EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3336-3355
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some Properties for Certain Subclasses of Spiral-Like
and Robertson Analytic Functions

Tamer M. Seoudy1,2

1 Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt
2 Department of Mathematics, Jamoum University College, Umm Al-Qura University, Makkah,
Saudi Arabia

Abstract. Making use of the definition of subordination, we introduce certain subclasses of spiral-
like and Robertson functions in the open unit disk and study some important results such as
convolution results, coefficients estimate, subordination properties and Fekete-Szego problems for
these subclasses. Further, some known and new outcomes which follow as special cases of our
outcomes are also mentioned.

2020 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Spiral-like function, Robertson function, convolution, subordination,
starlike, convex, Fekete-Szegö problem

1. Introduction

Denote A the family of all analytic functions of the form:

ψ(ξ) = ξ +
∞∑
j=2

ρjξ
j (1)

in U = {ξ ∈ C : |ξ| < 1}. Let Ω be the family of analytic functions ω (ξ) in U that satisfy
the conditions ω(0) = 0 and |ω (ξ)| < 1 (ξ ∈ U). If ψ (ξ) and ϕ (ξ) are analytic in U, we
say that ψ (ξ) is subordinate to ϕ (ξ), written ψ(ξ) ≺ ϕ(ξ) if there exists ω (ξ) ∈ Ω , such
that ψ(ξ) = ϕ(ω(ξ)) (ξ ∈ U) (see [6] and [13]).

For functions ψ (ξ) given by (1) and ϕ (ξ) given by

ϕ(ξ) = ξ +

∞∑
j=2

σjξ
j , (2)

DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5435

Email addresses: tms00@fayoum.edu.eg, tmsaman@uqu.edu.sa (T. M. Seoudy)

https://www.ejpam.com 3336 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3337

the convolution of the functions ψ (ξ) and ϕ (ξ) is defined by

(ψ ∗ ϕ) (ξ) = ξ +
∞∑
j=2

ρj σjξ
j = (ϕ ∗ ψ) (ξ). (3)

For |γ| < π
2 and −1 ≤ D < C ≤ 1, a function ψ (ξ) of A is said to be in Sγ [C,D] if it

it satisfies the following subordination condition:

eiγ
ξψ′ (ξ)

ψ (ξ)
≺ cos γ

(
1 + Cξ

1 +Dξ

)
+ i sin γ, (4)

also, let Kγ [C,D] denote the subfamily of all functions ψ (ξ) in A satisfying the condition
that ξψ′ (ξ) ∈ Sγ [C,D]. Sγ [C,D] and Kγ [C,D] are the subfamilies of spirallike and
Robertson functions respectively studied by several authors earlier ([15], [4, 5]). We note
that S0 [C,D] = S [C,D], K0 [C;D] = K [C;D] with −1 ≤ D < C ≤ 1, where the
subfamilies S [C,D] and K [C;D] of Janowski functions are introduced and studied by
many authors (see [1], [2], [7], [9], [10], [21] and [20]). Also, we have S0 [1− 2λ,−1] = S (λ)
and K0 [1− 2λ,−1] = K (λ) with 0 ≤ λ < 1, where S∗ (λ) and K (λ) denote the subfamilies
of A that consists, respectively, of starlike of order λ and convex of order λ in U (see [17]
and [19]).

Making use of the subordination, we combine the subfamilies Sγ [C,D] and Kγ [C,D]
into a new subfamily SKγ [α, β;C,D] of A as follows:

Definition 1. A function ψ (ξ) ∈ A is said to be in the subfamily SKγ [α, β;C,D] if it
satisfies the following condition:

eiγ
[
(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)

]
≺ cos γ

(
1 + Cξ

1 +Dξ

)
+ i sin γ (5)

(
ξ ∈ U;α, β ≥ 0; |γ| < π

2
;−1 ≤ D < C ≤ 1

)
.

We note that

(i) SKγ [α, 0;C,D] = Sγ [C,D] (see [15])

Sγ [C,D] =

{
ψ (ξ) ∈ A : eiγ

[
ξψ′ (ξ)

ψ (ξ)

]
≺ cos γ

(
1 + Cξ

1 +Dξ

)
+ i sin γ

}
;

(ii) SKγ [0, β;C,D] = Kγ [C,D] (see [4, 5])

Kγ [C,D] =

{
ψ (ξ) ∈ A : eiγ

[
1 +

ξψ′′ (ξ)

ψ′ (ξ)

]
≺ cos γ

(
1 + Cξ

1 +Dξ

)
+ i sin γ

}
;

(iii) SKγ [α, β; 1− 2λ,−1] = SKγ (α, β;λ) (0 ≤ λ < 1)

SKγ (α, β;λ) =

{
ψ (ξ) ∈ A : ℜ

{
eiγ

[
(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)

]}
> λ cos γ

}
;



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3338

(iv) SKγ [α, 0; 1− 2λ,−1] = Sγ (λ) (0 ≤ λ < 1) (see [12] and [11])

Sγ (λ) =
{
ψ (ξ) ∈ A : ℜ

[
eiγ

ξψ′ (ξ)

ψ (ξ)

]
> λ cos γ

}
;

(v) SKγ [α, 0; 1,−1] = S (γ) (see [23])

S (γ) =

{
ψ (ξ) ∈ A : ℜ

[
eiγ

ξψ′ (ξ)

ψ (ξ)

]
> 0

}
;

(vi) SKγ [0, β; 1− 2λ,−1] = Kγ (λ) (0 ≤ λ < 1) (see [12] and [11])

Kγ (λ) =

{
ψ (ξ) ∈ A : ℜ

{
eiγ

[
1 +

ξψ′′ (ξ)

ψ′ (ξ)

]}
> λ cos γ

}
;

(vii) SKγ [0, β; 1,−1] = K (γ) (see [23])

K (γ) =

{
ψ (ξ) ∈ A : ℜ

{
eiγ

[
1 +

ξψ′′ (ξ)

ψ′ (ξ)

]}
> 0

}
;

(viii) SK0 [α, 0;C,D] = S [C,D] (see [9] and [10])

S [C,D] =

{
ψ (ξ) ∈ A :

ξψ′ (ξ)

ψ (ξ)
≺ 1 + Cξ

1 +Dξ

}
;

(ix) SK0 [0, β;C,D] = K [C,D] (see [9], [10] and [2])

K [C,D] =

{
ψ (ξ) ∈ A : 1 +

ξψ′′ (ξ)

ψ′ (ξ)
≺ 1 + Cξ

1 +Dξ

}
;

(x) SK0 [α, β;C,D] = SK [α, β;C,D]

SK [α, β;C,D] =

{
ψ (ξ) ∈ A :

(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)
≺ 1 + Cξ

1 +Dξ

}
;

(xi) SK0 [α, β; 1− 2λ,−1] = SK (α, β;λ) (0 ≤ λ < 1)

SK (α, β;λ) =

{
ψ (ξ) ∈ A : ℜ

(
(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)

)
> λ

}
,

SK (α, 0;λ) = S (λ) and SK (α, β;λ) = K (λ) (see [17]);

(xii) SK0 [α, β; (1− 2λ) η,−η] = SK (α, β;λ, η) (0 ≤ λ < 1, 0 < η ≤ 1)

SK (α, β;λ, η) =

ψ (ξ) ∈ A :

∣∣∣∣∣∣
(α+β)ξψ′(ξ)+βξ2ψ′′(ξ)

αψ(ξ)+βξψ′(ξ) − 1

(α+β)ξψ′(ξ)+βξ2ψ′′(ξ)
αψ(ξ)+βξψ′(ξ) + 1− 2λ

∣∣∣∣∣∣ < η

 ,

SK (α, 0;λ, η) = S (λ, η) and SK (α, β;λ, η) = K (λ, η) (see [8]).



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3339

The aim of the present investigation is to define a general subfamily SKγ [α, β;C,D]
of spirallike and Robertson functions. We then investigate some convolution properties,
membership characterizations, coefficient estimates and subordination result for this sub-
family. Furthermore, Fekete-Szegö problems and several inequalities are studied. Various
corollaries and consequences of most of our outcomes are connected with earlier outcomes
related to the field of investigation here.

2. Convolution Properties

We suppose throughout this paper that α, β ≥ 0, |χ| = 1, −1 ≤ D < C ≤ 1, |γ| < π
2 ,

ξ ∈ U and ψ (ξ) ∈ A given by (1).

Theorem 1. ψ (ξ) ∈ SKγ [α, β;C,D] if and only if

1

ξ

ψ (ξ) ∗
ξ −

(
α−β
α+β + α+2β

α+β Λ
)
ξ2 + α

α+βΛξ
3

(1− ξ)3

 ̸= 0, (6)

where Λ is given by

Λ = Λ (χ, γ, C,D) =
(1 +Dχ) eiγ + (C −D) cos γχ

(C −D) cos γχ
. (7)

Proof. If ψ (ξ) ∈ SKγ [α, β;C,D], then there is a function ω (ξ) ∈ Ω such that

eiγ
[
(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)

]
= cos γ

(
1 + Cω (ξ)

1 +Dω (ξ)

)
+ i sin γ, (8)

hence

eiγ
[
(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)

]
̸= cos γ

(
1 + Cχ

1 +Dχ

)
+ i sin γ (|χ| = 1) ,

which is equivalent to

1

ξ

{
(1 +Dχ) eiγ

[
(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

]
̸= 0

−
[
eiγ + (C cos γ + i D sin γ)χ

] [
αψ (ξ) + βξψ′ (ξ)

]}
̸= 0 (9)

It is easy to verify that

ψ (ξ) ∗ ξ

1− ξ
= ψ (ξ) , (10)

ψ (ξ) ∗ ξ

(1− ξ)2
= ξψ′ (ξ) , (11)

and

ψ (ξ) ∗ 2ξ2

(1− ξ)3
= ξ2ψ′′ (ξ) . (12)



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3340

Using (10),(11) and (12) in (9), we obtain

1

ξ

{
(1 +Dχ) eiγ

[
ψ (ξ) ∗ (α+ β) ξ

(1− ξ)2
+ ψ (ξ) ∗ 2βξ2

(1− ξ)3

]
−

[
eiγ + (C cos γ + i D sin γ)χ

] [
ψ (ξ) ∗ αξ

1− ξ
+ ψ (ξ) ∗ βξ

(1− ξ)2

]}

= (α+β)(D−C) cos γχ
ξ

ψ (ξ) ∗
ξ−

(
α−β
α+β

+α+2β
α+β

[
(1+Dχ)eiγ+(C−D) cos γχ

(C−D) cos γχ

])
ξ2+ α

α+β

[
(1+Dχ)eiγ+(C−D) cos γχ

(C−D) cos γχ

]
ξ3

(1−ξ)3


= (α+β)(D−C) cos γχ

ξ

ψ (ξ) ∗
ξ −

(
α−β
α+β + α+2β

α+β Λ
)
ξ2 + α

α+βΛξ
2

(1− ξ)3

 ̸= 0

which shows the necessary condition of Theorem 1.
Reversely, since, the assumption (9) is equivalent to (6), we get that

eiγ
[
(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)

]
̸= cos γ

(
1 + Cχ

1 +Dχ

)
+ i sin γ, (13)

if we denote

φ (ξ) = eiγ
[
(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)

]
and

ψ (ξ) = cos γ

(
1 + Cξ

1 +Dξ

)
+ i sin γ,

the relation (13) proves that φ (U) ∩ ψ (∂U) = ∅. Thus, the simply-connected domain
φ (U) is subset of a connected component of C\ψ (∂U). From here, using the fact that
φ (0) = ψ (0) = eiγ together with the univalence ψ (ξ), it follows that φ (ξ) subordinate
to ψ (ξ), which leads in fact the subordination (7), i.e. ψ (ξ) ∈ SKγ [α, β;C,D]. This
completes Theorem 1.

Putting γ = 0 in Theorem 1, we get

Corollary 1. ψ (ξ) ∈ SK [α, β;C,D] if and only if

1

ξ

ψ (ξ) ∗
ξ −

(
α−β
α+β + α+2β

α+β Λ1

)
ξ2 + α

α+βΛ1ξ
3

(1− ξ)3

 ̸= 0,

where Λ1 is given by

Λ1 =
1 + Cχ

(C −D)χ
. (14)



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3341

Putting β = 0 in Theorem 1, we get

Corollary 2. [4] ψ (ξ) ∈ Sγ [C,D] if and only if

1

ξ

[
ψ (ξ) ∗ ξ − Λξ2

(1− ξ)2

]
̸= 0,

where Λ is given by (7).

Putting α = 0 in Theorem 1, we get

Corollary 3. [5, Lemma 3 with n = 1] ψ (ξ) ∈ Kγ [C,D] if and only if

1

ξ

[
ψ (ξ) ∗ ξ − (2Λ− 1) ξ2

(1− ξ)3

]
̸= 0,

where Λ is given by (7).

Taking C = 1− 2λ (0 ≤ λ < 1) and D = −1 in Theorem 1, we get

Corollary 4. ψ (ξ) ∈ SKγ (α, β;λ) if and only if

1

ξ

ψ (ξ) ∗
ξ −

(
α−β
α+β + α+2β

α+β Λ2

)
ξ2 + α

α+βΛ2ξ
3

(1− ξ)3

 ̸= 0,

where

Λ2 =
(1− χ) eiγ + 2 (1− λ) cos γχ

2 (1− λ) cos γχ
.

Theorem 2. ψ (ξ) ∈ SKγ [α, β;C,D] if and only if

1−
∞∑
j=2

(
α+ βj

α+ β

)
(j − 1) (1 +Dχ) eiγ − (C −D) cos γχ

(C −D) cos γχ
ρjξ

j−1 ̸= 0. (15)

Proof. From Theorem 1, we have ψ (ξ) ∈ SKγ [α, β;C,D] if and only if

1

ξ

ψ (ξ) ∗
ξ −

(
α−β
α+β + α+2β

α+β Λ
)
ξ2 + α

α+βΛξ
3

(1− ξ)3

 ̸= 0 (16)

for all Λ given by (7). The left hand side of (16) can be written as

1

ξ

[
ψ(ξ) ∗

(
αΛ
α+β

ξ
1−ξ +

α+β−αΛ
α+β

ξ

(1−ξ)2 + β(1−Λ)
α+β

2ξ

(1−ξ)3

)]

=
1

ξ

[
αΛ

α+ β
ψ (ξ) + α+β−αΛ

α+β ξψ′(ξ) +
β (1− Λ)

α+ β
ξ2ψ′′ (ξ)

]



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3342

= 1−
∞∑
j=2

(
β (Λ− 1)

α+ β
j2 +

(α− β) Λ− α

α+ β
j − αΛ

α+ β

)
ρjξ

j−1

= 1−
∞∑
j=2

(
α+ βj

α+ β

)
(j − 1) (1 +Dχ) eiγ − (C −D) cos γχ

(C −D) cos γχ
ρjξ

j−1.

Hence, the proof is completed.

Letting γ = 0 in Theorem 2, we obtain

Corollary 5. ψ (ξ) ∈ SK [α, β;C,D] if and only if

1−
∞∑
j=2

(
α+ βj

α+ β

)
(j − 1) (1 +Dχ)− (C −D)χ

(C −D)χ
ρjξ

j−1 ̸= 0.

Taking β = 0 in Theorem 2, we get

Corollary 6. ψ (ξ) ∈ Sγ [C,D] if and only if

1−
∞∑
j=2

(j − 1) (1 +Dχ) eiγ − (C −D) cos γχ

(C −D) cos γχ
ρjξ

j−1 ̸= 0.

Taking α = 0 in Theorem 2, we get

Corollary 7. ψ (ξ) ∈ Kγ [C,D] if and only if

1−
∞∑
j=2

j
(j − 1) (1 +Dχ) eiγ − (C −D) cos γχ

(C −D) cos γχ
ρjξ

j−1 ̸= 0.

Taking C = 1− 2λ (0 ≤ λ < 1) and D = −1 in Theorem 2, we get

Corollary 8. ψ (ξ) ∈ SKγ (α, β;λ) if and only if

1−
∞∑
j=2

(
α+ βj

α+ β

)
(j − 1) (1− χ) eiγ − 2 (1− λ) cos γχ

2 (1− λ) cos γχ
ρjξ

j−1 ̸= 0. (17)

3. Membership characterizations

Now we obtain several sufficient conditions for the subfamily SKγ [α, β;C,D].

Theorem 3. Let ψ (ξ) ∈ A and let µ be a real number with 0 ≤ µ < 1. If∣∣∣∣(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)
− 1

∣∣∣∣ ≤ 1− µ (ξ ∈ U) , (18)

then ψ (ξ) ∈ SKγ [α, β;C,D] provided that

|γ| ≤ cos−1

[
(1− µ) (1−D)

C −D

]
. (19)



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3343

Proof. From (18) it follows that

(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)
= 1 + (1− µ)ω (ξ) ,

where ω (ξ) ∈ Ω. We have

ℜ
{
eiγ

(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)

}
= ℜ

{
eiγ

}
+ (1− µ)ℜ

{
eiγω (ξ)

}
≥ cos γ − (1− µ)

∣∣eiγω (ξ)
∣∣

> cos γ − (1− µ)

≥
(
1− C

1−D

)
cos γ

provided that |γ| ≤ cos−1
[
(1−µ)(1−D)

C−D

]
. Thus, the proof is completed.

Putting µ = 1− (C−D) cos γ
(1−D) in Theorem 3, we obtain

Corollary 9. If ψ (ξ) ∈ A with∣∣∣∣(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)
− 1

∣∣∣∣ ≤ (C −D) cos γ

(1−D)
(ξ ∈ U) , (20)

then ψ (ξ) ∈ SKγ [α, β;C,D].

Putting γ = 0 in Corollary 9, we obtain

Corollary 10. If ψ (ξ) ∈ A with∣∣∣∣(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)
− 1

∣∣∣∣ ≤ C −D

1−D
(ξ ∈ U) ,

then ψ (ξ) ∈ SK [α, β;C,D].

Putting β = 0 in Corollary 9, we obtain

Corollary 11. If ψ (ξ) ∈ A with∣∣∣∣ξψ′ (ξ)

ψ (ξ)
− 1

∣∣∣∣ ≤ (C −D) cos γ

(1−D)
(ξ ∈ U) ,

then ψ (ξ) ∈ Sγ [C,D].

Putting α = 0 in Corollary 9, we obtain



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3344

Corollary 12. If ψ (ξ) ∈ A with∣∣∣∣ξ2ψ′′ (ξ)

ξψ′ (ξ)

∣∣∣∣ ≤ (C −D) cos γ

(1−D)
(ξ ∈ U) ,

then ψ (ξ) ∈ Kγ [C,D].

Taking C = 1− 2λ (0 ≤ λ < 1) and D = −1 in Corollary 9, we obtain

Corollary 13. If ψ (ξ) ∈ A with∣∣∣∣(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)
− 1

∣∣∣∣ ≤ (1− λ) cos γ (ξ ∈ U) ,

then ψ (ξ) ∈ SKγ (α, β;λ).

In the next theorem, we obtain a coefficients theorem for SKγ [α, β;C,D].

Theorem 4. ψ (ξ) ∈ SKγ [α, β;C,D] if

∞∑
j=2

(
α+ βj

α+ β

)
[(1−D) (j − 1) + (C −D) cos γ] |ρj | ≤ (C −D) cos γ. (21)

Proof. From Corollary 9, it suffices to prove that (20) is satisfied. We have

∣∣∣∣(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)
− 1

∣∣∣∣ =

∣∣∣∣∣∣∣∣∣
∞∑
j=2

(
α+βj
α+β

)
(j − 1) ρjξ

j−1

1 +
∞∑
j=2

(
α+βj
α+β

)
ρjξj−1

∣∣∣∣∣∣∣∣∣
<

∞∑
j=2

(
α+βj
α+β

)
(j − 1) |ρj |

1−
∞∑
j=2

(
α+βj
α+β

)
|ρj |

.

The last expression is bounded above by (C−D) cos γ
(1−D) , if

∞∑
j=2

(
α+ βj

α+ β

)
(j − 1) |ρj | ≤

(C −D) cos γ

(1−D)

1− ∞∑
j=2

(
α+ βj

α+ β

)
|ρj |


which is equivalent to

∞∑
j=2

(
α+ βj

α+ β

)
[(1−D) (j − 1) + (C −D) cos γ] |ρj | ≤ (C −D) cos γ.

This completes the Theorem 4.

Putting γ = 0 in Theorem 4, we obtain



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3345

Corollary 14. ψ (ξ) ∈ SK [α, β;C,D] if

∞∑
j=2

(
α+ βj

α+ β

)
[(1−D) (j − 1) + C −D] |ρj | ≤ C −D. (22)

Putting β = 0 in Theorem 4, we obtain

Corollary 15. ψ (ξ) ∈ Sγ [C,D] if

∞∑
j=2

[(1−D) (j − 1) + (C −D) cos γ] |ρj | ≤ (C −D) cos γ. (23)

Putting α = 0 in Theorem 4, we obtain

Corollary 16. ψ (ξ) ∈ Kγ [C,D] if

∞∑
j=2

j [(1−D) (j − 1) + (C −D) cos γ] |ρj | ≤ (C −D) cos γ. (24)

Taking C = 1− 2λ (0 ≤ λ < 1) and D = −1 in Theorem 4, we obtain

Corollary 17. ψ (ξ) ∈ SKγ (α, β;λ) if

∞∑
j=2

(
α+ βj

α+ β

)
[j − 1 + (1− λ) cos γ] |ρj | ≤ (1− λ) cos γ. (25)

4. Subordination result

Before proving our subordination result for SKγ [α, β;C,D], we shall make use the
following definitions and a lemma.

Definition 2. [24] We say that a complex sequence {σj}∞j=1 is a subordinating factor

sequence (SFS) if, whenever ψ (ξ) = ξ +
∞∑
j=2

ρjξ
j is univalent (analytic) and convex in U,

we have
∞∑
j=1

ρj σj ≺ ψ (ξ) (ρ1 = 1; ξ ∈ U) . (26)

Lemma 1. [24] The complex sequence {σj}∞j=1 is a subordinating factor sequence (SFS)
if and only if

ℜ

1 + 2

∞∑
j=1

σj ξ
j

 > 0 (ξ ∈ U) . (27)



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3346

Theorem 5. Let ψ (ξ) ∈ SKγ [α, β;C,D] satisfy the coefficient inequality (21) and let
ϕ (ξ) ∈ K, then (

α+2β
α+β

)
[1−D + (C −D) cos γ]

2
[
(C −D) cos γ +

(
α+2β
α+β

)
[1−D + (C −D) cos γ]

] (ψ ∗ ϕ) (ξ) ≺ ϕ (ξ) (28)

and

ℜ{ψ (ξ)} > −
(C −D) cos γ +

(
α+2β
α+β

)
[1−D + (C −D) cos γ](

α+2β
α+β

)
[1−D + (C −D) cos γ]

. (29)

The constant factor (
α+2β
α+β

)
[1−D + (C −D) cos γ]

2
[
(C −D) cos γ +

(
α+2β
α+β

)
[1−D + (C −D) cos γ]

]
in (28) cannot be replaced by a larger number.

Proof. Let ψ (ξ) ∈ SKγ [α, β;C,D] satisfy the coefficient inequality (21) and suppose
that

ϕ (ξ) = ξ +

∞∑
j=2

σjξ
j ∈ K.

Then, by Definition 2, the condition (28) will hold true if
(
α+2β
α+β

)
[1−D + (C −D) cos γ]

2
[
(C −D) cos γ +

(
α+2β
α+β

)
[1−D + (C −D) cos γ]

]ρj


∞

j=1

is a subordinating factor sequence, with σ1 = 1. From Lemma 1, it is equivalent to the
inequality

ℜ

1 +
∞∑
j=1

(
α+2β
α+β

)
[1−D + (C −D) cos γ]

(C −D) cos γ +
(
α+2β
α+β

)
[1−D + (C −D) cos γ]

ρj ξ
j

 > 0 (ξ ∈ U) . (30)

By noting the fact that(
α+ βj

α+ β

)[
(1−D) (j − 1) + (C −D) cos γ

(C −D) cos γ

]
is an increasing for j ≥ 2. In view of (21), when |ξ| = r < 1, we have

ℜ

1 +

(
α+2β
α+β

)
[1−D + (C −D) cos γ]

(C −D) cos γ +
(
α+2β
α+β

)
[1−D + (C −D) cos γ]

∞∑
j=1

ρj ξ
j





T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3347

= ℜ

1 +

(
α+2β
α+β

)
[1−D + (C −D) cos γ]

(C −D) cos γ +
(
α+2β
α+β

)
[1−D + (C −D) cos γ]

ξ

+

∞∑
j=2

(
α+2β
α+β

)
[1−D + (C −D) cos γ] ρj ξ

j

(C −D) cos γ +
(
α+2β
α+β

)
[1−D + (C −D) cos γ]


≥ 1−

(
α+2β
α+β

)
[1−D + (C −D) cos γ]

(C −D) cos γ +
(
α+2β
α+β

)
[1−D + (C −D) cos γ]

r

−

∞∑
j=2

(
α+jβ
α+β

)
[(1−D) (j − 1) + (C −D) cos γ] |ρj | rj

(C −D) cos γ +
(
α+2β
α+β

)
[1−D + (C −D) cos γ]

≥ 1−

(
α+2β
α+β

)
[1−D + (C −D) cos γ]

(C −D) cos γ +
(
α+2β
α+β

)
[1−D + (C −D) cos γ]

r

− (C −D) cos γ

(C −D) cos γ +
(
α+2β
α+β

)
[1−D + (C −D) cos γ]

r

= 1− r > 0 (|ξ| = r < 1) .

This proves (30) and (28). The inequality (29) follows from (28) by letting

ϕ (ξ) =
ξ

1− ξ
= ξ +

∞∑
j=2

ξj ∈ K.

The sharpness of the multiplying factor in (28) can be established by considering a function

Ψ (ξ) = ξ − (C −D) cos γ(
α+2β
α+β

)
[1−D + (C −D) cos γ]

ξ2.

Clearly Ψ ∈ SKγ [α, β;C,D] satisfy (21). Using (28) we infer that(
α+2β
α+β

)
[1−D + (C −D) cos γ]

2
{
(C −D) cos γ +

(
α+2β
α+β

)
[1−D + (C −D) cos γ]

}Ψ(ξ) ≺ ξ

1− ξ
,

and it follows that

min
|ξ|≤r


(
α+2β
α+β

)
[1−D + (C −D) cos γ]

2
{
(C −D) cos γ +

(
α+2β
α+β

)
[1−D + (C −D) cos γ]

}ℜ{Ψ(ξ)}

 = −1

2
.



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3348

This shows that the constant(
α+2β
α+β

)
[1−D + (C −D) cos γ]

2
{
(C −D) cos γ +

(
α+2β
α+β

)
[1−D + (C −D) cos γ]

}
cannot be replaced by any larger one.

For γ = 0 in Theorem 5, we get

Corollary 18. Let ψ (ξ) ∈ SK [α, β;C,D] satisfy the coefficient inequality (22) and let
ϕ (ξ) ∈ K, then (

α+2β
α+β

)
(1− 2D + C)

2
[
C −D +

(
α+2β
α+β

)
(1− 2D + C)

] (ψ ∗ ϕ) (ξ) ≺ ϕ (ξ) (31)

and

ℜ{ψ (ξ)} > −
C −D +

(
α+2β
α+β

)
(1− 2D + C)(

α+2β
α+β

)
(1− 2D + C)

. (32)

The constant factor (
α+2β
α+β

)
(1− 2D + C)

2
[
C −D +

(
α+2β
α+β

)
(1− 2D + C)

]
in (31) cannot be replaced by a larger number.

Taking β = 0 in Theorem 5, we get

Corollary 19. Let ψ (ξ) ∈ Sγ [C,D] satisfy the coefficient inequality (23) and let ϕ (ξ) ∈
K, then

1−D + (C −D) cos γ

2 [2 (C −D) cos γ + 1−D]
(ψ ∗ ϕ) (ξ) ≺ ϕ (ξ) (33)

and

ℜ{ψ (ξ)} > −2 (C −D) cos γ + 1−D

1−D + (C −D) cos γ
. (34)

The constant factor
1−D + (C −D) cos γ

2 [2 (C −D) cos γ + 1−D]

in (33) cannot be replaced by a larger number.

Taking α = 0 in Theorem 5, we get



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3349

Corollary 20. Let ψ (ξ) ∈ Kγ [C,D] satisfy the coefficient inequality (24) and let ϕ (ξ) ∈
K, then

1−D + (C −D) cos γ

3 (C −D) cos γ + 2 (1−D)
(ψ ∗ ϕ) (ξ) ≺ ϕ (ξ) (35)

and

ℜ{ψ (ξ)} > −3 (C −D) cos γ + 2 (1−D)

2 [1−D + (C −D) cos γ]
. (36)

The constant factor
1−D + (C −D) cos γ

3 (C −D) cos γ + 2 (1−D)

in (35) cannot be replaced by a larger number.

Taking C = 1− 2λ (0 ≤ λ < 1) and D = −1 in Theorem 5, we get

Corollary 21. Let ψ (ξ) ∈ SKγ (α, β;λ) satisfy the coefficient inequality (25) and let
ϕ (ξ) ∈ K, then (

α+2β
α+β

)
[1 + (1− λ) cos γ]

2
{
(1− λ) cos γ +

(
α+2β
α+β

)
[1 + (1− λ) cos γ]

} (ψ ∗ ϕ) (ξ) ≺ ϕ (ξ) (37)

and

ℜ{ψ (ξ)} > −
(1− λ) cos γ +

(
α+2β
α+β

)
[1 + (1− λ) cos γ](

α+2β
α+β

)
[1 + (1− λ) cos γ]

. (38)

The constant factor (
α+2β
α+β

)
[1 + (1− λ) cos γ]

2
{
(1− λ) cos γ +

(
α+2β
α+β

)
[1 + (1− λ) cos γ]

}
in (37) cannot be replaced by a larger number.

5. Fekete-Szegö problems

The Fekete-Szegö problem consists in finding upper-bounds for
∣∣ρ3 − µρ22

∣∣ for various
subfamilies of analytic functions (see [3], [16], [18] and [22]). In order to get upper-bounds
for

∣∣ρ3 − µρ22
∣∣ for the subfamily SKγ [α, β;C,D] the next lemma is required.

Lemma 2. [14, p.108]Let ω ∈ Ω be given by

ω (ξ) =

∞∑
j=1

ωj ξ
j (ξ ∈ U) .



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3350

Then
|ω1| ≤ 1, |ω2| ≤ 1− |ω1|2 , (39)

and ∣∣ω2 − ν ω2
1

∣∣ ≤ max {1, |ν|} , (40)

for any complex number ν ∈ C. The functions ω(ξ) = ξ and ω(ξ) = ξ2or one of their
rotations show that both inequalities (39) and (40) are sharp.

First we obtain upper-bounds for
∣∣ρ3 − µρ22

∣∣ with µ ∈ R.

Theorem 6. Let ψ (ξ) ∈ SKγ [α, β;C,D] and let µ ∈ R. Then

∣∣ρ3 − µρ22
∣∣ ≤


(α+β)(C−D) cos γ

2(α+3β)

[
−D + (C −D)

(
1− 2µ(α+β)(α+3β)

(α+2β)2

)]
(µ ≤ ϑ1)

(α+β)(C−D) cos γ
2(α+3β) (ϑ1 ≤ µ ≤ ϑ2)

(α+β)(C−D) cos γ
2(α+3β)

[
D − (C −D)

(
1− 2µ(α+β)(α+3β)

(α+2β)2

)]
(µ ≥ ϑ2)

(41)
where

ϑ1 =
(α+ 2β)2 (C − 2D − 1)

2 (α+ β) (α+ 3β) (C −D)
(42)

ϑ2 =
(α+ 2β)2 (C − 2D + 1)

2 (α+ β) (α+ 3β) (C −D)
. (43)

Proof. Suppose that ψ (ξ) is in SKγ [α, β;C,D]. Then, from the definition of the
subclass SKγ [α, β;C,D], there exists

ω (ξ) = ω1ξ + ω2ξ
2 + ω3ξ

3 + ... ∈ Ω

such that

eiγ
[
(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)

]
= cos γ

(
1 + Cω (ξ)

1 +Dω (ξ)

)
+ i sin γ (ξ ∈ U) . (44)

We have

eiγ
[
(α+ β) ξψ′ (ξ) + βξ2ψ′′ (ξ)

αψ (ξ) + βξψ′ (ξ)

]
= eiγ + eiγ

(
α+ 2β

α+ β

)
ρ2ξ

+eiγ

[
2 (α+ 3β)

α+ β
ρ3 −

(α+ 2β)2

(α+ β)2
ρ22

]
ξ2 + .... . (45)

and

cos γ
(

1+Cω(ξ)
1+Dω(ξ)

)
+i sin γ = eiγ+(C −D) cos γω1ξ+(C −D) cos γ

(
ω2 −Dω2

1

)
ξ2+... . (46)

By using (45) and (46), equating the coefficients of ξ and ξ2 on both sides of (44), we
have

ρ2 =
(α+ β) (C −D) e−iγ cos γ

α+ 2β
ω1 (47)



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3351

and

ρ3 =
(α+ β) (C −D) e−iγ cos γ

2 (α+ 3β)

[
ω2 +

(
−D + (C −D) e−iγ cos γ

)
ω2
1

]
. (48)

It follows∣∣ρ3 − µρ22
∣∣ ≤ (α+β)(C−D) cos γ

2(α+3β)

{
|ω2|+

∣∣∣−D + (C −D) e−iγ cos γ
[
1− 2µ(α+β)(α+3β)

(α+2β)2

]∣∣∣ |ω1|2
}

Making use of Lemma 2 we have∣∣ρ3 − µρ22
∣∣ ≤ (α+β)(C−D) cos γ

2(α+3β)

{
1 +

(∣∣∣−D + (C −D) e−iγ cos γ
[
1− 2µ(α+β)(α+3β)

(α+2β)2

]∣∣∣− 1
)

|ω1|2
}

or ∣∣ρ3 − µρ22
∣∣ ≤ (α+β)(C−D) cos γ

2(α+3β)

[
1 +

(√
D2 +Π(Π− 2D) cos2 γ − 1

)
|ω1|2

]
, (49)

where

Π = (C −D)

[
1− 2µ (α+ β) (α+ 3β)

(α+ 2β)2

]
. (50)

Denote by

H (x, y) = 1 +
(√

D2 +Π(Π− 2D)x2 − 1
)
y2

where x = cos γ, y = |ω1| and (x, y) : [0, 1]× [0, 1]. Simple calculation shows that H (x, y)
does not have a local maximum at any interior point of the rectangle (0, 1)× (0, 1). Thus,
the maximum must be attained at a boundary point. Since H (x, 0) = 1, H (0, y) =
1 + (|D| − 1) y2 ≤ 1 and H (1, 1) = |Π−D|, it follows that the maximal value of H (x, y)
may be H (0, 0) = 1 or H (1, 1) = |Π−D|. Hence, from (49) we obtain∣∣ρ3 − µρ22

∣∣ ≤ (α+β)(C−D) cos γ
2(α+3β) max {1, |Π−D|} , (51)

where Π is given by (50). Consider first the case |Π−D| ≥ 1. If µ ≤ ϑ1, where ϑ1 is given
by (42), then Π ≥ 1 +D and from (51) we obtain∣∣ρ3 − µρ22

∣∣ ≤ (α+β)(C−D) cos γ
2(α+3β)

[
−D + (C −D)

(
1− 2µ(α+β)(α+3β)

(α+2β)2

)]
which is the first part of the inequality (41). If µ ≥ ϑ2, where ϑ2 is given by (43), then
Π ≤ D − 1 and it follows from (51) that∣∣ρ3 − µρ22

∣∣ ≤ (α+β)(C−D) cos γ
2(α+3β)

[
D − (C −D)

(
1− 2µ(α+β)(α+3β)

(α+2β)2

)]
and this is the third part of (41).

Next, suppose ϑ1 ≤ µ ≤ ϑ2. Then, |Π−D| ≤ 1 and thus, from (51) we obtain∣∣ρ3 − µρ22
∣∣ ≤ (α+β)(C−D) cos γ

2(α+3β)

which is the second part of the inequality (41).

For γ = 0 in Theorem 6, we obtain



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3352

Corollary 22. Let ψ (ξ) ∈ SK [α, β;C,D] and let µ ∈ R. Then

∣∣ρ3 − µρ22
∣∣ ≤


(α+β)(C−D)

2(α+3β)

[
−D + (C −D)

(
1− 2µ(α+β)(α+3β)

(α+2β)2

)]
(µ ≤ ϑ1)

(α+β)(C−D)
2(α+3β) (ϑ1 ≤ µ ≤ ϑ2)

(α+β)(C−D)
2(α+3β)

[
D − (C −D)

(
1− 2µ(α+β)(α+3β)

(α+2β)2

)]
(µ ≥ ϑ2)

where ϑ1 and ϑ2 are given by (42) and (43).

Taking β = 0 in Theorem 6, we obtain

Corollary 23. Let ψ (ξ) ∈ Sγ [C,D] and let µ ∈ R. Then

∣∣ρ3 − µρ22
∣∣ ≤


(C−D) cos γ

2 [−D + (C −D) (1− 2µ)] (µ ≤ ϑ1)
(C−D) cos γ

2 (ϑ1 ≤ µ ≤ ϑ2)
(C−D) cos γ

2 [D − (C −D) (1− 2µ)] (µ ≥ ϑ2)

where

ϑ3 =
C − 2D − 1

2 (C −D)
, ϑ4 =

C − 2D + 1

2 (C −D)
.

Taking α = 0 in Theorem 6, we obtain

Corollary 24. Let ψ (ξ) ∈ Kγ [C,D] and let µ ∈ R. Then

∣∣ρ3 − µρ22
∣∣ ≤


(C−D) cos γ

6

[
−D + (C −D)

(
1− 3

2µ
)]

(µ ≤ ϑ5)
(C−D) cos γ

6 (ϑ5 ≤ µ ≤ ϑ6)
(C−D) cos γ

6

[
D − (C −D)

(
1− 3

2µ
)]

(µ ≥ ϑ6)

where

ϑ5 =
2 (C − 2D − 1)

3 (C −D)
, ϑ6 =

2 (C − 2D + 1)

3 (C −D)
.

Taking C = 1− 2λ (0 ≤ λ < 1) and D = −1 in Theorem 6, we obtain

Corollary 25. Let ψ (ξ) ∈ SKγ (α, β;λ) and let µ ∈ R. Then

∣∣ρ3 − µρ22
∣∣ ≤


(α+β)(1−λ) cos γ

(α+3β)

[
1 + 2 (1− λ)

(
1− 2µ(α+β)(α+3β)

(α+2β)2

)]
(µ ≤ ϑ7)

(α+β)(1−λ) cos γ
(α+3β) (ϑ7 ≤ µ ≤ ϑ8)

(α+β)(1−λ) cos γ
(α+3β)

[
−1− 2 (1− λ)

(
1− 2µ(α+β)(α+3β)

(α+2β)2

)]
(µ ≥ ϑ8)

where

ϑ7 =
(α+ 2β)2 (1− λ)

2 (α+ β) (α+ 3β) (1− λ)

ϑ8 =
(α+ 2β)2 (2− λ)

2 (α+ β) (α+ 3β) (1− λ)
.



T. M. Seoudy / Eur. J. Pure Appl. Math, 17 (4) (2024), 3336-3355 3353

We consider the Fekete-Szegö problem for the subclass SKγ [α, β;C,D] with complex
parameter µ ∈ C.

Theorem 7. Let ψ (ξ) ∈ SKγ [α, β;C,D] and let µ ∈ C. Then,∣∣ρ3 − µρ22
∣∣ ≤ (α+β)(C−D) cos γ

2(α+3β) max
{
1,
∣∣∣D + (C −D) e−iγ cos γ

[
2µ(α+β)(α+3β)

(α+2β)2
− 1

]∣∣∣} .
(52)

Proof. Assume that ψ (ξ) ∈ SKγ [α, β;C,D]. Making use of (47) and (48) we obtain∣∣ρ3 − µρ22
∣∣ ≤ (α+β)(C−D) cos γ

2(α+3β)

∣∣∣ω2 −
(
D + (C −D) e−iγ cos γ

[
2µ(α+β)(α+3β)

(α+2β)2
− 1

])
ω2
1

∣∣∣
The inequality (52) follows as an application of Lemma 2 with

ν = D + (C −D) e−iγ cos γ
[
2µ(α+β)(α+3β)

(α+2β)2
− 1

]
.

For γ = 0 in Theorem 7, we obtain

Corollary 26. Let ψ (ξ) ∈ SK [α, β;C,D] and let µ ∈ C. Then,∣∣ρ3 − µρ22
∣∣ ≤ (α+β)(C−D)

2(α+3β) max
{
1,
∣∣∣D + (C −D)

[
2µ(α+β)(α+3β)

(α+2β)2
− 1

]∣∣∣} .
Taking β = 0 in Theorem 7, we obtain

Corollary 27. Let ψ (ξ) ∈ Sγ [C,D] and let µ ∈ C. Then,∣∣ρ3 − µρ22
∣∣ ≤ (C−D) cos γ

2 max
{
1,
∣∣D + (C −D) e−iγ cos γ (2µ− 1)

∣∣} .
Taking α = 0 in Theorem 7, we obtain

Corollary 28. Let ψ (ξ) ∈ Kγ [C,D] and let µ ∈ C. Then,∣∣ρ3 − µρ22
∣∣ ≤ (C−D) cos γ

6 max
{
1,
∣∣D + (C −D) e−iγ cos γ

(
3
2µ− 1

)∣∣} .
Taking C = 1− 2λ (0 ≤ λ < 1) and D = −1 in Theorem 6, we obtain

Corollary 29. Let ψ (ξ) ∈ SKγ (α, β;λ) and let µ ∈ C. Then,∣∣ρ3 − µρ22
∣∣ ≤ (α+β)(1−λ) cos γ

(α+3β) max
{
1,
∣∣∣−1 + 2 (1− λ) e−iγ cos γ

[
2µ(α+β)(α+3β)

(α+2β)2
− 1

]∣∣∣} .
6. Conclusions

In our present investigation, we have defined a general subclass SKγ [α, β;C,D] of
spirallike and Robertson analytic functions. For functions belonging to this subclass, we
have derived some interesting results such as convolution properties, membership char-
acterizations, coefficient estimates, subordination result and the Fekete-Szegö estimates.
Furthermore, interesting corollaries and particular cases are shown for each of those results
for particular choices of parameters found in the definition of this subclass. Our results are
connected with those in several earlier works, which are related to the Geometric Function
Theory. Moreover, these results can be extended to multivalent functions and meromophic
functions.



REFERENCES 3354

References

[1] O P Ahuja. Families of analytic functions related to ruscheweyh derivatives and
subordinate to convex functions. Yokohama Math., 41:39–50, 1993.

[2] M K Aouf and T M Seoudy. Classes of analytic functions related to the dziok-
srivastava operator. Integral Transforms Spec. Funct., 22(6):423–430, 2011.

[3] M K Aouf and T M Seoudy. Fekete–szegö problem for certain subclass of analytic
functions with complex order defined by q–analogue of ruscheweyh operator. Con-
structive Math. Anal., 3(1):36–44, 2020.

[4] S S Bhoosnurmath and M V Devadas. Subclasses of spirallike functions defined by
subordination. J. of Analysis, Madras, 4:173–183, 1996.

[5] S S Bhoosnurmath and M V Devadas. Subclasses of spirallike functions defined by
ruschweyh derivatives. Tamkang J. Math., 28:59–65, 1997.

[6] T Bulboacă. Differential Subordinations and Superordinations, Recent Results. House
of Scientific Book Publ., Cluj-Napoca, 2005.

[7] R M Goel and B S Mehrok. On the coefficients of a subclass of starlike functions.
Indian J. Pure Appl. Math., 12:634–647, 1981.

[8] V P Gupta and P K Jain. Certain classes of univalent functions with negative coef-
ficients ii. Bull. Austral. Math. Soc., 15(3):467–473, 1976.

[9] W Janowski. Some extremal problems for certain families of analytic functions. Bull.
Polish Acad. Sci., 21:17–25, 1973.

[10] W Janowski. Some extremal problems for certain families of analytic functions. Ann.
Polon. Math., 28:297–326, 1973.

[11] F R Keogh and E P Merkes. A coefficient inequality for certain classes of analytic
functions. Proc. Amer. Math. Soc., 20:8–12, 1969.

[12] R J Libera. Univalent a-spiral functions. Canad. J. Math., 19:449–456, 1967.

[13] S S Miller and P T Mocanu. Differential Subordination: Theory and Applications,
Series on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225.
Marcel Dekker Inc., New York and Basel, 2000.

[14] Z Nehari. Conformal mapping. McGraw-Hill, New-York, 1952.

[15] S V Nikitin. A class of regular functions, current problems in function theory (russian).
Rostov-Gos. Univ. Rostov-on-Don., 188:143–147, 1987.



REFERENCES 3355

[16] H Orhan, D Raducanu, M Caglar, and M Rayram. Coefficient estimates and other
properties for a class of spirallike functions associated with a differential operator.
Abstr. Anal. Appl., Art. ID 415319:1–7, 2013.

[17] M S Robertson. On the theory of univalent functions. Ann. Math., 37:374–408, 1936.

[18] T M Seoudy. Convolution results and fekete-szegö inequalities for certain classes of
symmetric q-starlike and symmetric q-convex functions. J. Math., Art. ID 8203921:1–
11, 2022.

[19] H. Silverman. Univalent functions with negative coefficients. Proc. Amer. Math. Soc.,
51:109–116, 1975.

[20] H Silverman and E M Silvia. Subclasses of starlike functions subordinate to convex
functions. Canad. J. Math., 1:48–61, 1985.

[21] H Silverman, E M Silvia, and D Telage. Convolution conditions for convexity, star-
likeness and spiral-likeness. Math. Z., 162:125–130, 1978.

[22] H M Srivastava, T M Seoudy, and M K Aouf. A generalized conic domain and its
applications to certain subclasses of multivalent functions associated with the basic
(or q-) calculus. AIMS Math., 6:6580–6602, 2021.

[23] L Špaček. Contribution à la theorie des fonctions univalents. Cas. Mat. Fys., 62:12–
19, 1932.

[24] H S Wilf. Subordinating factor sequence for convex maps of the unit circle. Proc.
Amer. Math. Soc., 12:689–693, 1961.