EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5841
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Properties of Bazilevič Functions Involving q-analogue
of the Generalized M-series

Kadhavoor R. Karthikeyan1,∗, Dharmaraj Mohankumar2, Daniel Breaz3,∗

1 Department of Applied Mathematics and Science, College of Engineering,
National University of Science & Technology, Muscat P.O. Box 620, Oman
2 Department of Mathematics for Innovation, Saveetha School of Engineering,
Saveetha Institute of Medical and Technical Sciences(SIMATS), Thandalam, Chennai, 602
105, India
3 Department of Mathematics, “1 Decembrie 1918” University of Alba Iulia, Alba Iulia,
510009, Romania

Abstract. With primary motive to unify and extend the various well-known studies, we define a
new family of differential operator using the q-analogue of the generalized M -series. The gener-
alized M -series unifies two well-known and extensively used special functions namely generalized
hypergeometric function and Mittag-Leffler function. Making use of the defined operator, we define
a new family of analytic functions expressed as a combination two differential characterizations.
The combination of differential characterizations involving the operator not only unifies studies
of starlike, convex, Bazilevič and α-convex function classes, it extends to new classes. Estimates
involving the initial coefficients of the functions, which belong to the defined function class are
our main results. Some examples along with graphs have been used to establish the inclusion and
closure properties.

2020 Mathematics Subject Classifications: 30C45; 30C50, 30C55

Key Words and Phrases: Generalized M -series, Mittag-Leffler function, generalized hyper-
geometric function, quantum calculus, analytic function, univalent function, Schwarz function,
Bazilevič functions, subordination, coefficient inequalities, Fekete-Szegő inequality.

1. Introduction

Fractional calculus has been a very important tool in the mathematical analysis irre-
spective of whether the study involves theoretical aspects or applications oriented. It is
not a new calculus, in fact it is as old as classical calculus. But its has acquired the inter-
ests of several researchers since it fits very well in modelling of problems involving natural

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5841

Email addresses: karthikeyan@nu.edu.om (K. R. Karthikeyan),
dmohankumarmaths@gmail.com (D. Mohankumar), dbreaz@uab.ro (D. Breaz)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



K. R. Karthikeyan, D. Mohankumar, D. Breaz / Eur. J. Pure Appl. Math, 18 (1) (2025), 5841 2 of 19

phenomena. Its applications can be found in various scientific disciplines like biology,
chemistry, physics, acoustics, materials science, fluid mechanics and dynamical systems.
Mittag-Leffler function is one such special function which cannot be avoided if one has too
delve into the field fractional calculus.

Fractional q-calculus is an extension of the classical fractional calculus aimed at the
discretization, unification and generalization. It mainly unifies the study of continuous
and discrete analysis. Quantum calculus is essentially motivated by the concept of finite
difference rescaling. To be precise, it is nothing but a ratio that is similar to one used
in Newton’s divided difference table. The primary reason for this calculus to be relevant
even today is due to the fact the all the concepts of classical calculus cannot be translated
to quantum calculus, even the basic chain rule needs adaptation. Also the interest are
because quantum computing models involves lots of mathematics. It has a lot of appli-
cations in different mathematical areas such as number theory, combinatorics, orthogonal
polynomials, basic hyper-geometric functions and the theory of relativity. Refer to Kac
and Cheung [27] for its definition and basic properties. Refer to [2, 3, 11, 12] for its recent
developments.

The Meijer G-function and Fox’s H-function are most generalized function to which
nearly all special functions will form to be their special cases. A brief overview of the
some special functions which helps in unification with the generalized M -series (see [45,
Eq. 1]).

Let Λ, R, C and N denote the unit disc, set of real numbers, set of complex numbers
and set of natural numbers respectively. We denote Θ to be the class of functions χ(ξ)
analytic in Λ with the normalization χ(0) = χ′(0) − 1 = 0 which will result in a series of
the form

χ(ξ) = ξ +
∞∑
n=2

φnξ
n, (ξ ∈ Λ; φn ∈ C) . (1)

For κi ∈ C (i = 1, . . . , r) and σj ∈ C\Z−
0 = {0, −1, . . .} (j = 1, . . . , s), the Fox–Wright

function rΨs, which is defined by (see ([51, Equation 1.6]), ([53, p. 19]) and ([54, p. 21]))

rΨs

[
(κ1, A1) . . . (κr, Ar)
(σ1, B1) . . . (σs, Bs)

; ξ

]
=

∞∑
n=0

∏r
i=1 Γ(κj +Ajn)∏s
j=1 Γ(σj +Bjn)

ξn

n!
. (2)

where Re(Ai) > 0, (i = 1, . . . , r) and Re(Bj) > 0 ∈ C (j = 1, . . . , s) with 1 +

Re
(∑s

j=1Bj −
∑r

i=1Ai

)
≥ 0. For discussion on the convergence of the series (2), re-

fer to Srivastava ([52], Definition 2).
Lin and Srivastava [32, eq. 8] introduced a following generalization of the well-known

Hurwitz–Lerch zeta function ϕk, ϵκ, σ(ξ,m, κ) given by

ϕk, ϵκ, σ(ξ,m, κ) =

∞∑
n=0

(κ)kn
(σ)ϵn

ξn

(n+ κ)m
,

where κ ∈ C; σ, κ ∈ C \ Z−
0 ; k, ϵ ∈ R+; k < ϵ when m, ξ ∈ C; k = ϵ and m ∈ C when

|ξ| < 1; k = ϵ and Re(m − κ + σ) > 1 when |ξ| = 1. These types of generalizations are



K. R. Karthikeyan, D. Mohankumar, D. Breaz / Eur. J. Pure Appl. Math, 18 (1) (2025), 5841 3 of 19

very common as it not only unifies studies, it extends various studies. In some cases, such
generalizations requires adaptation or deviations and it has proved to be very useful tool
in analysis.

The study of generalized M -series is one such generalization which has proved to be an
important tool in the studies pertaining to duality theory. The generalized M -series[45,
Eq. 1] (also see [55]) defined to unify the studies pertaining to Mittag-Leffler function and
Gaussian hypergeometric function, is given by

rMη,θ
s (ξ) = rMη,θ

s (κ1, . . . , κr;σ1, . . . , σs; ξ) =

∞∑
n=0

(κ1)n . . . (κr)n
(σ1)n . . . (σs)n

ξn

Γ(nη + θ)
, (3)

ξ, η, θ ∈ C,Re(η) > 0 and (κi)n, (σj)n are the well-known Pochhammer symbol. Further
the primary condition for the existence of the series (3) is that the denominator terms
σ′js, (j = 1, 2, . . . s) are never zero or negative integer. Whereas if any of the numerator
terms κ′js, (j = 1, 2, . . . r) is zero or negative integer, then the infinite series terminates
to be polynomial in ξ. Note that generalized M -series can be represented in terms of the
Fox–Wright function as follow:

rMη,θ
s (ξ) = ε r+1Ψs+1

[
(κ1, 1) . . . (κr, 1), (1, 1)
(σ1, 1) . . . (σs, 1)(θ, η)

; ξ

]
, ε =

r∏
j=1

Γ(σj)/
s∏

j=1

Γ(κj)

Hence the convergence of the series (3) is same as that of Fox–Wright function.
The q-analogue of the generalized M -series (3) was studied by Shimelis and Suthar

[46–48] and is defined by

rMη,θ
s (q, ξ) =

∞∑
n=0

(qκ1 ; q)n . . . (q
κr ; q)n

(qσ1 ; q)n . . . (q
σs ; q)n

[
(−1)n qn(n−1)/2

]s−r+1 ξn

Γq(nη + θ)
, (4)

where Γq(.) is the q-gamma function and (qκi ; q)n, (qσj ; q)n ; κi, σj ̸= 0,−1, . . . (i =
1, 2, . . . r; j = 1, 2, . . . , s) are a q-analogue of Pochhammer symbol. Again, the conver-
gence details of (4) can be found in Srivastava ([52], Definition 2).

The primary aim of this study is unification, extension and discertization. For this
purpose, we will define a new family of linear operator involving the q-analogue of the
generalized M -series. In this study, we aim to analyze the behaviour and geometrical
implications when two differential characterization are expressed as convex combination.
We will obtain the coefficient inequalities which would in turn help us understand algebraic
and geometric properties.

1.1. New family of generalized differential operators and its special cases.

Corresponding to a function Gη, θ
r, s (κ1, σ1; ξ) defined by

Gη, θ
r, s (κ1, σ1; ξ) := ξΓ(θ)

[
rMη,θ

s+1(κ1, . . . , κr;σ1, . . . , σs, 1; ξ)
]
. (5)



K. R. Karthikeyan, D. Mohankumar, D. Breaz / Eur. J. Pure Appl. Math, 18 (1) (2025), 5841 4 of 19

We now define the following operator Dm
λ (κ1, σ1; η, θ)χ : Λ −→ Λ by

D0
λ(κ1, σ1; η, θ)χ = χ(ξ) ∗ Gη, θ

r, s (κ1, σ1; ξ)

D1
λ(κ1, σ1; η, θ)χ = (1 − λ)

(
χ(ξ) ∗ Gη, θ

r, s (κ1, σ1; ξ)
)

+ λ ξ
(
χ(ξ) ∗ Gη, θ

r, s (κ1, σ1; ξ)
)′

(6)

Dm
λ (κ1, σ1; η, θ)χ = D1

λ

[
Dm−1

λ (κ1, σ1; η, θ)χ
]
. (7)

Here ∗ denotes the Hadamard product or convolution. If χ ∈ Θ, then from (6) and (7) we
may easily deduce that

Dm
λ (κ1, σ1; η, θ)χ = ξ +

∞∑
n=2

[
1 + (n− 1)λ

]m (κ1)n−1 . . . (κr)n−1

(σ1)n−1 . . . (σs)n−1

Γ(θ)φnξ
n

(n− 1)!Γ(η(n− 1) + θ)

(8)
where κj ∈ C (j = 1, . . . , r); σj ∈ C \ Z−

0 = {0, −1, . . .} (j = 1, . . . , s); m ∈ N0; λ ≥
0; η, θ ∈ C and Re(η) > 0. Letting η = 0 in (8), we get the operator studied by Selvaraj
and Karthikeyan [42, Eq. 1.5]. Letting r = 2, s = 1, κ1 = σ1 and κ2 = 1 in (8), we get
the operator

Dm
λ (η, θ)χ(ξ) = ξ +

∞∑
n=2

[
1 + (n− 1)λ

]m Γ(θ)φnξ
n

Γ(η(n− 1) + θ)
. (9)

The operator Dm
λ (η, θ)χ was introduced by Elhaddad et al. [25, Eq 1.6] and was further

studied by Mashwan et al. [35, Eq. 16]. For the choice of m = 0 in (8), the operator
Dm

λ (κ1, σ1; η, θ)χ reduces to the well-known Dziok- Srivastava operator [22]. The operator
recently introduced by Breaz et al. [13, 14] (also see [58]), Cağlar et al [19] and Cang
and Liu [17] are closely related to the operator Dm

λ (κ1, σ1; η, θ)χ, in fact we could have
obtained the same operators if we had defined the equation (5) in the form

Gη, θ
r, s (κ1, σ1; ξ) :=

Γ(η + θ)
∏r

j=1 Γ(σj)∏s
j=1 Γ(κj)

[
rMη,θ

s (κ1, . . . , κr;σ1, . . . , σs; ξ) −
1

Γ(θ)

]
.

Also many (well known and new) integral and differential operators can be obtained by
specializing the parameters involved in Dm

λ (κ1, σ1; η, θ)χ.

1.2. q-analogue of the operator Dm
λ (κ1, σ1; η, θ)χ.

The q-analogue of the operator Dm
λ (κ1, σ1; η, θ)χ will be of the form

Jm
λ (κ1, σ1; η, θ; q, ξ)χ = ξ+

∞∑
n=2

[
1−λ+[n]qλ

]m (κ1; q)n−1 . . . (κr; q)n−1

(q; q)n−1 (σ1; q)n−1 . . . (σs; q)n−1

Γq(θ)φnξ
n

Γq(η(n− 1) + θ)

(10)
For κi = qci , σj = qdj , ci ∈ C, dj ∈ C \ Z−1

0 , (i = 1, . . . , r; j = 1, . . . , s) and q → 1−
in (10), the operator Jm

λ (κ1, σ1; η, θ; q, ξ)χ reduces to the operator Dm
λ (c1, d1; η, θ)χ.



K. R. Karthikeyan, D. Mohankumar, D. Breaz / Eur. J. Pure Appl. Math, 18 (1) (2025), 5841 5 of 19

Letting η = 0 in Jm
λ (κ1, σ1; η, θ; q, ξ)χ, we get the operator introduced and studied by

Reddy et al [40, Eq. 8]. Further setting η = m = 0 in (10), we get the operator introduced
and studied by Darus in [21, Eq. 3]. The q-Calson-Shaffer operator [43], q-Ruscheweyh
derivative operator [30], q-Salagean operators [7] and various other operators involving
Mittag-Leffler function are the special cases of the operator Jm

λ (κ1, σ1; η, θ; q, ξ)ξ.
The study of various subclasses of analytic functions involving with various special

functions was spotlighted after De Branges’s used it in the proof of Bieberbach conjecture.
It should be noted that various convolution properties studied by Ruscheweyh in [41] was
a stimulant to study this duality theory. After finding several applications of convolutions
in various fields, he posed many questions which led to the development in this duality
theory. Refer to [6, 15, 16, 24, 29, 31] for the recent developments pertaining to this duality
theory

1.3. Short introduction to geometric function theory.

We call F (see [18]) to denote the class of functions with normalization p(0) = 1 which
satisfies Re (p(ξ)) > 0, ξ ∈ Λ. We denote the subclasses of Θ namely starlike and convex
functions which satisfies the following respective differential inclusions

ξχ′(ξ)

χ(ξ)
∈ F and

(ξχ′(ξ))′

χ′(ξ)
∈ F .

We denote the class of starlike and convex functions by S∗ and C respectively. Expressing
the analytic characterizations of S∗ and C as a convex combination, Mocanu studied the
so-called δ-convex functions defined by

(1 − δ)
ξχ′(ξ)

χ(ξ)
+ δ

(ξχ′(ξ))′

χ′(ξ)
∈ F , (χ ∈ Θ; 0 ≤ δ ≤ 1).

Here we will denote the class of δ-convex functions as MC(δ).
Ma-Minda [34] obtained the coefficient estimates of a class of function Ψ ∈ F which

are starlike with respect to 1 and has a series expansion of the form

Ψ(ξ) = 1 + ψ1ξ + ψ2ξ
2 + ψ3ξ

3 + · · · , (ψ1 > 0; ξ ∈ Λ). (11)

Motivated by

S∗(Ψ) :=

{
χ ∈ Θ;

ξχ′(ξ)

χ(ξ)
≺ Ψ(ξ)

}
and

C(Ψ) :=

{
χ ∈ Θ;

(ξχ′(ξ))′

χ′(ξ)
≺ Ψ(ξ)

}
.

Replacing the Ma-Minda function Ψ in S∗(Ψ) and C(Ψ) with some special functions,
several authors studied interesting subclasses of starlike and convex functions. Here we
will tabulate only a few of them which was studied for class of analytic functions.



K. R. Karthikeyan, D. Mohankumar, D. Breaz / Eur. J. Pure Appl. Math, 18 (1) (2025), 5841 6 of 19

Conic Region Ψ(ξ) Reference

Right-half of the
lemniscate of
Bernoulli

√
1 + ξ Sokó l[49, 50]

Left-half of the
lemniscate of
Bernoulli

√
2 −

(√
2 − 1

)√ 1−ξ

1+2(
√
2−1)ξ

Mendiratta et al.
[36]

Cardioid 1 + 4ξ
3 + 2ξ2

3 Sharma et al. [44]

Crescent or Lune
shape

ξ +
√

1 + ξ2 Raina and
Sokó l[39]

Limacon 1 +
√

2ξ + ξ2

2 Cho et al. [20]

Nephroid 1 + ξ − ξ3

3 Wani and Swami-
nathan [56]

Table 1: Study of subclasses of analytic functions impacted by conic regions

1.4. New Subclass of Analytic Functions.

Motivated by [1, 8–10, 28], we define the following.

Definition 1. For ω ≥ 0 and δ ∈ C such that Re(δ) > 0, a function χ belongs to the class
BSm,ω

λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)) if it satisfies{
(1 − δ)

(
Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

ξ

)ω

+ δ
ξ1−ωJm

λ (κ1, σ1; η, θ; q, ξ)χ
′(ξ)[

Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

]1−ω

}
≺ Ψ(ξ),

(12)
where Ψ(ξ) ∈ F has a power series representation of the form (11).

Remark 1. Now we will discuss few special cases of our class BSm,ω
λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)):

(i) Letting r = 2, s = 1, κ1 = σ1, σ2 = q and q → 1− in (12), we get the class
Mm,ω

λ (η, θ; δ; k; ρ) which satisfies the condition

(1 − δ)

(
Dm

λ (η, θ)χ(ξ)

ξ

)ω

+ δ
ξ1−ωDm

λ (η, θ)χ′(ξ)[
Dm

λ (η, θ)χ(ξ)
]1−ω ∈ Fk(ρ)

where Dm
λ (η, θ)χ is given by (9) and Fk(ρ) consist of functions in F satisfying∫ 2π

0

∣∣∣∣Re p(ξ) − ρ

1 − ρ

∣∣∣∣ dν ≤ kπ, (ξ = reiν ; k ≥ 2; 0 ≤ ρ < 1).

The class Mm,ω
λ (η, θ; δ; k; ρ) was studied by Ahuja et al. [1, Definition 1].



K. R. Karthikeyan, D. Mohankumar, D. Breaz / Eur. J. Pure Appl. Math, 18 (1) (2025), 5841 7 of 19

(ii) Setting r = 2, s = 1, κ1 = σ1, σ2 = q, m = η = 0 and q → 1− in Definition 1, we
get

BS(δ; Ψ) =

{
χ ∈ Θ : (1 − δ)

(
χ(ξ)

ξ

)ω

+ δ
ξχ′(ξ)

χ(ξ)

(
χ(ξ)

ξ

)ω

≺ Ψ(ξ)

}
.

(iii) Letting δ = 1 and ψ(ξ) = (1 + ξ)/(1 − ξ) in BS(δ; Ψ), we get the famous Bazilevič
class.

Specializing the function Ψ(ξ) in the Definition 1, we can obtain the classes studied by
various authors.

We will need the following lemmas to establish our main results.

Lemma 1. [23, Theorem 1] If L(ξ) = 1 +
∞∑
r=1

ℓrξ
r ∈ F , and ρ ∈ C, then

|ℓε − ρℓrℓε−r| ≤ 2 max {1; |2ρ− 1|} ,

for all 1 ≤ r ≤ ε− 1.

Note that the above results is generalization of the well-known results of Ma-Minda
[34, p. 162] and Livingston [33, Lemma 1].

Lemma 2. [26] Let g be convex in Λ, with g(0) = d, ϱ ̸= 0 and Re(ϱ) > 0. Suppose that
∆(ξ) is analytic Λ, which is given by

∆(ξ) = d+ dnξ
n + dn+1ξ

n+1 + · · · , ξ ∈ Λ. (13)

If

∆(ξ) +
ξ∆

′
(ξ)

ϱ
≺ g(ξ),

then
∆(ξ) ≺ q(ξ) ≺ g(ξ),

where

q(ξ) =
ϱ

n ξϱ/n

∫ ξ

0
g(t) t(ϱ/n)−1dt.

The function q is convex and is the best (a, n)-dominant.

2. Inclusion Relations and Integral Representations

Theorem 1. Let the function Ψ(ξ) defined as in (11) be convex univalent in Λ. Let
χ ∈ BSm,ω

λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)) with Re(δ) > 0 and ω ̸= 0, then(
Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

ξ

)ω

≺ q(ξ) =
ω

δ
ξ

−ω
δ

∫ ξ

0
t
ω
δ
−1Ψ(t)dt ≺ Ψ(ξ). (14)



K. R. Karthikeyan, D. Mohankumar, D. Breaz / Eur. J. Pure Appl. Math, 18 (1) (2025), 5841 8 of 19

and for ω = 0, we have

Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ) = ξ exp

{∫ ξ

0

Ψ[w(t)] − 1

δt
dt

}
, (15)

where w(ξ) is the Schwarz function and q(ξ) is the best dominant.

Proof. Let h(ξ) be defined by

h(ξ) =

(
Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

ξ

)ω

, ξ ∈ Λ. (16)

Then the function h(ξ) is of the form h(ξ) = 1 + c1ξ + c2ξ
2 + · · · and is analytic in Λ.

Differentiating both sides of (16) and by simplifying, we have

(1− δ)
(
Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

ξ

)ω

+ δ
ξ1−ωJm

λ (κ1, σ1; η, θ; q, ξ)χ
′(ξ)[

Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

]1−ω = h(ξ) +
δ

ω
ξh′(ξ).

(17)
By hypothesis χ ∈ BSm,ω

λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)), so from Definition 1 we have

h(ξ) +
δ

ω
ξh′(ξ) ≺ Ψ(ξ).

Applying Lemma 2 to (17) with ϱ = ω
δ and n = 1, we get(

Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

ξ

)ω

≺ ω

δ
ξ

−ω
δ

∫ ξ

0
t
ω
δ
−1Ψ(t)dt ≺ Ψ(ξ). (18)

Hence the proof (14). Letting ω = 0 in (12), we get

d

dz
log

[
Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

ξ

]
=

Ψ[w(ξ)] − 1

δξ
.

On integrating the above expression, we get the result (15).

Remark 2. Note that for the result (15), the necessity of Ψ(ξ) to be convex is not required.

Remark 3. Now we will discuss the benefits of studying the class involving an operator.
Letting ω = 1 in Theorem 1, (14) will become

Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ) ≺ 1

δ
ξ1−

1
δ

∫ ξ

0
t
ω
δ
−1Ψ(t)dt. (19)

Now setting r = 2, s = 1, κ1 = σ1, σ2 = q, η = 0 in (19), we can have

(1 − λ)χ(ξ) + λξχ′(ξ) ≺ 1

δ
ξ1−

1
δ

∫ ξ

0
t
ω
δ
−1Ψ(t)dt = k(ξ). (20)



K. R. Karthikeyan, D. Mohankumar, D. Breaz / Eur. J. Pure Appl. Math, 18 (1) (2025), 5841 9 of 19

By Lemma 2, the function k(ξ) is convex provided Ψ(ξ) is convex univalent. Now let us
suppose that k(ξ) is convex univalent in Λ. Then we have the following cases: For λ = 0,
we have χ(ξ) = k[w(ξ)], where w(ξ) is the Schwarz function. For 0 < λ ≤ 1, we get

χ(ξ) =
1

λ
ξ1−

1
λ

∫ ξ

0
u

1
λ
−2k [w(u)] du,

where k(ξ) is given as in (20).

Remark 4. Theorem 1 is not valid for δ = 0. From (17), it can be easily seen that if
δ = 0 we can get (

Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

ξ

)ω

≺ Ψ(ξ).

Corollary 1. Let χ ∈ BS(δ) with Re(δ) > 0 , then we have(
χ(ξ)

ξ

)ω

≺ q(ξ) =
ω

δ
ξ

−ω
δ

∫ ξ

0
t
ω
δ
−1

(
1 + t

1 − t

)
dt ≺ 1 + ξ

1 − ξ
, (ω ̸= 0),

χ(ξ) = ξ exp

{∫ ξ

0

2w(t)

δt [1 − w(t)]
dt

}
, (ω = 0),

where q(ξ) is the best dominant and w(ξ) is the Schwarz function.

Proof. Clearly Ψ(ξ) = 1+ξ
1−ξ maps Λ onto a convex domain. So letting r = 2, s = 1,

κ1 = σ1, σ2 = q, m = η = 0, Ψ(ξ) = (1+ ξ)/(1− ξ) in Theorem 1, we can get the assertion
of the Corollary.

3. Coefficient Inequalities

We will obtain the bounds for the initial coefficients and solution to the Fekete-Szegő
problem for χ ∈ BSm,ω

λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)).

Theorem 2. Let χ(ξ) ∈ BSm,ω
λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)) and ω, δ be chosen such (ω + (n −

1)δ) ̸= 0, for n = 2, 3, 4, . . ., then we have

|φ2| ≤
ψ1

|(ω + δ) Γ2|
(21)

|φ3| ≤
ψ1

|(ω + 2δ)Γ3|
max

{
1;

∣∣∣∣ψ2

ψ1
− (ω − 1)(ω + 2δ)ψ1

2(ω + δ)2

∣∣∣∣} (22)

and for all ρ ∈ C

∣∣φ3 − ρφ2
2

∣∣ ≤ ψ1

|(ω + 2δ)Γ3|
max

{
1;

∣∣∣∣∣ψ2

ψ1
− (ω − 1)(ω + 2δ)ψ1

2(ω + δ)2
− ρψ1 (ω + 2δ) Γ3

(ω + δ)2 Γ2
2

∣∣∣∣∣
}
. (23)

The inequalities are sharp.



K. R. Karthikeyan, D. Mohankumar, D. Breaz / Eur. J. Pure Appl. Math, 18 (1) (2025), 5841 10 of 19

Proof. As χ(ξ) ∈ BSm,ω
λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)), by (12), we have

(1− δ)
(
Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

ξ

)ω

+ δ
ξ1−ωJm

λ (κ1, σ1; η, θ; q, ξ)χ
′(ξ)[

Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

]1−ω = Ψ [w(ξ)] . (24)

Equivalently, for an arbitrary function ϑ of the form ϑ(ξ) = 1 +
∑∞

k=1 ϑnξ
n ∈ F , the

function w(ξ) can be written in the form by

ϑ(ξ) =
1 + w(ξ)

1 − w(ξ)
, ξ ∈ Λ.

The right side of (24) will be of the form

Ψ[w(ξ)] = 1 +
ϑ1ψ1

2
ξ +

ψ1

2

[
ϑ2 −

ϑ21
2

(
1 − ψ2

ψ1

)]
ξ2 + · · · . (25)

The left hand side of (24) will be of the form

(1 − δ)

(
Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

ξ

)ω

+ δ
ξ1−ωJm

λ (κ1, σ1; η, θ; q, ξ)χ
′(ξ)[

Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

]1−ω

= 1 + (ω + δ)φ2Γ2ξ + (ω + 2δ)

[
φ3Γ3 +

(ω − 1)φ2
2Γ

2
2

2

]
ξ2 + · · · . (26)

From (26) and (25), we obtain

φ2 =
ϑ1ψ1

2 (ω + δ) Γ2
(27)

and

φ3 =
ψ1

2(ω + 2δ)Γ3

[
ϑ2 −

ϑ21
2

(
1 − ψ2

ψ1
+

(ω − 1)(ω + 2δ)ψ1

2(ω + δ)2

)]
. (28)

Equation (21) can be obtained by applying |ϑ1| ≤ 2 ([38, p. 41]) in (27). Using Lemma 1
in (28), we get (22).

Now to prove (23), we consider

∣∣φ3 − ρφ2
2

∣∣ =

∣∣∣∣∣ ψ1

2(ω + 2δ)Γ3

[
ϑ2 −

ϑ21
2

(
1 − ψ2

ψ1
+

(ω − 1)(ω + 2δ)ψ1

2(ω + δ)2

)]
− ρϑ21ψ

2
1

4 (ω + δ)2 Γ2
2

∣∣∣∣∣
=

∣∣∣∣∣ ψ1

2(ω + 2δ)Γ3

[
ϑ2 −

ϑ21
2

(
1 − ψ2

ψ1
+

(ω − 1)(ω + 2δ)ψ1

2(ω + δ)2
+
ρψ1 (ω + 2δ) Γ3

(ω + δ)2 Γ2
2

)] ∣∣∣∣∣.

=

∣∣∣∣∣ ψ1

2(ω + 2δ)Γ3

[
ϑ2 −

ϑ21
2

+
ϑ21
2

(
ψ2

ψ1
− (ω − 1)(ω + 2δ)ψ1

2(ω + δ)2
− ρψ1 (ω + 2δ) Γ3

(ω + δ)2 Γ2
2

)] ∣∣∣∣∣.



K. R. Karthikeyan, D. Mohankumar, D. Breaz / Eur. J. Pure Appl. Math, 18 (1) (2025), 5841 11 of 19

≤ ψ1

2 |(ω + 2δ)Γ3|

[
2 +

|ϑ1|2

2

(∣∣∣∣∣ψ2

ψ1
− (ω − 1)(ω + 2δ)ψ1

2(ω + δ)2
− ρψ1 (ω + 2δ) Γ3

(ω + δ)2 Γ2
2

∣∣∣∣∣− 1

)]
. (29)

Denoting

B :=

∣∣∣∣∣ψ2

ψ1
− (ω − 1)(ω + 2δ)ψ1

2(ω + δ)2
− ρψ1 (ω + 2δ) Γ3

(ω + δ)2 Γ2
2

∣∣∣∣∣ ,
if B ≤ 1, from (29) we obtain ∣∣φ3 − ρφ2

2

∣∣ ≤ ψ1

|(ω + 2δ)Γ3|
. (30)

Further, if B ≥ 1 from (29) we deduce

∣∣φ3 − ρφ2
2

∣∣ ≤ ψ1

|(ω + 2δ)Γ3|

(∣∣∣∣∣ψ2

ψ1
− (ω − 1)(ω + 2δ)ψ1

2(ω + δ)2
− ρψ1 (ω + 2δ) Γ3

(ω + δ)2 Γ2
2

∣∣∣∣∣
)
. (31)

An examination of the proof shows that the equality for (30) holds if ϑ1 = 0, ϑ2 = 2.

Equivalently, by Lemma 1 we have Ψ(ξ2) = Ψ2(ξ) =
1 + ξ2

1 − ξ2
. Therefore, the extremal

function of the class BSm,ω
λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)) is given by

(1 − δ)

(
Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

ξ

)ω

+ δ
ξ1−ωJm

λ (κ1, σ1; η, θ; q, ξ)χ
′(ξ)[

Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

]1−ω = Ψ2(ξ
2).

Similarly, the equality for (31) holds if ϑ2 = 2. Equivalently, by Lemma 1 we have

Ψ(ξ) = Ψ1(ξ) =
1 + ξ

1 − ξ
. Therefore, the extremal function in BSm,ω

λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)) is

given by

(1 − δ)

(
Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

ξ

)ω

+ δ
ξ1−ωJm

λ (κ1, σ1; η, θ; q, ξ)χ
′(ξ)[

Jm
λ (κ1, σ1; η, θ; q, ξ)χ(ξ)

]1−ω = Ψ1(ξ),

and the proof of the theorem is complete.

Corollary 2. If χ(ξ) ∈ BS(δ; Ψ)( see Remark1 (2)) and ω, δ be chosen such (ω + (n −
1)δ) ̸= 0, for n = 2, 3, 4, . . ., then we have

|φ2| ≤
ψ1

|(ω + δ)|
,

|φ3| ≤
ψ1

|(ω + 2δ)|
max

{
1;

∣∣∣∣ψ2

ψ1
− (ω − 1)(ω + 2δ)ψ1

2(ω + δ)2

∣∣∣∣}
and for all ρ ∈ C



K. R. Karthikeyan, D. Mohankumar, D. Breaz / Eur. J. Pure Appl. Math, 18 (1) (2025), 5841 12 of 19

∣∣φ3 − ρφ2
2

∣∣ ≤ ψ1

|(ω + 2δ)|
max

{
1;

∣∣∣∣ψ2

ψ1
− (ω − 1)(ω + 2δ)ψ1

2(ω + δ)2
− ρψ1 (ω + 2δ)

(ω + δ)2

∣∣∣∣} .
The inequalities are sharp.

Corollary 3. If χ(ξ) ∈ BS( see Remark1 (3)), then we have

|φ2| ≤
2

|(ω + 1)|
, |φ3| ≤

2

|(ω + 2)|
max

{
1;

∣∣∣∣1 − (ω − 1)(ω + 2)

(ω + 1)2

∣∣∣∣}
and for all ρ ∈ C

∣∣φ3 − ρφ2
2

∣∣ ≤ 2

|(ω + 2)|
max

{
1;

∣∣∣∣1 − (ω − 1)(ω + 2)

(ω + 1)2
− 2ρ (ω + 2)

(ω + 1)2

∣∣∣∣} .
The inequalities are sharp.

Corollary 4. [57, Theorem 3.1.] If χ(ξ) = ξ + φ2ξ
2 + φ3ξ

3 + · · · ∈ S∗(ψ), then for all
ρ ∈ C we have ∣∣φ3 − ρφ2

2

∣∣ ≤ ψ1

2
max

{
1;

∣∣∣∣L1 +
ψ2

ψ1
− 2ρψ1

∣∣∣∣} .
The inequality is sharp for the function χ∗ given by

χ∗(ξ) =


ξ exp

∫ ξ

0

Ψ(t) − 1

t
dt, if

∣∣∣ψ1 +
ψ2

ψ1
− 2ρψ1

∣∣∣ ≥ 1,

ξ exp

∫ ξ

0

Ψ(t2) − 1

t
dt, if

∣∣∣ψ1 +
ψ2

ψ1
− 2ρψ1

∣∣∣ ≤ 1.

(32)

Proof. In Theorem 2, taking ω = 0, δ = 1, r = 2, s = 1, κ1 = σ1, κ2 = q, m = η = 0
and q → 1−, we get the inequality

∣∣φ3 − ρφ2
2

∣∣ ≤

ψ1

2
, if

∣∣∣ψ1 +
ψ2

ψ1
− 2ρψ1

∣∣∣ ≤ 1,

ψ1

2

∣∣∣∣ψ1 +
ψ2

ψ1
− 2ρψ1

∣∣∣∣ , if
∣∣∣ψ1 +

ψ2

ψ1
− 2ρψ1

∣∣∣ ≥ 1.

Finally, following a similar technique to that for the sharpness of Theorem 3.1 of [57],
we obtain (32).

Letting ω = 0, δ = 1, r = 2, s = 1, κ1 = σ1, σ2 = q, m = η = 0, Ψ(ξ) = (1 + ξ)/(1− ξ)
and q → 1− in Theorem 2, we get

Corollary 5. If χ(ξ) ∈ S∗, then the bounds of the initial coefficients of χ are given by

|φ2| ≤ 2, |φ3| ≤ 3.

and the Fekete-Szegö inequality for ρ ∈ C is given by∣∣φ3 − ρφ2
2

∣∣ ≤ max {1, |4ρ− 3|} .



K. R. Karthikeyan, D. Mohankumar, D. Breaz / Eur. J. Pure Appl. Math, 18 (1) (2025), 5841 13 of 19

4. Coefficient Estimates of χ−1(ξ)

The inverse χ−1, defined by χ−1(χ(ξ)) = ξ, ξ ∈ Λ and χ(χ−1(t)) = t, (|t| < r; r ≥ 1/4)
where

g(t) = χ−1(t) = t− φ2t
2 + (2φ2

2 − φ3)t
3 −

(
5φ2

2 − 5φ2φ3 + φ4

)
t4 + · · · . (33)

The coefficient inequalities of the inverse functions BSm,ω
λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)) are

valid only for the functions which are univalent.

Theorem 3. Let χ ∈ BSm,ω
λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)) and let χ−1 be the inverse of χ defined

by

χ−1(t) = t+
∞∑
k=2

bkt
k, (|t| < r; r ≥ 1/4),

then we have

|b2| ≤
ψ1

|(ω + δ) Γ2|
and

|b3| ≤
ψ1

|(ω + 2δ)Γ3|
max

{
1;

∣∣∣∣∣ψ2

ψ1
− (ω − 1)(ω + 2δ)ψ1

2(ω + δ)2
− 2ψ1 (ω + 2δ) Γ3

(ω + δ)2 Γ2
2

∣∣∣∣∣
}

Also, for all τ ∈ C

∣∣b3 − τb22
∣∣ ≤ ψ1

|(ω + 2δ)Γ3|
max

{
1;

∣∣∣∣∣ψ2

ψ1
− (ω − 1)(ω + 2δ)ψ1

2(ω + δ)2
− (τ − 2)ψ1 (ω + 2δ) Γ3

(ω + δ)2 Γ2
2

∣∣∣∣∣
}
,

where ω and δ be chosen such that ω + δ ̸= 0, ω + 2δ ̸= 0.

Proof. From χ(ξ) = ξ +
∑∞

n=2 φnξ
n and (33), we have

b2 = −φ2 and b3 = 2φ2
2 − φ3.

The estimate for |b2| = |φ2| can be got by taking modulus of (27). Letting ρ = 2 in (23),
we get |b3|. To find the Fekete-Szegő inequality for the inverse function, consider∣∣b3 − τb22

∣∣ =
∣∣2φ2

2 − φ3 − τφ2
2

∣∣ =
∣∣φ3 − (τ − 2)φ2

2

∣∣ .
Changing ρ = (τ − 2) in the (23), we get the desired result.



K. R. Karthikeyan, D. Mohankumar, D. Breaz / Eur. J. Pure Appl. Math, 18 (1) (2025), 5841 14 of 19

5. Logarithmic Coefficients for Functions Belonging to
BSm,ω

λ,q (κ1, σ1; η, θ; δ; Ψ(ξ))

Logarithmic coefficients took the spotlight when Milin in [37] studied its properties
which would imply the bounds of the Taylor coefficients of univalent functions. For detailed
study, refer to [4, 5].

If χ is analytic in Λ, with χ(ξ)
ξ ̸= 0 for all ξ ∈ Λ, then the well-known logarithmic

coefficients ϕn := ϕn(χ), n ∈ N, of χ are given by

log
χ(ξ)

ξ
= 2

∞∑
n=1

ϕnξ
n, ξ ∈ Λ, log 1 = 0. (34)

Now we will add additional criterion to the class BSm,ω
λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)), so that

logarithmic coefficients of BSm,ω
λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)) is well-defined. That is, we let

T Sm,ω
λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)) = BSm,ω

λ,q (κ1, σ1; η, θ; δ; Ψ(ξ))∩
{
χ is analytic in Λ : χ(ξ)

ξ ̸= 0, ξ ∈ Λ
}

.

Note that for all functions T Sm,ω
λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)), the relation (34) is well-defined.

Theorem 4. If χ(ξ) ∈ T Sm,ω
λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)) with the logarithmic coefficients given

by (34), then we have

|ϕ1| ≤
ψ1

2 |(ω + δ) Γ2|
, (35)

|ϕ2| ≤
ψ1

2 |(ω + 2δ)Γ3|
max

{
1;

∣∣∣∣∣ψ2

ψ1
− (ω − 1)(ω + 2δ)ψ1

2(ω + δ)2
− ψ1 (ω + 2δ) Γ3

2 (ω + δ)2 Γ2
2

∣∣∣∣∣
}
, (36)

and

∣∣ϕ2 − µϕ21
∣∣ ≤ ψ1

2 |(ω + 2δ)Γ3|
max

{
1;

∣∣∣∣∣ψ2

ψ1
− (ω − 1)(ω + 2δ)ψ1

2(ω + δ)2
− (1 + µ)ψ1 (ω + 2δ) Γ3

2 (ω + δ)2 Γ2
2

∣∣∣∣∣
}
.

(37)

Proof. From χ(ξ) = ξ +
∑∞

n=2 φnξ
n and equating the first two coefficients of relation

(34), we get

ϕ1 =
φ2

2
, ϕ2 =

1

2

(
φ3 −

φ2
2

2

)
.

Using (27)) and (28) in the above equation and applying Lemma 1, we obtain (35) and
(36). To obtain (37), consider

∣∣ϕ2 − µϕ21
∣∣ =

1

2

[
φ3 −

(1 + µ)

2
φ2
2

]
.

Changing ρ = 1+µ
2 in (23), we get the desired result.



K. R. Karthikeyan, D. Mohankumar, D. Breaz / Eur. J. Pure Appl. Math, 18 (1) (2025), 5841 15 of 19

6. Conclusions

We have defined an operator which is most generalized and whose definition is not
straightforward. Using the defined operator, we defined a subclass of analytic functions
whose analytic characterization is associated with the class of Bazilevič functions. Though
one has to be content with the parameters involved, but it helps in specializing most of
the subclass of the univalent function theory. Some subordination properties and bounds
of the initial coefficient are our main results.

Further the questions arises regarding the inclusions and radius problems. In detail,

(i) The functions belongs to the classes BSm,ω
λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)) need not be univa-

lent, so for what radius of the disc |z| < r and for what values of the parameters
would the functions in BSm,ω

λ,q (κ1, σ1; η, θ; δ; Ψ(ξ)) be univalent.

(ii) Theorem 1 is not valid if the ordinary derivatives is replaced with a quantum deriva-
tive. Are there any equivalent condition for which Theorem 1 remains valid if the
ordinary derivatives is replaced with a quantum derivative.

Funding

This research study received no external funding.

Declaration

Authors declare no conflicts of interest.

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