

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6349
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Fekete-Szegö Inequality Estimate for Analytic Functions
Using Sǎlǎgean-Difference Operator and Leaf-Like

Domain

Avaya Naik1,∗, Sushree Chinmayee Sahoo1

1 P.G. Department of Mathematics Fakir Mohan University,Balasore, Odisha, India

Abstract. This paper investigates the Fekete-Szegö inequality for subclasses of analytic func-
tions in the unit disk, including starlike, convex, bounded turning, and close-to-convex functions
of complex order. Employing the Sǎlǎgean-difference operator, we derive sharp bounds for the
functional |b3−γb22| and extend these results to leaf-like domains. Our findings generalize classical
inequalities, offering new insights into coefficient constraints and geometric properties in complex
analysis.

2020 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Analytic function, Univalent functions,Fekete-Szegö inequality,Leaf
like domain

1. Introduction

The theory of univalent functions, a cornerstone of geometric function theory, explores
the properties of analytic functions that are injective in the open unit disk. A fundamental
problem in this field is to estimate the coefficients of such functions, as these coefficients
encode critical geometric and analytic information about their mappings. Among the clas-
sical results, the Fekete-Szegö inequality stands out as a powerful tool for a normalized
analytic function. The Fekete-Szegö inequality was first proposed by Hungarian Mathe-
maticians Michael Fekete and Gaber Szegö in 1933 (see[1]). Since then, the various au-
thors were investigated and obtained the Fekete-Szegö inequalities for different subclasses
(see[2][3][4],[5],[6],[7],[8],[9],[10])This inequality has been extensively studied for various
subclasses of univalent functions, such as starlike, convex, and close-to-convex functions,
due to its applications in understanding extremal problems and conformal mappings.

In recent decades, differential and integral operators have enriched the study of uni-
valent functions by generalizing classical subclasses and introducing new geometric con-
straints. One such operator is the Sǎlǎgeandifferential operator, introduced by Sǎlǎgean

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6349

Email addresses: avayanaik@gmail.com (A.Naik), chinmayee144@gmail.com (S. C. Sahoo)

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


A.Naik, S. C. Sahoo / Eur. J. Pure Appl. Math, 18 (3) (2025), 6349 2 of 14

in 1983 (see[11]), which iteratively applies weighted combinations of the function and its
derivatives to generate new classes of analytic functions. Building on this foundation,
Al-Oboudi (see [12]) proposed a generalized difference operator, later adapted to form the

Salagean-difference operator, denoted Dβ
λ .

Parallel to operator-based studies, the exploration of non-standard domains has gained
traction in complex analysis.The leaf-like domains, defined by mappings exhibit unique
boundary properties that deviate from the circular or convex shapes typically associated
with starlike or convex functions. These domains, named for their resemblance to a leaf?s
contour, challenge traditional coefficient bounds and inspire new inequalities tailored to
their geometry. The interplay between differential operators and such domains is partic-
ularly intriguing, as it allows researchers to investigate how operator-induced transforma-
tions affect mappings onto complex regions.

The research conducted by Srivastava et al.(see [13]),Murugusundaramoorthy (see
[14]), Orhan and Cot̆irlă ([15] ), Al-Sadi (see[16]), Panigrahi et al.([17]) advances the the-
oretical foundations of these subclasses. Several scholars, including Al-Sadi (see[16]) and
Srivastava et al.(see [13]), focused on deriving constraints for the initial coefficients, which
are crucial for understanding the evolution and structure of these functions. The Fekete-
Szegö functional was the focus of several investigations, including those by Srivastava et
al. (see [13]) and Panigrahi et al. (see[17]), which provided upper estimates and inequal-
ities for these specialized subclasses. And the Geometric structures like leaf-shaped do-
mains Panigrahi et al.(see[17]) and crescent-shaped areas Murugusundaramoorthy,(see[14])
demonstrate how these functions can be connected to specific geometric curves and figures,
impacting their analytical behavior.

Recently Kavita P et al.(see[18]) obtained significant inequality for starlike, bounded
turning, and close-to-convex functions by using Hohlov operator and taking leaf like do-
main in account.

Motivated by these developments, this paper derives new Fekete-Szegöinequality esti-
mates for subclasses of analytic functions defined using the Sǎlǎgean-difference operator
Dβ

λ and associated with leaf-like domains. Specifically, we consider functions in classes
such as starlike, convex, bounded turning, and close-to-convex functions of complex order,
extending classical results to these generalized settings. Our main objective is to obtain
sharp bounds for the functional |b3−γb22| for functions h(ζ) satisfying Dβ

λh(ζ) ∈ S∗
℘ where

S∗
℘, denotes a starlike class relative to a specific subordination. Additionally, we explore

how these bounds adapt to mappings onto leaf-like domains, offering insights into their
geometric implications.

The novelty of this work lies in its integration of the Sǎlǎgean-difference operator with
leaf-like domains, a combination that has not been extensively studied in the context of the
Fekete-Szegö problem. By leveraging known results on coefficient bounds (see [19][20]) and
introducing new techniques for handling the operators symmetry, we establish inequalities
that generalize and sharpen existing estimates. These findings not only enhance our
understanding of univalent function behavior but also pave the way for applications in
conformal mapping and operator theory.

Let A be the family of function h which are analytic in the open unit dick ∆ = {ζ :


A.Naik, S. C. Sahoo / Eur. J. Pure Appl. Math, 18 (3) (2025), 6349 3 of 14

|ζ| < 1} normalized by the conditions h(0) = 0 and h′(0) = 1 with the series expansion of
the form

h(ζ) = ζ +

∞∑
n=2

bnζ
n (1)

Let S be a subclass of A consists of ”schilit functions”. The sufficient conditions for a
function h ∈ A to belong to the classes S∗ (starlike functions) and Sc (convex functions)
are well established results in geometric function theory. These foundational conditions
can be trace back to the pioneering contribution of W.odzimierz̀erański, Robertson, and
Hummel in the mid-20th century. Alexander (see[21]) introduced the necessary and suffi-
cient condition for a function h ∈ A to be in S∗ is that

Re
{ζh′(ζ)

h(ζ)

}
> 0, (ζ ∈ ∆)

Similarly, the necessary and sufficient condition for a function h ∈ A to be in Sc is that

Re
{
1 +

ζh′′(ζ)

h′(ζ)

}
> 0, (ζ ∈ ∆)

In 1964, Robertson [22] introduced a generalized class of starlike and convex functions of
complex order. Subsequently, Miller and Mocanu [23] incorporated similar conditions in
their work to extend classical results related to starlike and convex functions. These condi-
tions were formulated using differential subordinations, a powerful technique for analyzing
functional inequalities within geometric function theory.

These classes have been thoroughly examined, and various characteristics have been
identified, such as coefficient bounds, growth and distortion theorems, radii of starlike-
ness and convexity, as well as properties related to convolution, Hadamard products, and
subordination. These functions facilitate the development of new function classes, aid in
modeling intricate geometric forms, and contribute to a deeper understanding of geometric
properties. Let α ∈ C, a function h ∈ A is in the class of starlike functions of complex
order α and denoted by S∗(α), [24] if and only if

h(ζ)

ζ
̸= 0 and Re

(
1 +

1

α

{
ζh′(ζ)

h(ζ)
− 1

})
> 0, (ζ ∈ ∆) (2)

A function h ∈ A is in the class of convex functions of complex order α and denoted
by C(α),[25] if and only if

h′(ζ) ̸= 0 and Re

(
1 +

1

α

{
ζh′′(ζ)

h′(ζ)

})
> 0, (ζ ∈ ∆) (3)

A function h ∈ A is in the class of convex functions of complex order α and denoted
by K(α),[25] if and only if

Re

(
1 +

1

α
(h′(ζ)− 1)

)
> 0 (ζ ∈ ∆) (4)


A.Naik, S. C. Sahoo / Eur. J. Pure Appl. Math, 18 (3) (2025), 6349 4 of 14

For a function h ∈ A the class of analytic functions with h(0) = 0 and h′(0) = 1, the

operator is defined as Dβ
λ : A −→ A as follows:

D0
λh(ζ) = h(ζ)

D1
λh(ζ) = ζh′(ζ) +

λ

2
[h(ζ)− h(−ζ)− 2ζ] (λ ∈ R)

= ζ +
∞∑
n=2

[
n+

λ

2
(1 + (−1)n+1)

]
bnζ

n

D2
λh(ζ) = D1(D1

λh(ζ))

In general, for β ∈ N0 = {0, 1, 2, 3, · · · },

Dβ
λh(ζ) = D1

λ(D
β−1
λ h(ζ)) = ζ +

∞∑
n=2

[
n+

λ

2
(1 + (−1)n+1)

]β
bnζ

n (ζ ∈ ∆) (5)

This operator, characterized by its parameter λ and β order offers a versatile framework for
studying function behavior under symmetric transformations, bridging classical differential
operators and modern geometric constraints. The Operator Dβ

λ is known as the Sǎlǎgean-
difference operator in literature (see [26],[27]). This operator is a modified Dunkel operator

of complex variables(see [28],[29]). When λ = 0,Dλ = Dβ
0 = Dβ is known as the Sǎlǎgean-

differential operator (see [11]).

Example 1.

h(ζ) = ζ +
ζ2

2
+

ζ3

8
+

ζ4

48
+

ζ5

384
+ · · ·

Then

D1
1h(ζ) = ζ + ζ2 +

ζ3

2
+

ζ4

12
+

ζ5

64
+ · · ·

Example 2.

h(ζ) = ζ +
2ζ2

5
+

3ζ3

25
+

4ζ4

125
+ · · ·

Then

h(ζ) = ζ +
4ζ2

5
+

12ζ3

25
+

16ζ4

125
+ · · ·

2. Preliminaries

Let us define the bounded turning function with Sǎlǎgean- difference operator as R℘

which contains all the function h ∈ A and satisfying

Re
(
(Dβ

λh(ζ))
′), (ζ ∈ ∆) (6)


A.Naik, S. C. Sahoo / Eur. J. Pure Appl. Math, 18 (3) (2025), 6349 5 of 14

Similarly starlike function with Sǎlǎgean- difference operator S∗
℘ which maps |∆| < 1

conformally on to starlike domain and statisfying

Re

(
ζ(Dβ

λh(ζ))
′

Dβ
λh(ζ)

)
(ζ ∈ ∆) (7)

Let us define starlike function of complex order with Sǎlǎgean- difference operator S∗
℘(α)

which maps |∆| < 1 conformally onto starlike domain of complex order and satisfying

Re

(
1 +

1

α

[
ζ(Dβ

λh(ζ))
′

Dβ
λh(ζ)

− 1

])
> 0 (ζ ∈ ∆) (8)

Let us define convex function of complex order with Sǎlǎgean- difference operator Sc
℘(α),

which maps |ζ| < 1 conformally onto convex domain of complex order and satisfying

Re

(
1 +

1

α

[
ζ(Dβ

λh(ζ))
′′

(Dβ
λh(ζ))

′
− 1

])
> 0 (ζ ∈ ∆) (9)

Let us define close to convex function with Sǎlǎgean- difference operator K℘(α), which
maps |ζ| < 1 conformally onto closed convex domain of complex order and satisfying

Re
{
1 +

1

α
[(Dβ

λh(ζ))
′ − 1]

}
, (ζ ∈ ∆) (10)

Raina and Sokol [30] and Haripriya [31] explored the function h(ζ) = ζ + (1 + ζ3)
1
3 ,

which has symmetry with respect to the real axis. Real part of this function is positive
with conditions h(0) = h′(0) = 1, and it maps the unit disc onto analytic and univalent
region which has the shape of leaf-like domain.This leaf-like domain can model complex
shapes with smooth boundaries. In general, a ”leaf” is a smooth submanifold or region
of a manifold that resembles ”sheets” within a system with layers. Particular subsets, or
”leaves,” inside a manifold are referred to as leaf-like domains when discussing foliations
or decompositions of the manifold into simpler structures. In certain geometric contexts,
such as the study of dynamical systems or the theory of foliations, examining the behavior
of the manifold within these leaves can provide crucial insights into the general topology
and geometry of the space.

The result of following Lemmas are applied in our main theorems.

Lemma 1. Let P denote the class of function denoted by p such that p(ζ) = d1ζ + d2ζ
2+

d3ζ
3 + · · · be an analytic function in the region R with the property that p(0) = 1 then

|dn| ≤ 2 for all n ≥ 1 and |d2 −
d21
2 |. P is the class of all such function which has the

property of positive real part.

Lemma 2. Let the analytic function p(ζ) = d1ζ+d2ζ
2+d3ζ

3+ · · · which have the positive
real part, then |d2 − αd21| ≤ 2max1, |2α− 1| here α is the complex number.

Functions p(ζ) = 1+ζ2

1−ζ2
and p(ζ) = 1+ζ

1−ζ provides the sharp results.


A.Naik, S. C. Sahoo / Eur. J. Pure Appl. Math, 18 (3) (2025), 6349 6 of 14

3. Main Results

Theorem 1. If h ∈ A is the form given by (1) belongs S∗
℘ and γ is a real number then

|b3 − γb22| ≤


1

2(3+λ)β
if p(ζ) = 1+ζ2

1−ζ2

1
2(3+λ)β

∣∣∣2γ(3+λ)β

22β
− 1
∣∣∣ if p(ζ) = 1+ζ2

1−ζ2

(11)

Proof. If h ∈ S∗Mα , then for the schwarz function w with w(0) = 0 and |w(ζ)| ≤ 1
and concept of sub-ordination property of equation (7), we have

ζ(Dβ
λh(ζ))

′

Dβ
λh(ζ)

= w(ζ) + 3
√
1 + (w(ζ))3 (12)

we have

p(ζ) =
1 + w(ζ)

1− w(ζ)
= 1 + d1ζ + d2ζ

2 + d3ζ
3 + · · ·

w(ζ) =
1 + p(ζ)

1− p(ζ)

on simplifying right hand side of equation (25) we get

w(ζ) + 3
√
1 + (w(ζ))3 = 1 +

d1
2
ζ +

(
d2
2

− d21
4

)
ζ2 +

(
d3
2

− d1d2
2

+
d31
6

)
ζ3+ (13)(

d4
2

− d22
4

− d1d3
2

+
d21d2
2

− d41
8

)
ζ4 + · · ·

From left hand side of (25) we get

ζ(Dβ
λh

′(ζ))

Dβ
λh(ζ)

= 1 + 2βb2ζ +

(
2(3 + λ)βb3 − 22βb22

)
ζ2 +

(
3(4β)b4 − 3(2β)(3 + λ)βb2b3 + 23βb32

)
ζ3

(14)

+

(
4(5 + λ)βb5 − 4(2β)4βb2b4 − 2(3 + λ)2βb23 − 24βb42 + 4(22β)(3 + λ)βb22b3

)
ζ4 + · · ·

Now from (12),(13) and (14)

1 + 2βb2ζ +

(
2(3 + λ)βb3 − 22βb22

)
ζ2 +

(
3(4β)b4 − 3(2β)(3 + λ)βb2b3 + 23βb32

)
ζ3

+

(
4(5 + λ)βb5 − 4(2β)4βb2b4 − 2(3 + λ)2βb23 − 24βb42 + 4(22β)(3 + λ)βb22b3

)
ζ4 + · · ·

= 1 +
d1
2
ζ +

(
d2
2

− d21
4

)
ζ2 +

(
d3
2

− d1d2
2

+
d31
6

)
ζ3+


A.Naik, S. C. Sahoo / Eur. J. Pure Appl. Math, 18 (3) (2025), 6349 7 of 14(
d4
2

− d22
4

− d1d3
2

+
d21d2
2

− d41
8

)
ζ4 + · · ·

On equating the coefficients, we get

b2 =
d1

2(2β)

b3 =
d2

4(3 + λ)β

b3 − γb22 =
1

4(3 + λ)β

(
d2 −

γ(3 + λ)βd21
22β

)
Applying the lemma (2) we get

|b3 − γb22| ≤
1

2(3 + λ)β
max

{
1,

∣∣∣∣2γ(3 + λ)β

22β

∣∣∣∣} (15)

This Completes the proof

Theorem 2. If h ∈ A is the form given by (1) belongs RM℘ and γ is a real number then

|b3 − γb22| ≤


1

3(3+λ)β
if p(ζ) = 1+ζ2

1−ζ2

1
3(3+λ)β

∣∣∣3γ(3+λ)β

4(2)2β

∣∣∣ if p(ζ) = 1+ζ2

1−ζ2

(16)

Proof. If RM℘ , then for the schwarz function w with w(0) = 0 and |w(ζ)| ≤ 1

(Dβ
λh(ζ))

′ = w(ζ) + 3
√
1 + (w(ζ))3 (17)

(Dβ
λh(ζ))

′ = 1 + 2(2β)b2ζ + 3(3 + λ)βb3ζ
2 + 4(4β)b4ζ

3 + 5(5 + λ)βb5ζ
5 + · · · (18)

From equation (13) and (18) we have

1 + 2(2β)b2ζ + 3(3 + λ)βb3ζ
2 + 4(4β)b4ζ

3 + 5(5 + λ)βb5ζ
5 + · · ·

= 1 +
d1
2
ζ +

(
d2
2

− d21
4

)
ζ2 +

(
d3
2

− d1d2
2

+
d31
6

)
ζ3+(

d4
2

− d22
4

− d1d3
2

+
d21d2
2

− d41
8

)
ζ4 + · · ·

On contrasting similar terms

b2 =
d1

4(2β)

b3 =
1

3(3 + λ)β

(
d2
2

− d21
4

)


A.Naik, S. C. Sahoo / Eur. J. Pure Appl. Math, 18 (3) (2025), 6349 8 of 14

on streamlining and using lemma (2) we get

|b3 − γb22| ≤
1

3(3 + λ)β
max

{
1,

∣∣∣∣3γ(3 + λ)β

4(22β)

∣∣∣∣} (19)

Theorem 3. If h ∈ A is the form given by (1) belongs S∗M℘(α) and γ is a real number
then

|b3 − γb22| ≤


α

2(3+λ)β
if p(ζ) = 1+ζ2

1−ζ2

α
2(3+λ)β

∣∣∣(2γ(3+λ)β

22β
− 1

)
α
∣∣∣ if p(ζ) = 1+ζ2

1−ζ2

(20)

Proof.

1 +
1

α

(
ζ(Dβ

λh(ζ))
′

Dβ
λh(ζ)

− 1

)
= w(ζ) + 3

√
1 + (w(ζ))3 (21)

and

1 +
1

α

(
ζ(Dβ

λh(ζ))
′

Dβ
λh(ζ)

− 1

)
(22)

= 1 +
1

α

[
2βb2ζ +

(
2(3 + λ)βb3 − 22βb22

)
ζ2 +

(
3(4β)b4 − 3(2β)(3 + λ)βb2b3 + 23βb32

)
ζ3

+
(
4(5 + λ)βb5 − 4(2β)4βb2b4 − 2(3 + λ)2βb23 − 24βb42 + 4(22β)(3 + λ)βb22b3

)
ζ4 + · · ·

]
from equation (13),(21) and (22)

1 +
1

α

[
2βb2ζ +

(
2(3 + λ)βb3 − 22βb22

)
ζ2 +

(
3(4β)b4 − 3(2β)(3 + λ)βb2b3 + 23βb32

)
ζ3

+
(
4(5 + λ)βb5 − 4(2β)4βb2b4 − 2(3 + λ)2βb23 − 24βb42 + 4(22β)(3 + λ)βb22b3

)
ζ4 + · · ·

]
= 1 +

d1
2
ζ +

(
d2
2

− d21
4

)
ζ2 +

(
d3
2

− d1d2
2

+
d31
6

)
ζ3+(

d4
2

− d22
4

− d1d3
2

+
d21d2
2

− d41
8

)
ζ4 + · · ·

By equating the like term we have

b2 =
d1α

2(2β)

b3 =
α

4(3 + λ)β

(
d2 −

d21
2
(1− α)

)


A.Naik, S. C. Sahoo / Eur. J. Pure Appl. Math, 18 (3) (2025), 6349 9 of 14

On simplifying by lemma (2) we get

|b3 − γb22| ≤
α

2(3 + λ)β
max

{
1,

∣∣∣∣(2γ(3 + λ)β

22β
− 1

)
α

∣∣∣∣}
As a result, we get the desired outcomes.

Theorem 4. If h ∈ A is the form given by (1) belongs KM℘ and γ is a real number then

|b3 − γb22| ≤


α

3(3+λ)β
if p(ζ) = 1+ζ2

1−ζ2

α
3(3+λ)β

∣∣∣3γα(3+λ)β

4(2)2β
− 1
∣∣∣ if p(ζ) = 1+ζ2

1−ζ2

Proof. If KM℘, then for the schwarz function w with w(0) = 0 and |w(ζ)| ≤ 1

1 +
1

α

(
(Dβ

λh(ζ))
′ − 1

)
= w(ζ) + 3

√
1 + (w(ζ))3 (23)

1+
1

α

(
(Dβ

λh(ζ))
′−1

)
= 1+

1

α

(
2(2β)b2ζ+3(3+λ)βb3ζ

2+4(4β)b4ζ
3+5(5+λ)βb5ζ

5+ · · ·
)

(24)
From equation (13),(23) and (24)

1 +
1

α

(
2(2β)b2ζ + 3(3 + λ)βb3ζ

2 + 4(4β)b4ζ
3 + 5(5 + λ)βb5ζ

5 + · · ·
)

= 1 +
d1
2
ζ +

(
d2
2

− d21
4

)
ζ2 +

(
d3
2

− d1d2
2

+
d31
6

)
ζ3+(

d4
2

− d22
4

− d1d3
2

+
d21d2
2

− d41
8

)
ζ4 + · · ·

On contrasting similar terms

b2 =
d1α

4(2β)

b3 =
α

6(3 + λ)β

(
d2 −

d21
2

)
on streamlining and using lemma (2) we get

|b3 − γb22| ≤
α

3(3 + λ)β
max

{
1,

∣∣∣∣3γα(3 + λ)β

4(22β)

∣∣∣∣}
As a result, we get the desired outcomes.


A.Naik, S. C. Sahoo / Eur. J. Pure Appl. Math, 18 (3) (2025), 6349 10 of 14

Theorem 5. If h ∈ A is the form given by (1) belongs ScM℘(α) and γ is a real number
then

|b3 − γb22| ≤


α

6(3+λ)β
if p(ζ) = 1+ζ2

1−ζ2

α
6(3+λ)β

∣∣∣3γα(3+λ)β

2(22β)
− α

∣∣∣ if p(ζ) = 1+ζ2

1−ζ2

(25)

Proof.

1 +
1

α

(
ζ(Dβ

λh(ζ))
′′

(Dβ
λh(ζ))

′

)
= w(ζ) + 3

√
1 + (w(ζ))3 (26)

and

1 +
1

α

(
ζ(Dβ

λh(ζ))
′′

(Dβ
λh(ζ))

′

)
= (27)

1 +
1

α

[
2(2β)b2ζ +

(
6(3 + λ)βb3 − 4(22β)b22

)
ζ2+(

12(4β)a4 − 18(2β)(3 + λ)βb2b3 + 8(23β)b32

)
ζ3 + · · ·

]
From equation (13),(26) and (27), we have

b2 =
d1α

4(2β)

b3 =
α

6(3 + λ)β

(
d2
2

− d21
4
(1− α)

)
On simplifying by lemma (2) we get

|b3 − γb22| ≤
α

6(3 + λ)β
max

{
1,

∣∣∣∣(3γα(3 + λ)β

2(22β)
− α

)∣∣∣∣}
As a result, we get the desired outcomes.

Remark 1. case-I: If p(ζ) = 1+ζ
1−ζ then in this case d1 = d2 = d3 = · · · = 2.

Case-II: If p(ζ) = 1+ζ2

1−ζ2
then in this case d1 = d3 = d5 = · · · = 0 and d2 = d4 = d6 = · · · =

2.

On taking consideration of these above instance we get the results of above theorems.

4. special cases

Remark 2. If we take λ = 0 in Dβ
λh(ζ), it will reduce to Sǎlǎgean-deferential Operator

and ,β = 1 then 2β = 2, 1
(3+λ)β

= 1
3 so that in theorem

(1),(2),(3),(4),(5) we find the coresponding results of Sǎlǎgean deference Operator.


A.Naik, S. C. Sahoo / Eur. J. Pure Appl. Math, 18 (3) (2025), 6349 11 of 14

Corollary 1. Let λ = 0, β = 1 , If h ∈ S∗M℘, then

|b3 − γb22| ≤


1
6 if p(ζ) = 1+ζ2

1−ζ2

1
6

∣∣∣3γ2 − 1
∣∣∣ if p(ζ) = 1+ζ2

1−ζ2

Corollary 2. Let λ = 0, β = 1 , If h ∈ RM℘, then

|b3 − γb22| ≤


1
9 if p(ζ) = 1+ζ2

1−ζ2

1
9

∣∣∣9γ16 ∣∣∣ if p(ζ) = 1+ζ2

1−ζ2

Corollary 3. Let λ = 0, β = 1 , If h ∈ S∗M℘(α), then

|b3 − γb22| ≤


α
6 if p(ζ) = 1+ζ2

1−ζ2

α
6

∣∣∣(3γ
2 − 1

)
α
∣∣∣ if p(ζ) = 1+ζ2

1−ζ2

Corollary 4. Let λ = 0, β = 1 , If h ∈ KM℘(α), then

|b3 − γb22| ≤


α
9 if p(ζ) = 1+ζ2

1−ζ2

α
9

∣∣∣9γα16 − 1
∣∣∣ if p(ζ) = 1+ζ2

1−ζ2

Corollary 5. Let λ = 0, β = 1 . If h ∈ ScM℘(α), then

|b3 − γb22| ≤


α
18 if p(ζ) = 1+ζ2

1−ζ2

α
18

∣∣∣9γα8 − α
∣∣∣ if p(ζ) = 1+ζ2

1−ζ2

Remark 3. If we take λ = 1, in Dβ
λh(ζ) then it will reduce to Al-Oboudi differential

operator, β = 1 then 2β = 2 and 1
(3+λ)β

= 1
4 so that in theorem

(1), (2), (3), (4), (5) we find the coresponding results of Al-Oboudi differential operator.

Corollary 6. Let λ = 1, β = 1 , If h ∈ S∗M℘, then

|b3 − γb22| ≤

{
1
8 if p(ζ) = 1+ζ2

1−ζ2

1
8

∣∣2γ − 1
∣∣ if p(ζ) = 1+ζ2

1−ζ2

Corollary 7. Let λ = 1, β = 1 , If h ∈ RM℘, then

|b3 − γb22| ≤


1
12 if p(ζ) = 1+ζ2

1−ζ2

1
12

∣∣∣3γ4 ∣∣∣ if p(ζ) = 1+ζ2

1−ζ2

Corollary 8. Let λ = 1, β = 1 , If h ∈ S∗M℘(α), then

|b3 − γb22| ≤

{
α
8 if p(ζ) = 1+ζ2

1−ζ2

α
8

∣∣(2γ − 1)α
∣∣ if p(ζ) = 1+ζ2

1−ζ2


A.Naik, S. C. Sahoo / Eur. J. Pure Appl. Math, 18 (3) (2025), 6349 12 of 14

Corollary 9. Let λ = 1, β = 1 , If h ∈ KM℘(α), then

|b3 − γb22| ≤


α
12 if p(ζ) = 1+ζ2

1−ζ2

α
12

∣∣∣3γα4 − 1
∣∣∣ if p(ζ) = 1+ζ2

1−ζ2

Corollary 10. Let λ = 1, β = 1 . If h ∈ ScM℘(α), then

|b3 − γb22| ≤


α
24 if p(ζ) = 1+ζ2

1−ζ2

α
24

∣∣∣3γα2 − α
∣∣∣ if p(ζ) = 1+ζ2

1−ζ2

5. Conclusion

In conclusion, this work expands our understanding of the Fekete-Szegö inequality
by applying it to a larger class of holomorphic functions, specifically starlike, bounded
turning, and close-to-convex functions of complex order. By considering the Sǎlǎgean-
difference operator and leaf-like domains, we have created new inequalities that expand
on conventional findings. These findings improve our understanding of how these functions
behave in geometric function theory and complex analysis.Additionally, we examine spe-
cific instances of the deferential operator and provide strict limitations on the coefficients,
providing useful information for further study.

Acknowledgements

The authors would like to express sincere gratitude to the editorial board and anony-
mous reviewers of the European Journal of Pure and Applied Mathematics for their valu-
able comments and suggestions, which helped to improve the quality and clarity of this
research article.

Conflict of Interest

The authors declare that there are no conflicts of interest.

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