EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5869
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Certain Subclass of Multivalently Bazilevič and
Non-Bazilevič Functions Involving the Lemniscate of

Bernoulli

Tamer M. Seoudy1,∗, Amnah E. Shammaky2

1 Department of Mathematics, Jamoum University College, Umm Al-Qura University, Makkah,
Saudi Arabia
2 Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia

Abstract. Making use of the principle of subordination, we define a certain subclass of p−valently
Bazilevič and non-Bazilevič functions associated with the Lemniscate of Bernoulli. Also, subordi-
nation results, convolution properties, coefficients estimate and Fekete–Szegö inequalities for this
subclass are derived.

2020 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Analytic functions, subordination, convolution, Bazilevič function,
non-Bazilevič function, Fekete–Szegö problem

1. Introduction

Let H (U) be the class of all analytic functions in U = {ξ ∈ C : |ξ| < 1}. For χ, ρ ∈
H (U), we say that χ (ξ) is subordinate to ρ (ξ), written χ ≺ ρ in U or χ(ξ) ≺ ρ(ξ) (ξ ∈ U),
if there exists a Schwarz function ω (ξ), which (by definition) is analytic in U with ω (0) = 0
and |ω (ξ)| < 1(ξ ∈ U) such that χ(ξ) = ρ(ω(ξ)) (ξ ∈ U). In addition, if ρ (ξ) is a univalent
function in U, then we have the following equivalence (see [1] and [2]):

χ(ξ) ≺ ρ(ξ) (ξ ∈ U) ⇐⇒ χ(0) = ρ(0) and χ(U) ⊂ ρ(U).

Also, let Ap denote the subclass of H (U) consisting of functions of the form:

χ (ξ) = ξp +

∞∑
k=p+1

ϱkξ
k (p ∈ N = {1, 2, 3, ...} ; ξ ∈ U) , (1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5869

Email addresses: tmsaman@uqu.edu.sa (T.M. Seoudy), aeshamakhi@jazan.edu.sa (A.E. Shammaky)

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



T.M. Seoudy, A.E. Shammaky / Eur. J. Pure Appl. Math, 18 (2) (2025), 5869 2 of 14

which are p-valent in U with Ap = A. Sokól and Stankiewicz [3] defined the class SL∗

consisting of analytic functions χ ∈ A satisfying the next inequality∣∣∣∣∣
[
ξχ′ (ξ)

χ (ξ)

]2
− 1

∣∣∣∣∣ < 1,

which is equivalent to
ξχ′ (ξ)

χ (ξ)
≺ q (ξ) =

√
1 + ξ

where the function
q (ξ) =

√
1 + ξ (ξ ∈ U) (2)

maps U into the domain O =
{
w ∈ C : ℜ{w} > 0,

∣∣w2 − 1
∣∣ < 1

}
and its boundary ∂O is

the right-half of the lemniscate of Bernoulli
(
x2 + y2

)2−2
(
x2 − y2

)
= 0. Several geometric

properties of SL∗ were studied by many authors (see, for example, [4–7]).
Using the principle of differential subordination and the function q (ξ) =

√
1 + ξ of the

Bernoulli domain of lemniscate, we now define a new subclass BN p (λ, α, β) of Bazilevič
and non-Bazilevič functions as follows:

Definition 1. A function χ ∈ Ap is said to be the subclass BN p (λ, α, β) when it satisfies
the next subordination condition:(

1− α− β

α+ β
λ

)[
χ (ξ)

ξp

]α−β

+
α− β

α+ β
λ
ξχ′ (ξ)

pχ (ξ)

[
χ (ξ)

ξp

]α−β

≺
√
1 + ξ (3)

all the powers are principal values and throughout the paper unless otherwise mentioned
the real parameters λ, α, β are constrained as α ̸= β, p ∈ N and ξ ∈ U.

We note that

(i) BN p (λ, α, 0) = Bp (λ, α) =
{
χ ∈ Ap : (1− λ)

(
χ(ξ)
ξp

)α
+ λ ξχ′(ξ)

pχ(ξ)

(
χ(ξ)
ξp

)α
≺

√
1 + ξ

}
(see [8]);

(ii) BN p (λ, 0, β) = Np (λ, β) =

{
χ ∈ Ap : (1 + λ)

(
ξp

χ(ξ)

)β
− λ ξχ′(ξ)

pχ(ξ)

(
ξp

χ(ξ)

)β
≺

√
1 + ξ

}
;

(iii) BN 1 (λ, α, 0) = B (λ, α) =
{
χ ∈ A : (1− λ)

(
χ(ξ)
ξ

)α
+ λ ξχ′(ξ)

χ(ξ)

(
χ(ξ)
ξ

)α
≺

√
1 + ξ

}
(see [8]);

(iv) BN 1 (λ, 0, β) = N (λ, β) =

{
χ ∈ A : (1 + λ)

(
ξ

χ(ξ)

)β
− λ ξχ′(ξ)

χ(ξ)

(
ξ

χ(ξ)

)β
≺

√
1 + ξ

}
;

(v) BN p (λ, 1, 0) = Bp (λ) =
{
χ ∈ Ap : (1− λ) χ(ξ)

ξp + λ χ′(ξ)
pξp−1 ≺

√
1 + ξ

}
and B1 (λ) =

B (λ) =
{
χ ∈ A : (1− λ) χ(ξ)

ξ + λχ′ (ξ) ≺
√
1 + ξ

}
(see [8]);



T.M. Seoudy, A.E. Shammaky / Eur. J. Pure Appl. Math, 18 (2) (2025), 5869 3 of 14

(vi) BN p (λ, 0, 1) = Np (λ) =
{
χ ∈ Ap : (1 + λ) ξp

χ(ξ) − λ ξp+1χ′(ξ)
pχ2(ξ)

≺
√
1 + ξ

}
andN1 (λ) =

N (λ) =
{
χ ∈ A : (1 + λ) ξ

χ(ξ) − λ ξ2χ′(ξ)
χ2(ξ)

≺
√
1 + ξ

}
;

(vii) BN p (1, α, 0) = Bp (α) =
{
χ ∈ Ap :

ξχ′(ξ)
pχ(ξ)

(
χ(ξ)
ξp

)α
≺

√
1 + ξ

}
and B1 (α) = B (α) ={

χ ∈ A : ξχ′(ξ)
χ(ξ)

(
χ(ξ)
ξ

)α
≺

√
1 + ξ

}
(see [8]);

(viii) BN p (−1, 0, β) = Np (β) =

{
χ ∈ Ap :

ξχ′(ξ)
pχ(ξ)

(
ξp

χ(ξ)

)β
≺

√
1 + ξ

}
andN1 (β) = N (β) ={

χ ∈ A : ξχ′(ξ)
χ(ξ)

(
ξ

χ(ξ)

)β
≺

√
1 + ξ

}
;

(ix) BN p (1, 0, 0) = SL∗
p =

{
χ ∈ Ap :

ξχ′(ξ)
pχ(ξ) ≺

√
1 + ξ

}
and SL∗

1 = SL∗ =
{
χ ∈ A : ξχ′(ξ)

χ(ξ) ≺
√
1 + ξ

}
.

In order to establish our main results, we need the following lemmas.

Lemma 1. [9] Let h (ξ) be univalent and convex the function in U with h (0) = 1. Suppose
also that ρ (ξ) given by

ρ (ξ) = 1 + c1ξ + c2ξ
2 + ... (4)

is analytic in U. If

ρ (ξ) +
ξρ′ (ξ)

γ
≺ h (ξ) (ℜ (γ) ≥ 0; γ ̸= 0; ξ ∈ U) , (5)

then

ρ (ξ) ≺ q (ξ) = γξ−γ

∫ ξ

0
h (t) tγ−1 dt ≺ h (ξ) ,

and q (ξ) is the best dominant.

Lemma 2. [10] For real or complex numbers a, b, c(c ̸= 0,−1,−2, ...) and ξ ∈ U,∫ 1

0
tb−1(1− t)c−b−1(1− tξ)−adt = Γ(b)Γ(c−b)

Γ(c) 2Ω1(a, b; c; ξ) (ℜ(c) > ℜ(b) > 0); (6)

2Ω1(a, b; c; ξ) = (1− ξ)−a
2Ω1

(
a, c− b; c;

ξ

ξ − 1

)
; (7)

Lemma 3. [11] Let χ (ξ) =
∞∑
k=1

ϱkξ
k be analytic in U and ρ (ξ) =

∞∑
k=1

bkξ
k be analytic and

convex in U. If χ ≺ ρ, then
|ϱk| < |b1| (k ∈ N) .



T.M. Seoudy, A.E. Shammaky / Eur. J. Pure Appl. Math, 18 (2) (2025), 5869 4 of 14

Lemma 4. [12] Let ρ (ξ) = 1 +
∞∑
k=1

ckξ
k ∈ P, i.e., let ρ be analytic in U and satisfy

ℜ{ρ (ξ)} > 0 for ξ ∈ U, then the following sharp estimate holds∣∣c2 − vc21
∣∣ ≤ 2max {1, |2v − 1|} for all v ∈ C. (8)

The result is sharp for the functions given by

ρ(ξ) =
1 + ξ2

1− ξ2
or ρ(ξ) =

1 + ξ

1− ξ
.

Lemma 5. [12] If ρ (ξ) = 1 +
∞∑
k=1

ckξ
k ∈ P, then

∣∣c2 − νc21
∣∣ ≤


−4ν + 2 if ν ≤ 0,
2 if 0 ≤ ν ≤ 1,
4ν − 2 if ν ≥ 1,

(9)

when v < 0 or ν > 1, the equality holds if and only if ρ(ξ) = (1 + ξ)/(1− ξ) or one of its
rotations. If 0 < ν < 1, then the equality holds if and only if ρ(ξ) = (1 + ξ2)/(1 − ξ2) or
one of its rotations. If ν = 0, the equality holds if and only if

ρ (ξ) =

(
1 + λ

2

)
1 + ξ

1− ξ
+

(
1− λ

2

)
1− ξ

1 + ξ
(0 ≤ λ ≤ 1)

or one of its rotations. If ν = 1, the equality holds if and only if ρ is the reciprocal of one
of the functions such that equality holds in the case of ν = 0.

Also the above upper bound is sharp, and it can be improved as follows when 0 < ν < 1:∣∣c2 − νc21
∣∣+ ν |c1|2 ≤ 2

(
0 ≤ ν ≤ 1

2

)
and ∣∣c2 − νc21

∣∣+ (1− ν) |c1|2 ≤ 2

(
1

2
≤ ν ≤ 1

)
.

In some literature, we found many works related to the subclasses of Bazilevi č or
non-Bazilevič analytic functions which are sometimes defined by linear operators. For
example, we can see those subclasses in the papers in [13–22]. The novelty in our paper
is that we have combined Bazilevič and non-Bazilevič analytic functions in one subclass
BN p (λ, α, β) to study some geometric properties such as subordination properties, inclu-
sion relationship, convolution result, coefficients estimate and Fekete–Szegö inequalities.

2. Geometric Properties for BN p (λ, α, β)

Theorem 1. If χ ∈ BN p (λ, α, β) with
λ

α+β > 0, then[
χ (ξ)

ξp

]α−β

≺ Q (ξ) = (1 + ξ)
1
2 2Ω1

(
−1

2
, 1;

p (α+ β)

λ
+ 1;

ξ

1 + ξ

)
≺
√

1 + ξ, (10)

where the function Q (ξ) is the best dominant.



T.M. Seoudy, A.E. Shammaky / Eur. J. Pure Appl. Math, 18 (2) (2025), 5869 5 of 14

Proof. Let

ρ (ξ) =

[
χ (ξ)

ξp

]α−β

(ξ ∈ U) . (11)

Then the function ρ(ξ) is of the form (4), analytic in U and ρ (0) = 1. By taking the
derivatives in the both sides of (11), we get(

1− α− β

α+ β
λ

)[
χ (ξ)

ξp

]α−β

+
α− β

α+ β
λ
ξχ′ (ξ)

pχ (ξ)

[
χ (ξ)

ξp

]α−β

= ρ (ξ) +
λξρ′ (ξ)

p (α+ β)
. (12)

Since χ ∈ BN p (λ, α, β), we have

ρ (ξ) +
λξρ′ (ξ)

p (α+ β)
≺
√
1 + ξ.

Now, by applying Lemma 1 for γ = p(α+β)
λ , we derive that[

χ (ξ)

ξp

]α−β

≺ Q (ξ) =
p (α+ β)

λ
ξ − p(α+β)

λ

∫ ξ

0
t

p(α+β)
λ

−1 (1 + t)
1
2 dt

=
p (α+ β)

λ

∫ 1

0
u

p(α+β)
λ

−1 (1 + ξu)
1
2 du

= (1 + ξ)
1
2 2Ω1

(
−1

2
, 1;

p (α+ β)

λ
+ 1;

ξ

1 + ξ

)
, (13)

where we have made a change of variables followed by the use of identities in Lemma 2
with a = −1

2 , b =
pα
λn and c = b+ 1. This finishes the proof of Theorem 1.

Taking β = 0 in Theorem 1, we get

Corollary 1. If χ ∈ Bp (λ, α) with
λ
α > 0, then[

χ (ξ)

ξp

]α
≺ Q2 (ξ) = (1 + ξ)

1
2 2Ω1

(
−1

2
, 1;

pα

λ
+ 1;

ξ

1 + ξ

)
≺
√

1 + ξ,

where Q2 (ξ) is the best dominant.

Taking α = 0 in Theorem 1, we get

Corollary 2. If χ ∈ Np (λ, β) with
λ
β > 0, then[

ξp

χ (ξ)

]β
≺ Q3 (ξ) = (1 + ξ)

1
2 2Ω1

(
−1

2
, 1;

pβ

λ
+ 1;

ξ

1 + ξ

)
≺
√
1 + ξ,

where Q3 (ξ) is the best dominant.



T.M. Seoudy, A.E. Shammaky / Eur. J. Pure Appl. Math, 18 (2) (2025), 5869 6 of 14

For a function χ ∈ Ap given by (1), the generalized Bernardi-Libera-Livingston integral
operator Lp,µ : Ap → Ap, with µ > −p, is defined by (see [23–26])

Lp,µχ(ξ) =
µ+ p

ξµ

ξ∫
0

tµ−1χ(t) dt (µ > −p) . (14)

It is easy to verify that for all χ ∈ Ap we have

ξ (Lp,µχ(ξ))
′ = (µ+ p)χ(ξ)− µLp,µχ(ξ). (15)

Theorem 2. If the function χ ∈ Ap satisfies the next subordination condition(
1− α− β

α+ β
λ

)[
Lp,µχ(ξ)

ξp

]α−β

+
α− β

α+ β
λ

χ (ξ)

Lp,µχ(ξ)

[
Lp,µχ(ξ)

ξp

]α−β

≺
√
1 + ξ, (16)

with λ
α+β > 0 and Lp,µ is the integral operator defined by (14), then[

Lp,µχ(ξ)

ξp

]α−β

≺ K(ξ) = (1 + ξ)
1
2 2Ω1

(
−1

2
, 1;

(α+ β) (p+ µ)

λ
+ 1;

ξ

1 + ξ

)
≺
√

1 + ξ,

where the function K is the best dominant of (16).

Proof. Let

ρ(ξ) =

[
Lp,µχ(ξ)

ξp

]α−β

(ξ ∈ U) , (17)

then ρ is analytic function in U. Differentiating (17) with respect to ξ and using (16) in
the resulting relation, we get(

1− α− β

α+ β
λ

)[
Lp,µχ(ξ)

ξp

]α−β

+
α− β

α+ β
λ

χ (ξ)

Lp,µχ(ξ)

[
Lp,µχ(ξ)

ξp

]α−β

= ρ(ξ) +
λξρ′ (ξ)

(α+ β) (p+ µ)
≺
√

1 + ξ.

Using the same method we used to prove Theorem 1, the remaining part of this theorem
can be derived in a similar way.

Theorem 3. χ ∈ BN p (λ, α, β) if and only if[
χ (ξ)

ξp

]α−β

∗

(
1−

[(
1+

λ
p(α+β)

)
e−iθ

(
1+
√

1+eiθ
)
+2

]
ξ+

[
e−iθ

(
1+
√

1+eiθ
)
+1

]
ξ2

(1−ξ)2

)
̸= 0. (18)



T.M. Seoudy, A.E. Shammaky / Eur. J. Pure Appl. Math, 18 (2) (2025), 5869 7 of 14

Proof. For any function χ ∈ Ap, we can confrim that[
χ (ξ)

ξp

]α−β

=

[
χ (ξ)

ξp

]α−β

∗ 1

1− ξ
(19)

and

ξχ′ (ξ)

pχ (ξ)

[
χ (ξ)

ξp

]α−β

=

[
χ (ξ)

ξp

]α−β

∗
1−

(
1− 1

p(α−β)

)
ξ

(1− ξ)2
. (20)

First, in order to prove that (18) holds, we will write (3) by using the principle of
subordination, that is,(

1− α− β

α+ β
λ

)[
χ (ξ)

ξp

]α−β

+
α− β

α+ β
λ
ξχ′ (ξ)

pχ (ξ)

[
χ (ξ)

ξp

]α−β

=
√

1 + w(ξ),

where w (ξ) is a Schwarz function, hence(
1− α− β

α+ β
λ

)[
χ (ξ)

ξp

]α−β

+
α− β

α+ β
λ
ξχ′ (ξ)

pχ (ξ)

[
χ (ξ)

ξp

]α−β

̸=
√
1 + eiθ, (21)

for all ξ ∈ U and 0 ≤ θ < 2π. From (19) and (20), the relation (21) may be written as[
χ (ξ)

ξp

]α−β

∗

1−√
1 + eiθ −

(
1− λ

p(α+β) − 2
√
1 + eiθ

)
ξ −

√
1 + eiθξ2

(1− ξ)2

 ̸= 0,

which is equivalent to[
χ (ξ)

ξp

]α−β

∗

[
1−

[(
1+ λ

p(α+β)

)
e−iθ

(
1+
√

1+eiθ
)
+2

]
ξ+

[
e−iθ

(
1+
√

1+eiθ
)
+1

]
ξ2

(1−ξ)2

]
̸= 0,

that is (18).
Reversely, let χ ∈ Ap satisfy the condition (18). Like it was previously shown, the

assumption (18) is equivalent to (20), that is,(
1− α− β

α+ β
λ

)[
χ (ξ)

ξp

]α−β

+
α− β

α+ β
λ
ξχ′ (ξ)

pχ (ξ)

[
χ (ξ)

ξp

]α−β

̸=
√
1 + eiθ (ξ ∈ U) . (22)

Denoting

φ(ξ) =

(
1− α− β

α+ β
λ

)[
χ (ξ)

ξp

]α−β

+
α− β

α+ β
λ
ξχ′ (ξ)

pχ (ξ)

[
χ (ξ)

ξp

]α−β

and ψ(ξ) =
√

1 + ξ,

the relation (22) could be written as φ(U) ∩ ψ(∂U) = ∅. Therefore, the simply connected
domain φ(U) is included in a connected component of C \ ψ(∂U). From this fact, using
that φ(0) = ψ(0) = 1 together with the univalence of the function ψ, it follows that
φ(ξ) ≺ ψ(ξ), that is χ ∈ BN p (λ, α, β).

Taking β = 0 in Theorem 1, we get



T.M. Seoudy, A.E. Shammaky / Eur. J. Pure Appl. Math, 18 (2) (2025), 5869 8 of 14

Corollary 3. χ ∈ Bp (λ, α) if and only if(
χ (ξ)

ξp

)α

∗

(
1−

[(
1+

λ
pα

)
e−iθ

(
1+
√

1+eiθ
)
+2

]
ξ+

[
e−iθ

(
1+
√

1+eiθ
)
+1

]
ξ2

(1−ξ)2

)
̸= 0.

Taking α = 0 in Theorem 1, we get

Corollary 4. χ ∈ Np (λ, β) if and only if(
ξp

χ (ξ)

)β

∗

(
1−

[(
1+

λ
pβ

)
e−iθ

(
1+
√

1+eiθ
)
+2

]
ξ+

[
e−iθ

(
1+
√

1+eiθ
)
+1

]
ξ2

(1−ξ)2

)
̸= 0.

Theorem 4. If χ (ξ) given by (1) belongs to BN p (λ, α, β), then

|ϱp+1| ≤
|α+ β| p

2 |α− β| |p (α+ β) + λ|
. (23)

Proof. Combining (1) and (3), we obtain(
1− α− β

α+ β
λ

)[
χ (ξ)

ξp

]α−β

+
α− β

α+ β
λ
ξχ′ (ξ)

pχ (ξ)

[
χ (ξ)

ξp

]α−β

= 1 +
(α− β) [p (α+ β) + λ]

p (α+ β)
ϱp+1ξ + .... ≺

√
1 + ξ = 1 +

1

2
ξ − 1

8
ξ2 + ... . (24)

An application of Lemma 3 to (24) yields∣∣∣∣(α− β) [p (α+ β) + λ]

p (α+ β)
ϱp+1

∣∣∣∣ < 1

2
. (25)

Thus, from (25), we easily obtain (23) asserted by Theorem 4.

Taking β = 0 in Theorem 1, we get

Corollary 5. If χ (ξ) given by (1) belongs to Bp (λ, α), then

|ϱp+1| ≤
p

2 |pα+ λ|
.

Taking α = 0 in Theorem 1, we get

Corollary 6. If χ (ξ) given by (1) belongs to Np (λ, β), then

|ϱp+1| ≤
p

2 |pβ + λ|
.



T.M. Seoudy, A.E. Shammaky / Eur. J. Pure Appl. Math, 18 (2) (2025), 5869 9 of 14

3. Fekete-Szegö Problem for BN p (λ, α, β)

In this section we study the Fekete–Szegö inequalities for the class BN p (λ, α, β). It
is worth noting that many authors have been investigated the Fekete-Szegö problem for
several subclasses of analytic functions (see, for instance [27–32]).

Theorem 5. If χ given by (1) belongs to the class BN p (λ, α, β), then∣∣ϱp+2 − µa2p+1

∣∣ ≤ p|α+β|
2|α−β||p(α+β)+2λ| max

{
1;

1

4

∣∣∣1 + p(α+β)[p(α+β)+2λ](α−β+2µ−1)

(α−β)[p(α+β)+λ]2

∣∣∣} . (26)

Proof. If χ ∈ BN p (λ, α, β), then there is a Schwarz function ω in U such that(
1− α− β

α+ β
λ

)[
χ (ξ)

ξp

]α−β

+
α− β

α+ β
λ
ξχ′ (ξ)

pχ (ξ)

[
χ (ξ)

ξp

]α−β

=
√

1 + ω (ξ), (27)

Define the function g (ξ) by

g (ξ) =
1 + ω (ξ)

1− ω (ξ)
= 1 + c1ξ + c2ξ

2 + ... . (28)

Since ω (ξ) is a Schwarz function, we see that g ∈ P with g (0) = 1. Therefore,

√
1 + ω (ξ) =

√
2g (ξ)

g (ξ) + 1
= 1 +

1

4
c1ξ +

(
1

4
c2 −

5

32
c21

)
ξ2 + ... . (29)

Now by substituting (29) in (27), we have(
1− α− β

α+ β
λ

)[
χ (ξ)

ξp

]α−β

+
α− β

α+ β
λ
ξχ′ (ξ)

pχ (ξ)

[
χ (ξ)

ξp

]α−β

= 1+
c1
4
ξ +

(
c2
4

− 5c21
32

)
ξ2 + ... .

Equating the coefficients of ξ and ξ2 we obtain

ϱp+1 =
p (α+ β)

4 (α− β) [p (α+ β) + λ]
c1.

ϱp+2 =
p (α+ β)

4 (α− β) [p (α+ β) + 2λ]

[
c2 −

1

8

(
5 +

p (α+ β) (α− β − 1) [p (α+ β) + 2λ]

(α− β) [p (α+ β) + λ]2

)
c21

]
.

Therefore,

ϱp+2 − µϱ2p+1 =
p (α+ β)

4 (α− β) [p (α+ β) + 2λ]

{
c2 − vc21

}
, (30)

where

ν =
1

8

[
5 +

p (α+ β) [p (α+ β) + 2λ] (α− β + 2µ− 1)

(α− β) [p (α+ β) + λ]2

]
. (31)

Our result now follows by an application of Lemma 4. This completes the proof of Theorem
5.

Putting β = 0 in Theorem 5, we obtain the following.



T.M. Seoudy, A.E. Shammaky / Eur. J. Pure Appl. Math, 18 (2) (2025), 5869 10 of 14

Corollary 7. If χ given by (1) belongs to the class Bp (λ, α), then∣∣ϱp+2 − µϱ2p+1

∣∣ ≤ p

2 |pα+ 2λ|
max

{
1;

1

4

∣∣∣∣1 + p [pα+ 2λ] (α+ 2µ− 1)

(pα+ λ)2

∣∣∣∣} .
Putting α = 0 in Theorem 5, we obtain the following.

Corollary 8. If χ given by (1) belongs to the class Np (λ, β), then∣∣ϱp+2 − µϱ2p+1

∣∣ ≤ p

2 |pβ + 2λ|
max

{
1;

1

4

∣∣∣∣1 + p (pβ + 2λ) (β − 2µ+ 1)

(pβ + λ)2

∣∣∣∣} .

Theorem 6. Let

σ1 =
1

2

(
1− α+ β − 5 (α− β) [p (α+ β) + λ]2

p (α+ β) [p (α+ β) + 2λ]

)
,

σ2 =
1

2

(
1− α+ β +

3 (α− β) [p (α+ β) + λ]2

p (α+ β) [p (α+ β) + 2λ]

)
,

σ3 =
1

2

(
1− α+ β − (α− β) [p (α+ β) + λ]2

p (α+ β) [p (α+ β) + 2λ]

)
.

If χ given by (1) belongs to the class BN p (λ, α, β), then

∣∣ϱp+2 − µϱ2p+1

∣∣ ≤


p(α+β)
8(α−β)

[
− 1

[p(α+β)+2λ] −
p(α+β)(α−β+2µ−1)

(α−β)[p(α+β)+λ]2

]
(µ ≤ σ1)

p(α+β)
2(α−β)[p(α+β)+2λ] (σ1 ≤ µ ≤ σ2)
p(α+β)
8(α−β)

[
1

[p(α+β)+2λ] +
p(α+β)(α−β+2µ−1)

(α−β)[p(α+β)+λ]2

]
(µ ≥ σ2)

Further, if σ1 ≤ µ ≤ σ3, then∣∣ϱp+2 − µϱ2p+1

∣∣+ 1

2

[
5(α−β)[p(α+β)+λ]2

p(α+β)[p(α+β)+2λ] + α− β + 2µ− 1
]
|ϱp+1|2 ≤ p(α+β)

2(α−β)[p(α+β)+2λ] .

If σ3 ≤ µ ≤ σ2, then∣∣ϱp+2 − µϱ2p+1

∣∣+ 1

2

[
3(α−β)[p(α+β)+λ]2

p(α+β)[p(α+β)+2λ] − α+ β − 2µ+ 1
]
|ϱp+1|2 ≤ p(α+β)

2(α−β)[p(α+β)+2λ] .

Proof. Applying Lemma 5 to (30) and (31), we can get our results of Theorem 6.

Putting β = 0 in Theorem 6, we obtain the following.



T.M. Seoudy, A.E. Shammaky / Eur. J. Pure Appl. Math, 18 (2) (2025), 5869 11 of 14

Corollary 9. Let

σ4 =
1

2

(
1− α− 5 (pα+ λ)2

p (pα+ 2λ)

)
, σ5 =

1

2

(
1− α+

3 (pα+ λ)2

p (pα+ 2λ)

)
,

σ6 =
1

2

(
1− α− (pα+ λ)2

p (pα+ 2λ)

)
.

If χ given by (1) belongs to the class Bp (λ, α), then

∣∣ϱp+2 − µϱ2p+1

∣∣ ≤


−p
8

[
1

pα+2λ + p(α+2µ−1)

(pα+λ)2

]
(µ ≤ σ4)

p
2(pα+2λ) (σ4 ≤ µ ≤ σ5)
p
8

[
1

pα+2λ + p(α+2µ−1)

(pα+λ)2

]
(µ ≥ σ5)

Further, if σ4 ≤ µ ≤ σ6, then∣∣ϱp+2 − µϱ2p+1

∣∣+ 1

2

[
5 (pα+ λ)2

p (pα+ 2λ)
+ α+ 2µ− 1

]
|ϱp+1|2 ≤

p

2 (pα+ 2λ)
.

If σ6 ≤ µ ≤ σ5, then∣∣ϱp+2 − µϱ2p+1

∣∣+ 1

2

[
3 (pα+ λ)2

p (pα+ 2λ)
− α− 2µ+ 1

]
|ϱp+1|2 ≤

p

2 (pα+ 2λ)
.

Putting α = 0 in Theorem 6, we obtain the following result.

Corollary 10. Let

σ7 =
1

2

(
1 + β +

5 (pβ + λ)2

p (pβ + 2λ)

)
, σ8 =

1

2

(
1 + β − 3 (pβ + λ)2

p (pβ + 2λ)

)
,

σ9 =
1

2

(
1 + β +

(pβ + λ)2

p (pβ + 2λ)

)
.

If χ given by (1) belongs to the class Bp (λ, β), then

∣∣ϱp+2 − µϱ2p+1

∣∣ ≤


p
8

[
1

pβ+2λ + p(β−2µ+1)

(pβ+λ)2

]
(µ ≤ σ7)

− p
2(pβ+2λ) (σ7 ≤ µ ≤ σ8)

−p
8

[
1

pβ+2λ + p(β−2µ+1)

(pβ+λ)2

]
(µ ≥ σ8)

Further, if σ7 ≤ µ ≤ σ9, then∣∣ϱp+2 − µϱ2p+1

∣∣+ 1

2

[
−5 (pβ + λ)2

p (pβ + 2λ)
− β + 2µ− 1

]
|ϱp+1|2 ≤ − p

2 (pβ + 2λ)
.

If σ9 ≤ µ ≤ σ8, then∣∣ϱp+2 − µϱ2p+1

∣∣+ 1

2

[
−3 (pβ + λ)2

p (pβ + 2λ)
+ β − 2µ+ 1

]
|ϱp+1|2 ≤ − p

2 (pβ + 2λ)
.



T.M. Seoudy, A.E. Shammaky / Eur. J. Pure Appl. Math, 18 (2) (2025), 5869 12 of 14

4. Conclusion

In this presentation, we have defined the subclass of multivalently Bazilevič and Non-
Bazilevič functions that are subordinate to the function of the Bernoulli domain lemnis-
cate BN p (λ, α, β). We have investigated some interesting properties such as subordina-
tion results, convolution properties, coefficients estimate and Fekete-Szegö inequalities for
functions belonging to this subclass. This paper provides significant contributions to the
study of some geometric properties of the Bazilevič and Non-Bazilevič functions. It also
highlights the potential for future research to explore important geometric properties for
similar subclasses of analytic functions involving linear operators.

Funding

This research work was funded by Umm Al-Qura University, Saudi Arabia under grant
number: 25UQU4350561GSSR02.

Acknowledgements

The authors extend their appreciation to Umm Al-Qura University, Saudi Arabia for
funding this research work through grant number: 25UQU4350561GSSR02.

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