)

}
and

M =
1

[3]k+1
q Ψ(s)(1− γ + [2]q µ)

∣∣∣[3]k+1
q Ψ(s)(1− γ + [2]q µ)−

[2]2k+2
q Ψ2(s)(1− γ + µ)

{
γ +

(
pby2 + ra

(by)2

)
(1− γ + µ)

}∣∣∣∣ . (27)



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2527

Proof. Let g(z) ∈ LΣ(y, γ, µ, k, q,Ψ(s)). Then, for two analytic functions m(z) and
n(z) such that

m(0) = n(0) = 0

and
|m(z)| < 1 and |n(ω)| < 1, z, ω ∈ D.

Using Definition 7, we can write

z∂q(D
kgΨ(z)) + µz2∂2q (D

k
q gΨ(z))

(1− γ) z + γz∂q(Dk
q gΨ(z))

= 𭟋(y,m (z)) + 1− α (28)

and
ω∂q(D

k
q fΨ(ω)) + µω2∂2q (D

k
q fΨ(ω))

(1− γ)ω + γω∂q(Dk
q fΨ(ω))

= 𭟋(y, n (ω)) + 1− α. (29)

Following (10), (11), (12), and (13) in the proof of Theorem 1, one gets the following in
view of (28) and (29):

[2]k+1
q Ψ(s) (1− γ + µ) d2 = Υ2(y)m1, (30)

{
[3]k+1

q Ψ(s)(1− γ + µ [2]q)d3

− [2]2k+2
q Ψ2(s) (1− γ + µ) γd22

}
= Υ2(y)m2 +Υ3(y)m

2
1, (31)

− [2]k+1
q Ψ(s) (1− γ + µ) d2 = Υ2(y)n1 (32)

and

− [3]k+1
q Ψ(s)(1− γ + µ [2]q)d3 +

{
2 [3]k+1

q Ψ(s)(1− γ + µ [2]q)

− [2]2k+2
q Ψ2(s) (1− γ + µ) γd2

}
= Υ2(y)n2 +Υ3(y)n

2
1. (33)

The results (24)-(26) of this theorem now follow from (30)-(33) by applying the procedure
as in Theorem 1 with respect to (15)-(18).

Remark 8. The results obtained in Theorem 2 coincide with Theorem 2.1 of [42] for
k = 0, q → 1− and Ψ(s) = 1.



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2528

2.2. Coefficient estimates and Fekete-Szegö problem for the class BΣ(y, ξ, τ, k, q,Ψ(s))

We derive the estimates for the coefficients |d2 and |d3| and Fekete-Szegö problem in
the following result.

Theorem 3. Let g(z) of the form 1 is in BΣ(y, ξ, τ, k,Ψ(s)). Then

|d2| ≤
|b(y)|

√
|b(y)|√

[3]k+1
q Ψ(s)

(
ξτ
(
[2]q + 1

)
− 1
)
(by)2 −H (s, ξ, τ, b, y)

, (34)

|d3| ≤
(by)2

(2ξτ − 1)2 [2]2k+2
q Ψ2(s)

+
|b(y)|

[3]k+1
q Ψ(s)

(
ξτ
(
[2]q + 1

)
− 1
) , (35)

where

H (s, ξ, τ, b, y) =
{
[2]k+1

q Ψ2(s)(2ξτ (τ − 1)− 2ξτ + 1) (by)2 −

(2ξτ − 1) [2]2k+2
q Ψ2(s)

(
pby2 + ra

)}
.

For δ ∈ R

∣∣d3 − δd22
∣∣ ≤


|by|

[3]k+1
q Ψ(s)(ξτ([2]q+1)−1)

, |1− δ| ≤ Ω,

|by|3|1−δ|
|[3]k+1

q Ψ(s)(ξτ([2]q+1)−1)(by)2−H(s,ξ,τ,b,y)| , |1− δ| ≥ Ω,
(36)

where

Ω =

∣∣∣[3]k+1
q Ψ(s)

(
ξτ
(
[2]q + 1

)
− 1
) (
b2y2

)
−H (s, ξ, τ, b, y)

∣∣∣
4 [3]k+1

q Ψ(s)
(
ξτ
(
[2]q + 1

)
− 1
)
(b2y2)

.

Proof. Let g(z) ∈ BΣ(y, ξ, τ, k, q,Ψ(s)), we have

(1− ξ) + ξ
[
∂q(z∂q

(
Dk

q gΨ(z)
)
)
]τ

∂q(Dk
q gΨ(z))

= 𭟋(y,m (z)) + 1− α, z ∈ D (37)

and
(1− ξ) + ξ

[
∂q(ω∂q

(
Dk

q fΨ(ω)
)
)
]τ

∂q(Dk
q fΨ(ω))

= 𭟋(y, n (ω)) + 1− α, ω ∈ D. (38)

Solving the both side of (37) and (38), we get following equations:

(2ξτ − 1) [2]k+1
q Ψ(s)d2 = Υ2(y)m1, (39)

{
[3]k+1

q Ψ(s)
(
ξτ
(
[2]q + 1

)
− 1
)
d3 −



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2529

[2]k+1
q Ψ2(s)(2ξτ (τ − 1)− 2ξτ + 1)d22

}
= Υ2(y)m2 +Υ3(y)m

2
1, (40)

− (2ξτ − 1) [2]k+1
q Ψ(s)d2 = Υ2(y)n1 (41)

and

− [3]k+1
q Ψ(s)

(
ξτ
(
[2]q + 1

)
− 1
)
d3 +

(
2 [3]k+1

q Ψ(s)
(
ξτ
(
[2]q + 1

)
− 1
)

− [2]k+1
q Ψ2(s)(2ξτ (τ − 1)− 2ξτ + 1)

)
d22

= Υ2(y)n2 +Υ3(y)n
2
1. (42)

By using the same procedure of Theorem 3, we have the required result.

Remark 9. The results obtained in Theorem 3 coincide with Theorem 2.2 of [42], when
k = 0 and Ψ(s) = 1.

In the next section, we present some interesting consequences of our main results.

3. Corollaries and Consequences

Corollary 1. Let g(z) be in the family JΣ(y, k, q,Ψ(s)).Then

|d2| ≤
|by|

√
|by|√∣∣∣{(Υ2 (y))

2 [2]q [3]
k
q Ψ(s)

(
q − 1

2 + 1
2 [3]q

)
−B (s, q, y)

}∣∣∣
and

|d3| ≤
(by)2

[2]2kq Ψ2(s)
(
q − 1

2 + 1
2 [2]q µ

)2 +
|by|

[2]q [3]
k
q Ψ(s)

(
q − 1

2 + 1
2 [3]q

) ,
where

B (s, q, y) = [2]2kq Ψ2(s)

(
q − 1

2
+

1

2
[2]q

){
3

2
(Υ2 (y))

2 +Υ3(y)

(
q − 1

2
+

1

2
[2]q

)}
.

For δ ∈ R

∣∣d3 − δd22
∣∣ ≤


|by|

[2]q [3]
k
qΨ(s)(q− 1

2
+ 1

2
[3]q)

, |1− δ| ≤ J1,

|by|3|1−δ|
|{(Υ2(y))

2[2]q [3]
k
qΨ(s)(q− 1

2
+ 1

2
[3]q)−B(s,q,y)}| , |1− δ| ≥ J1,

where

J1 =

∣∣∣{[2]q [3]kq Ψ(s)
(
q − 1

2 + 1
2 [3]q

)
(Υ2(y))

2 −B (s, q, y)
}∣∣∣

[2]q [3]
k
q Ψ(s)

(
q − 1

2 + 1
2 [3]q

)
(Υ2(y))

2
.



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2530

Corollary 2. Let g(z) ∈ KΣ(y, k, q,Ψ(s)). Then

|d2| ≤
|by|

√
|by|√∣∣∣{(Υ2 (y))

2 [2]q [3]
k
q Ψ(s)

(
q + 1

2 [3]q µ
)
−B1 (s, q, y)

}∣∣∣ ,

|d3| ≤
(by)2

[2]2kq Ψ2(s)
(
q + 1

2 [2]q

)2 +
|by|

[2]q [3]
k
q Ψ(s)

(
q + 1

2 [3]q

) ,
where

B1 (s, q, y) = [2]2kq Ψ2(s)

(
q +

1

2
[2]q

){
(Υ2 (y))

2 +Υ3(y)

(
q +

1

2
[2]q

)}
.

For δ ∈ R

∣∣d3 − δd22
∣∣ ≤


|by|

[2]q [3]
k
qΨ(s)(q+ 1

2
[3]q)

, |1− δ| ≤ J2

|by|3|1−δ|
|{(Υ2(y))

2[2]q [3]
k
qΨ(s)(q+ 1

2
[3]q)−B1(s,q,y)}| , |1− δ| ≥ J2,

where

J2 =

∣∣∣{[2]q [3]kq Ψ(s)
(
q + 1

2 [3]q

)
(Υ2(y))

2 −B1 (s, q, y)
}∣∣∣

[2]q [3]
k
q Ψ(s)

(
q + 1

2 [3]q µ
)
(Υ2(y))

2
.

Corollary 3. Let g(z) ∈ LΣ(y, k, q,Ψ(s)). Then

|d2| ≤
|by|

√
|by|√∣∣∣{(Υ2 (y))

2 [2]q [3]
k
q Ψ(s)

(
q − 1

2 + [3]q

)
−B2 (s, q, y)

}∣∣∣ ,

|d3| ≤
(by)2

[2]2kq Ψ2(s)
(
q − 1

2 + [2]q

)2 +
|by|

[2]q [3]
k
q Ψ(s)

(
q − 1

2 + [3]q

) ,
where

B2 (s, q, y) = [2]2kq Ψ2(s)(q − 1

2
+ [2]q)

{
3

2
(Υ2 (y))

2 +Υ3(y)(q −
1

2
+ [2]q)

}
.

For δ ∈ R

∣∣d3 − δd22
∣∣ ≤


|by|

[2]q [3]
k
qΨ(s)(q− 1

2
+[3]q)

, |1− δ| ≤ J3,

|by|3|1−δ|
|{(Υ2(y))

2[2]q [3]
k
qΨ(s)(q− 1

2
+[3]q)−B2(s,q,y)}| , |1− δ| ≥ J3,

where

J3 =

∣∣∣{[2]q [3]kq Ψ(s)
(
q − 1

2 + [3]q

)
(Υ2(y))

2 −B2 (s, q, y)
}∣∣∣

[2]q [3]
k
q Ψ(s)

(
q − 1

2 + [3]q

)
(Υ2(y))

2
.



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2531

Corollary 4. Let g(z) be in the family MΣ(y, µ, k,Ψ(s)). Then

|d2| ≤
|by|

√
|by|√∣∣∣{(Υ2 (y))

2 [2]q [3]
k
q Ψ(s)

(
q + [3]q µ

)
−B3 (s, q, y)

}∣∣∣ ,

|d3| ≤
(by)2

[2]2kq Ψ2(s)
(
q + [2]q µ

)2 +
|by|

[2]q [3]
k
q Ψ(s)

(
q + [3]q µ

)
where

B3 (s, q, y) = [2]2kq Ψ2(s)
(
q + [2]q µ

){
(Υ2 (y))

2 +Υ3(y)
(
q + [2]q µ

)}
.

For δ ∈ R

∣∣d3 − δd22
∣∣ ≤


|by|

[2]q [3]
k
qΨ(s)(q+[3]qµ)

, |1− δ| ≤ J4,

|by|3|1−δ|
|{(Υ2(y))

2[2]q [3]
k
qΨ(s)(q+[3]qµ)−B3(s,q,y)}| , |1− δ| ≥ J4,

where

J4 =

∣∣∣{[2]q [3]kq Ψ(s)
(
q + [3]q µ

)
(Υ2(y))

2 −B3 (s, q, y)
}∣∣∣

[2]q [3]
k
q Ψ(s)

(
q + [3]q µ

) .

Corollary 5. Let g(z) ∈ NΣ(y, µ, k, q,Ψ(s)). Then

|d2| ≤
|b(y)|

√
|b(y)|√∣∣∣(Υ2 (y))

2 [3]k+1
q Ψ(s)

(
1 + [2]q µ

)
−B3 (s, q, y, µ)

∣∣∣ ,

|d3| ≤
∣∣b2y2∣∣

[2]2k+2
q Ψ2(s)(1 + µ)2

+
|b(y)|

2 [3]k+1
q Ψ(s)

(
1 + [2]q µ

) ,
where

B3 (s, q, y, µ) = [2]2k+2
q Ψ2(s)(1 + µ) {Υ3(y)(1 + µ)} .

For δ ∈ R

∣∣d3 − δd22
∣∣ ≤


|by|

[3]k+1
q Ψ(s)(1+[2]qµ)

, |1− δ| ≤M1,

|by|3|1−δ|
|2(by)2[3]k+1

q Ψ(s)(1+[2]qµ)−B3(s,q,y,µ)| , |1− δ| ≥M1,

where

M1 =
1

[3]k+1
q Ψ(s)

(
1 + [2]q µ

)
(by)2

∣∣∣[3]k+1
q Ψ(s)

(
1 + [2]q µ

)
(by)2 −B3 (s, q, y, µ)

∣∣∣ .



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2532

Corollary 6. Let g(z) ∈ OΣ(y, µ, k,Ψ(s)). Then

|d2| ≤
|b(y)|

√
|b(y)|√∣∣∣(Υ2 (y))

2 [3]k+1
q Ψ(s) [2]q µ− [2]2k+2

q Ψ2(s)µ
{
(Υ2 (y))

2 +Υ3(y)µ
}∣∣∣ ,

|d3| ≤
∣∣b2y2∣∣

[2]2k+2
q Ψ2(s)µ2

+
|b(y)|

2 [3]k+1
q Ψ(s) [2]q µ

.

For δ ∈ R

∣∣d3 − δd22
∣∣ ≤


|by|

[3]k+1
q Ψ(s)[2]qµ

, |1− δ| ≤M2,

|by|3|1−δ|
|2(by)2[3]k+1

q Ψ(s)[2]qµ−[2]2k+2
q Ψ2(s)µ{(by)2γ+(pby2+ra)µ}| , |1− δ| ≥M2,

where

M2 =
1

[3]k+1
q Ψ(s) [2]q µ

∣∣∣∣[3]k+1
q Ψ(s) [2]q µ− [2]2k+2

q Ψ2(s)µ

{
1 +

(
pby2 + ra

(by)2

)
µ

}∣∣∣∣ .
Corollary 7. Let g(z) ∈ PΣ(y, ξ, k,Ψ(s)). Then

|d2| ≤
|b(y)|

√
|b(y)|√

[3]k+1
q Ψ(s)

(
ξ
(
[2]q + 1

)
− 1
)
(by)2 −B4 (r, s, q, ξ)

and

|d3| ≤
(by)2

(2ξ − 1)2 [2]2k+2
q Ψ2(s)

+
|b(y)|

[3]k+1
q Ψ(s)

(
ξ
(
[2]q + 1

)
− 1
) ,

where

B4 (r, s, q, ξ) =
{
[2]k+1

q Ψ2(s)(1− 2ξ) (by)2

− (2ξ − 1) [2]2k+2
q Ψ2(s)

(
pby2 + ra

)}
.

For δ ∈ R

∣∣d3 − δd22
∣∣ ≤


|by|

[3]k+1
q Ψ(s)(ξ([2]q+1)−1)

, |1− δ| ≤ Ω1,

|by|3|1−δ|
|[3]k+1

q Ψ(s)(ξ([2]q+1)−1)(by)2−B4(r,s,q,ξ)| , |1− δ| ≥ Ω1,

where

Ω1 =

∣∣∣[3]k+1
q Ψ(s)

(
ξ
(
[2]q + 1

)
− 1
)
b2y2 −B4 (r, s, q, ξ)

∣∣∣
4 [3]k+1

q Ψ(s)
(
ξ
(
[2]q + 1

)
− 1
)
b2y2

.



N. K. Mishra, M. F. Khan, S. A. Lone / Eur. J. Pure Appl. Math, 17 (4) (2024), 2516-2537 2533

Corollary 8. Let g(z) be in the family QΣ(y, τ, k,Q,Ψ(s)). Then

|d2| ≤
|b(y)|

√
|b(y)|√

[3]k+1
q Ψ(s)

(
ξ
(
[2]q + 1

)
− 1
)
(by)2 −B5 (r, s, τ, y)

and

|d3| ≤
(by)2

(2τ − 1)2 [2]2k+2
q Ψ2(s)

+
|b(y)|

[3]k+1
q Ψ(s)

(
τ
(
[2]q + 1

)
− 1
) ,

where

B5 (r, s, τ, y) =
{
[2]k+1

q Ψ2(s)(2τ2 − 4τ + 1) (by)2

−
{
(2τ − 1) [2]2k+2

q Ψ2(s)
(
pby2 + ra

)
.

For δ ∈ R

∣∣d3 − δd22
∣∣ ≤


|by|

[3]k+1
q Ψ(s)(τ([2]q+1)−1)

, |1− δ| ≤ Ω2,

|by|3|1−δ|
|[3]k+1

q Ψ(s)(τ([2]q+1)−1)(by)2−B5(r,s,τ,y)| , |1− δ| ≥ Ω2,

where

Ω2 =

∣∣∣[3]k+1
q Ψ(s)

(
ξ
(
[2]q + 1

)
− 1
)
−B6 (r, s, τ, y)

∣∣∣
4 [3]k+1

q Ψ(s)
(
τ
(
[2]q + 1

)
− 1
)

and

B6 (r, s, τ, y) =
{
[2]k+1

q Ψ2(s)(2τ2 − 4τ + 1) −

(ξτ − 1) [2]2k+2
q Ψ2(s)

(
pby2 + qa

b2y2

)}
.

4. Conclusions

This research aims to introduce new subfamilies of bi-univalent functions within the
open unit disk, leveraging the connections between Horadam polynomials, modified Sig-
moid functions, and the principles of subordination. By utilizing the power of q-calculus,
quantum difference operators, and the modified Sigmoid function, we define and investi-
gate three novel subclasses of bi-univalent functions linked to Horadam polynomials. Our
study yields estimates for the Fekete-Szegö functional problems and the Taylor-Maclaurin
coefficients |d2| and |d3| for each of these subclasses. Furthermore, by examining the vari-
ables in our main results, we uncover additional original findings. This methodology paves
the way for the introduction of new subclasses of bi-univalent functions related to other
generating functions, such as Fibonacci numbers and square-root functions. By applying
the Faber polynomial technique, we can derive bounds for the nth coefficients of these sub-
classes, specifically the first two initial coefficients and Fekete-Szegö functional problems.



REFERENCES 2534

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at Saudi
Electronic University for funding this research (8399).

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