International Journal of Analysis and Applications

Volume 17, Number 6 (2019), 974-979

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-974

ON PROPERTIES OF CERTAIN ANALYTIC MULTIPLIER TRANSFORM OF

COMPLEX ORDER

DEBORAH OLUFUNMILAYO MAKINDE1,∗, SHAHRAM NAJAFZADEH2

1Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria

2Department of Mathematics, Payame Noor University, P. O. Box: 19395–3697, Tehran, Iran

∗Corresponding author: funmideb@yahoo.com

Abstract. The focus of this paper is to investigate the subclasses S∗C(γ,µ,α,λ; b), TS∗C(γ,µ,α,λ; b) =

T ∩ S∗C(γ,µ,α,λ; b) and obtain the coefficient bounds as well as establishing its relationship with certain

existing results in the literature.

1. Introduction

Let A be the class of normalized analytic functions f in the open unit disc U = {z ∈ C : |z| < 1} with

f(0) = f′(0) = 0 and of the form

f(z) = z +

∞∑
n=2

anz
n, an ∈ C, (1.1)

and S the class of all functions in A that are univalent in U. Also, the subclass of functions in A that are

of the form

f(z) = z −
∞∑
n=2

anz
n, an ≥ 0, (1.2)

is denoted by T and the subclasses S∗(α), C(γ) are given respectively by

S∗(α) =

{
f ∈ S : Re

(
zf′(z)

f(z)

)
> γ z ∈ U

}
(1.3)

Received 2019-07-05; accepted 2019-08-12; published 2019-11-01.

2010 Mathematics Subject Classification. 30C45.

Key words and phrases. analyticity; univalent; linear transformation; coefficient bounds.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

974

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-974


Int. J. Anal. Appl. 17 (6) (2019) 975

C(α) =

{
f ∈ S : Re

(
1 +

zf′′(z)

f′(z)

)
> γ z ∈ U,≥ γ < 1

}
. (1.4)

Moreover, the class TS∗(γ) denoted by T ∩ S∗(γ) which is the subclass of function f ∈ T such that f is

starlike of order γ and respectively, TC(γ) is the class of function f ∈ T such that f is convex of order γ.

An interesting unification of the classes S∗(α) and C(γ) denoted by S∗C(γ,β) which satisfies the condition

Re

{
zf′(z) + βz2f′(z)

βzf′(z) + (1 −β)f(z)

}
> γ 0 ≥ γ < 1,z ∈ U. (1.5)

has been extensively studied by different researchers, for example, see [6] and [1,2,3]. The special cases for

β = 0, 1 are given by S∗(γ) and C((γ)) respectively.

Furthermore, the class TS∗C(γ,β) which is the subclass of function f ∈ T such that f belongs the class

S∗C(γ,β), was studied by Altintas et al. and other researchers. For details see [ 3, 5, 6 ].

Using the unification in (5), Nizami Mustafa [6] introduced and investigated the class S∗C(γ,β; τ) and

TS∗C(γ,β; τ), 0 ≤ α < 1; β ∈ [0, 1]; τ ∈ C which he defined as follows

A function f ∈ S given by (1.1) is said to belong to the class S∗C(γ,β; τ) if the following condition is satisfied

Re

{
1 +

1

τ

[
zf′(z) + βz2f′(z)

βzf′(z) + (1 −β)f(z)
− 1
]}

> γ 0 ≥ γ < 1; β ∈ [0, 1]; τ ∈ C −{0},z ∈ U. (1.6)

Meanwhile, the author in [4] defined a linear transformation Dmα,λf by

Dmα,λf(z) = z +

∞∑
n=2

α

(
1 + λ(n + α− 2)

1 + λ(α− 1)

)m
anz

n, 0 ≤ λ ≤ 1; α ≥ 1; m ∈ N∪ 0 (1.7)

Motivated by the work of Mustafa in [6], we study the effect of the application of the linear operator Dmα,λf

on the unification of the classes of the functions S∗C(γ,β; τ).

Now, we define the class S∗C(γ,α,λ; b) to be class of functions f ∈ S which satisfies the condition

Re

{
1 +

1

b

[
z(Dmα,λf)

′(z) + µz2(Dmα,λf)
′′(z)

µz(Dmα,λf)
′(z) + (1 −µ)(Dmα,λf)(z)

− 1

]}
> γ, 0 ≥ γ < 1,z ∈ U; 0 ≤ λ,µ ≤ 1; α ≥ 1; m ∈ N∪0

(1.8)

Also, we denote by DT the subclass of the class of functions in (7) which is of the form

Dmα,λf(z) = z −
∞∑
n=2

α

(
1 + λ(n + α− 2)

1 + λ(α− 1)

)m
anz

n, 0 ≤ λ,µ ≤ 1; α ≥ 1; m ∈ N∪ 0 (1.9)

and denote by TS∗C(γ,µ,α,λ; b) = T ∩S∗C(γ,µ,α,λ; b) which is the class of functions f in (1.9) such that

f belong to the class S∗C(γ,µ,α,λ; b) = T ∩S∗C(γ,µ,α,λ; b).

In this paper, we investigate the subclasses S∗C(γ,µ,α,λ; b) and

TS∗C(γ,µ,α,λ; b) = T ∩S∗C(γ,µ,α,λ; b)



Int. J. Anal. Appl. 17 (6) (2019) 976

2. Coeffiecient bounds for the classes S∗Cλα(γ,µ; b) and TS
∗Cλα(γ,µ; b)

Theorem 2.1. Let f be as defined in (1.1). Then the function Dmα,λf belongs to the class S
∗C(γ,µ,α,λ; b),

0 ≥ γ < 1,z ∈ U; 0 ≤ λ,µ ≤ 1; α ≥ 1; m ∈ N∪ 0

if
∞∑
n=2

[
α

(
1 + λ(n + α− 2)

1 + λ(α− 1)

)m
[1 + µ(n− 1)][n + |b|(1 −γ) − 1]

]
|an| ≤ |b|(1 −γ)

The result is sharp for the function

Dmα,λf(z) = z +
|b|(1 −γ)(1 + λ(α− 1))m

α[1 + µ(n− 1)][n + |b|(1 −γ)](1 + λ(n + α− 2))m
zn n ≥ 2

Proof. By (1.8), f belong to the class S∗C(γ,µ,α,λ; b) if

Re

{
1 +

1

b

[
z(Dmα,λf)

′(z) + µz2(Dmα,λf)
′′(z)

µz(Dmα,λf)
′(z) + (1 −µ)(Dmα,λf)(z)

− 1

]}
> γ

It suffices to show that: ∣∣∣∣∣1b
[

z(Dmα,λf)
′(z) + µz2(Dmα,λf)

′′(z)

µz(Dmα,λf)
′(z) + (1 −µ)(Dmα,λf)(z)

− 1

]∣∣∣∣∣ < 1 −γ (2.1)
Simple computation in (2.1), using (1.7), we have:∣∣∣∣∣1b

[
z(Dmα,λf)

′(z) + µz2(Dmα,λf)
′′(z)

µz(Dmα,λf)
′(z) + (1 −µ)(Dmα,λf)(z)

− 1

]∣∣∣∣∣

=

∣∣∣∣∣∣1b

 z + ∑∞n=2 nα

(
1+λ(n+α−2)
1+λ(α−1)

)m
anz

n + µ
∑∞
n=2 n(n− 1)α

(
1+λ(n+α−2)
1+λ(α−1)

)m
anz

n

µz +
∑∞
n=2 µnα

(
1+λ(n+α−2)
1+λ(α−1)

)m
anzn + (1 −µ)

(
z +

∑∞
n=2 α

(
1+λ(n+α−2)
1+λ(α−1)

)m
anzn

) − 1


∣∣∣∣∣∣

=

∣∣∣∣∣∣1b

z + ∑∞n=2 nα[1 + µ(n− 1)]

(
1+λ(n+α−2)
1+λ(α−1)

)m
anz

n

z +
∑∞
n=2 α(1 + µ(n− 1))

(
1+λ(n+α−2)
1+λ(α−1)

)m
anzn

− 1



∣∣∣∣∣∣

≤
1

b


∑∞n=2 α(n− 1)[1 + µ(n− 1)]

(
1+λ(n+α−2)
1+λ(α−1)

)m
|an|

1 −
∑∞
n=2 α(1 + µ(n− 1))

(
1+λ(n+α−2)
1+λ(α−1)

)m
|an|




which is bounded by 1 −γ if∑∞
n=2 α(n− 1)[1 + µ(n− 1)]

(
1+λ(n+α−2)
1+λ(α−1)

)m
|an| ≤ |b|(1 −γ)1 −

∑∞
n=2 α(1 + µ(n− 1))

(
1+λ(n+α−2)
1+λ(α−1)

)m
|an|

which is equivalent to∑∞
n=2

[
α(n− 1)[1 + µ(n− 1)]

(
1+λ(n+α−2)
1+λ(α−1)

)m
+ α|b|(1 −γ)(1 + µ(n− 1))

(
1+λ(n+α−2)
1+λ(α−1)

)m]
|an| ≤ |b|(1 −γ)

Which implies that

∞∑
n=2

[
α

(
1 + λ(n + α− 2)

1 + λ(α− 1)

)m
[1 + µ(n− 1)][n + |b|(1 −γ) − 1]

]
|an| ≤ |b|(1 −γ) (2.2)



Int. J. Anal. Appl. 17 (6) (2019) 977

Thus, (2.1) is satisfied if (2.2) is satisfied. �

Corollary 2.1. Let f be as defined in (1) and the function Dmα,λf belongs to the class S
∗C(γ,µ,α,λ; b),

0 ≥ γ < 1,z ∈ U; 0 ≤ λ,µ ≤ 1; α ≥ 1; m ∈ N∪ 0. Then

|an| ≤
|b|(1 −γ)(1 + λ(α− 1))m

α[1 + µ(n− 1)][n + |b|(1 −γ) − 1](1 + λ(n + α− 2))m

Corollary 2.2. Let f be as defined in (1.1). Then the function Dmα,λf belongs to the class S
∗C(γ,µ, 1,λ,m; b),

0 ≥ γ < 1,z ∈ U; 0 ≤ λ,µ ≤ 1; m ∈ N∪ 0 if
∞∑
n=2

[(1 + λ(n− 1))m [1 + µ(n− 1)][n + |b|(1 −γ) − 1]] |an| ≤ |b|(1 −γ) (2.3)

The result is sharp for the function

Dmα,λf(z) = z +
|b|(1 −γ)

[1 + µ(n− 1)][n + |b|(1 −γ) − 1](1 + λ(n− 1))m
zn, n ≥ 2

Corollary 2.3. Let f be as defined in (1.1). Then the function Dmα,λf belongs to the class S
∗C(γ,µ, 1,λ, 1; b),

0 ≥ γ < 1,z ∈ U; 0 ≤ λ,µ ≤ 1; m ∈ N∪ 0 if
∞∑
n=2

[(1 + λ(n− 1)) [1 + µ(n− 1)][n + |b|(1 −γ) − 1]] |an| ≤ |b|(1 −γ) (2.4)

The result is sharp for the function

Dmα,λf(z) = z +
|b|(1 −γ)

[1 + µ(n− 1)][n + |b|(1 −γ) − 1](1 + λ(n− 1))
zn, n ≥ 2

Corollary 2.4. Let f be as defined in (1.1). Then the function Dmα,λf belongs to the class S
∗C(γ,µ, 1, 1, 1; b),

0 ≥ γ < 1,z ∈ U; 0 ≤ λ,µ ≤ 1; m ∈ N∪ 0 if
∞∑
n=2

[n[1 + µ(n− 1)][n + |b|(1 −γ) − 1]] |an| ≤ |b|(1 −γ) (2.5)

The result is sharp for the function

Dmα,λf(z) = z +
|b|(1 −γ)

n[1 + µ(n− 1)][n + |b|(1 −γ) − 1]
zn, n ≥ 2

Corollary 2.5. Let f be as defined in (1.1). Then the function Dmα,λf belongs to the class S
∗C(γ,µ, 1, 0, 1; b),

0 ≥ γ < 1,z ∈ U; 0 ≤ λ,µ ≤ 1; m ∈ N∪ 0 if
∞∑
n=2

[[1 + µ(n− 1)][n + |b|(1 −γ) − 1]] |an| ≤ |b|(1 −γ) (2.6)

The result is sharp for the function

Dmα,λf(z) = z +
|b|(1 −γ)

[1 + µ(n− 1)][n + |b|(1 −γ)−]
zn, n ≥ 2

This result agrees with the Theorem 2.1 in [6].



Int. J. Anal. Appl. 17 (6) (2019) 978

Corollary 2.6. Let f be as defined in (1.1). Then the function Dmα,λf belongs to the class S
∗C(γ, 0, 1,λ, 0; 1),

0 ≥ γ < 1,z ∈ U; 0 ≤ λ,µ ≤ 1; m ∈ N∪ 0 if
∞∑
n=2

[[1 + µ(n− 1)][n−γ]] |an| ≤ 1 −γ (2.7)

The result is sharp for the function

Dmα,λf(z) = z +
1 −γ

[1 + µ(n− 1)][n−γ]
zn, n ≥ 2

This result agrees with the Corollary 2.1 in [6].

Corollary 2.7. Let f be as defined in (1.1). Then the function Dmα,λf belongs to the class S
∗C(γ,µ, 1,λ, 0; 1),

0 ≥ γ < 1,z ∈ U; 0 ≤ λ,µ ≤ 1; m ∈ N∪ 0 if
∞∑
n=2

(n−γ)|an| ≤ 1 −γ (2.8)

The result is sharp for the function

Dmα,λf(z) = z +
1 −γ
n−γ

zn, n ≥ 2

This result agrees with the Corollary 2.2 in [6].

Theorem 2.2. Let f ∈ DT . Then the function Dmα,λf belongs to the class DTS
∗C(γ,µ,α,λ; b),

0 ≥ γ < 1,z ∈ U; 0 ≤ λ,µ ≤ 1; α ≥ 1; m ∈ N∪ 0 if and only if
∞∑
n=2

α(n− 1)[1 + µ(n− 1)][n + b(1 −γ)]
(
x

y

)m
|an| ≤ |b|(1 −γ)

Proof. We shall prove only the necessity part of the Theorem as the sufficiency proof is similar to the proof

of Theorem 1.

Let f be as defined in (1.1) and Dmα,λf belongs to the class TS
∗C(γ,µ,α,λ; b), 0 ≥ γ < 1,z ∈ U; 0 ≤

λ,µ ≤ 1; α ≥ 1; m ∈ N∪ 0; b ∈ R−{0}, we have

Re

{
1 +

1

b

[
z(Dmα,λf)

′(z) + µz2(Dmα,λf)
′′(z)

µz(Dmα,λf)
′(z) + (1 −µ)(Dmα,λf)(z)

− 1

]}
> γ (2.9)

Using (1.7) in (2.9) and by algebraic simplification, we have

Re



−
∑∞
n=2 α(n− 1)[1 + µ(n− 1)]

(
1+λ(n+α−2)
1+λ(α−1)

)m
anz

n

b
{
z −

∑∞
n=2 α(1 + µ(n− 1))

(
1+λ(n+α−2)
1+λ(α−1)

)m
anzn

}

 ≥ γ − 1

Choosing z to be real and z −→ 1, we have

−
∑∞
n=2 α(n− 1)[1 + µ(n− 1)]

(
1+λ(n+α−2)
1+λ(α−1)

)m
an

b
{

1 −
∑∞
n=2 α(1 + µ(n− 1))

(
1+λ(n+α−2)
1+λ(α−1)

)m
an

} ≥ γ − 1 (2.10)



Int. J. Anal. Appl. 17 (6) (2019) 979

b ∈ R−{0} implies that b could be greater or less than zero.

Let b > 0 in (19), we have

−
∞∑
n=2

α(n− 1)[1 + µ(n− 1)]
(
x

y

)m
an ≥ (γ − 1)b

{
1 −

∞∑
n=2

α(1 + µ(n− 1))
(
x

y

)m
an

}
(2.11)

where x = 1 + λ(n + α− 2) and y = 1 + λ(α− 1) From (20), we have
∞∑
n=2

α(n− 1)[1 + µ(n− 1)][n + b(1 −γ)]
(
x

y

)m
|an| ≤ b(1 −γ) (2.12)

Now suppose b < 0, which implies that b = −|b| and substituting b = −|b| in (19), we have∑∞
n=2 α(n− 1)[1 + µ(n− 1)]

(
x
y

)m
an

|b|
{

1 −
∑∞
n=2 α(1 + µ(n− 1))

(
x
y

)m
an

} ≥ (2.13)
∑∞
n=2 α(n− 1)[1 + µ(n− 1)]

(
x
y

)m
|an| ≥ (γ − 1)|b|

{
1 −

∑∞
n=2 α(1 + µ(n− 1))

(
x
y

)m
an

}
which implies

∞∑
n=2

α(n− 1)[1 + µ(n− 1)][n + b(1 −γ)]
(

1 + λ(n + α− 2)
1 + λ(α− 1)

)m
|an| ≥−b(1 −γ) (2.14)

From (21) and (23), the proof of the necessity is completed. �

References

[1] Altintas, O., On a subclass of certain starlike functions with negative coefficient. Math. Japon., 36 (1991), 489-495.

[2] Altintas, O., Irmak, H. and Srivastava, H. M., Fractional calculus and certain starlke functions with negative coefficients.

Comput. Math. Appl., 30 (1995), No. 2, 9-16.

[3] Altintas, O., Özkan, Ö. and Srivastava, H. M., Neighborhoods of a Certain Family of Multivalent Functions with Negative

Coefficients. Comput. Math. Appl., 47 (2004), 1667- 1672.

[4] Makinde, D.O., A new multiplier differential operator. Adv. Math., Sci. J., 7 (2018), no.2, 109 -114.

[5] Irmak, H., Lee, S. H. and Cho, N. E., Some multivalently starlike functions with negative coefficients and their subclasses

defined by using a differential operator. Kyungpook Math. J., 37 (1997), 43-51.

[6] Mustafa, N., The various properties of certain subclasses of analytic functions of complex order. arXiv:1704.04980

[math.CV], 2017.


	1. Introduction
	2. Coeffiecient bounds for the classes S*C(,;b) and TS*C(,;b)
	References