International Journal of Analysis and Applications
ISSN 2291-8639
Volume 13, Number 2 (2017), 170-177
http://www.etamaths.com

ON STABILITY OF CONVOLUTION OF JANOWSKI FUNCTIONS

KHALIDA INAYAT NOOR AND HUMAYOUN SHAHID∗

Abstract. In this paper, the classes S∗ [A, B] and C [A, B] are discussed in terms of dual sets. Using
duality, various geometric properties of mentioned class are analyzed. Problem of neighborhood as

well as stability of convolution of S∗ [A, B] and C [A, B] are studied. Some of our results generalize
previously known results.

1. Introduction

Let A be the class of all functions of the form

f (z) = z +

∞∑
n=2

anz
n, (1.1)

which are analytic in open unit disc E = {z ∈ C : |z| < 1}.
Let S ⊂A be the class of functions which are univalent and also S∗ (α) and C (α) be the well known

subclasses of S which, respectively consist of starlike and convex functions of order α. If f (z) and g (z)
are analytic in E, we say that f (z) is subordinate to g (z) , written as f ≺ g or f (z) ≺ g (z) if there
exist a Schwarz function w (z) which is analytic in E with w (0) = 0 and |w (z)| < 1 where z ∈ E, such
that f (z) = g (w (z)) , z ∈ E. Also, if g ∈ S, then

f (z) ≺ g (z) if and only if f (0) = g (0) and f (E) ⊂ g (E) .

A number of subclasses of analytic functions were introduced using subordination. In 1973, Janowski
[2] introduced the class P [A,B] which is defined as

P [A,B] =

{
p (z) : p (z) ≺

1 + Az

1 + Bz

}
,

where −1 ≤ B < A ≤ 1. Geometrically, p (E) is contained in the open disc centered on the real axis
having diameter end points 1−A

1−B and
1+A
1+B

with centered at 1−AB
1−B2 . For specific values of A and B we

obtain many known subclasses of P [A,B] . Some specific cases include

(i). P [1,−1] = P, the class of Caratheodory functions.
(ii). P [1 − 2α,−1] = P (α) , the class of Caratheodory functions of order α.
(iii). p (z) ∈ P [α, 0] satisfies the condition |p (z) − 1| < α, see [4].
Using P [A,B] , Janowski [2] introduced S∗ [A,B] and C [A,B] which are defined as

S∗ [A,B] =

{
f ∈A :

zf′ (z)

f (z)
≺

1 + Az

1 + Bz
, z ∈ E

}
and

C [A,B] =

{
f ∈A :

(zf′ (z))
′

f′ (z)
≺

1 + Az

1 + Bz
, z ∈ E

}
,

where −1 ≤ B < A ≤ 1. We note that Alexander relation holds between S∗ [A,B] and C [A,B].

Received 27th October, 2016; accepted 9th January, 2017; published 1st March, 2017.
2010 Mathematics Subject Classification. 30C45.
Key words and phrases. convex; univalent functions; convolution; coefficent result; neighbourhood; stability of con-

volution; Janowski functions.

c©2017 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

170



STABILITY OF CONVOLUTION OF JANOWSKI FUNCTIONS 171

The convolution (Hadamard) of two functions f (z) given by (1.1) and g (z) = z+
∞∑
n=2

bnz
n is defined

as

(f ∗g) (z) = (g ∗f) (z) = z +
∞∑
n=2

anbnz
n.

Let V ⊂A the dual set V ∗ (see [6]) is defined as following

V ∗ =

{
g ∈A :

(f ∗g) (z)
z

6= 0, ∀f ∈ A, z ∈ E
}
. (1.2)

Silverman et al. [8] proved that S∗ [A,B] = G∗, where G∗ represents the dual set of G defined in
(1.2) and G is given by

G =

{
g ∈A : g (z) =

z −Lz2

(1 −z)2

}
, (1.3)

where L = e
−iθ+A
A−B and θ ∈ [0, 2π]. Using the Alexander type relation, C [A,B] = H

∗ where

H =

{
h ∈A : h (z) =

z + (1 − 2L) z2

(1 −z)3

}
, (1.4)

where L is same as given in (1.3) and −1 ≤ B < A ≤ 1.

For f ∈A and is of form (1.1) and δ ≥ 0, the Nδ neighborhood of function f is defined as following
(see [7]).

Nδ (f) =

{
g (z) = z +

∞∑
n=2

bnz
n ∈ A :

∞∑
n=2

n |bn −an| ≤ δ

}
.

Ruscheweyh proved many inclusion results of Nδ (f) especially N1
4

(f) ⊂ S∗ for all f ∈ C.

For X, Y ⊂ A. The convolution is called stable univalent if there exist δ > 0 such that Nδ (f) ∗
Nδ (g) ⊂ S, where f ∈ X and g ∈ Y. The constant δ is defined as

δ (X ∗Y,Z) = sup{δ : Nδ (f) ∗Nδ (g) ⊂ Z} . (1.5)

In the current paper, we estimate the coefficient bounds of functions given in (1.3) and (1.4). Using
these estimates we discuss some interesting properties of Nδ (f) for different classes and inclusion
properties of Nδ (f) .

2. Preliminaries

To prove our main results, we need the following Lemmas.

Lemma 2.1. [8]. Let −1 ≤ B < A ≤ 1 and θ ∈ [0, 2π] . Then G∗ = S∗ [A,B] , where

G =

{
g ∈ A : g (z) =

z − e
−iθ+A
A−B z

2

(1 −z)2

}
.

Lemma 2.2. [8]. Let −1 ≤ B < A ≤ 1 and θ ∈ [0, 2π] . Then H∗ = C [A,B] , where

H =


h ∈ A : h (z) =

z +
(

1 − 2e
−iθ+A
A−B

)
z2

(1 −z)3


 . (2.1)

Lemma 2.3. [5]. Let Ψ be convex and g be starlike in E. Then, for F analytic in E with F (0) = 1,
Ψ∗Fg
Ψ∗g is contained in the convex hull of F (E) .



172 NOOR AND SHAHID

3. Main Results

Theorem 3.1. Let −1 ≤ B < A ≤ 1, then for h (z) = z +
∞∑
n=2

cnz
n ∈ G,∣∣∣∣n (1 + B) − (A + 1)A−B

∣∣∣∣ ≤ |cn| ≤ n (1 −B) + A− 1A−B ,
Proof. For h ∈ G, the coefficients can be written as

cn = n(1 −L) + L,

where L = e
−iθ+A
A−B and θ ∈ [0, 2π] . To find the maximum value of |cn (θ)| where θ varies from 0 to 2π,

consider

|cn (θ)|
2

=
(nB −A)2 + (n− 1)2 + 2 (n− 1) (nB −A) cos θ

(A−B)2
= φ (θ) ,

φ (θ) attains its maximum value at θ = π as φ′ (π) = −2(n−1)(nB−A)
(A−B)2 sin π = 0 and φ

′′ (π) =

−2(n−1)(nB−A)
(A−B)2 cos π < 0 as nB − A < 0.The maximum value of φ (θ) is φ (π) =

(
n(B−1)+1−A

A−B

)2
,

we note that φ (θ) ≤ φ (π) for all θ ∈ [0, 2π] . Substituting the value of φ (π) we obtain

|cn| ≤
n (1 −B) + A− 1

A−B
.

Now again consider φ (θ) and we note that φ (z) has its minimum at θ = 0 and φ (0) =
(
n(B+1)−(A+1)

A−B

)2
.

Thus we obtain

|cn| ≥
∣∣∣∣n (B + 1) − (A + 1)A−B

∣∣∣∣ .
This completes the proof. �

Applying the Alexander type relation between set G and H we obtain following

Corollary 3.1. Let −1 ≤ B < A ≤ 1, then for h (z) = z +
∞∑
n=2

cnz
n ∈ H,∣∣∣∣n [n (B + 1) − (A + 1)]A−B

∣∣∣∣ ≤ |cn| ≤ n [n (1 −B) + A− 1]A−B .
Corollary 3.2. Let −1 ≤ B < A ≤ 1 and let f (z) = z + λzn, n ≥ 2. Then f ∈ S∗ [A,B] if and only if

|λ| ≤
A−B

n (1 −B) + A− 1
. (3.1)

Proof. Let f (z) = z + λzn where λ is given in inequality (3.1) and then for g ∈ G, consider∣∣∣∣(f ∗g) (z)z
∣∣∣∣ ≥ 1 −|λ| |cn|zn−1, z ∈ E.

Now using Theorem 3.1 and value of λ given in (3.1), we obtain∣∣∣∣(f ∗g) (z)z
∣∣∣∣ > 0, z ∈ E.

Hence f ∈ S∗ [A,B] . Conversely, now consider f (z) = z + λzn ∈ S∗ [A,B] and let g (z) = z +
∞∑
n=2

n(1−B)+A−1
A−B z

n and

(f ∗g) (z)
z

= 1 + λ
n (1 −B) + A− 1

A−B
zn−1 6= 0.

If |λ| > A−B
n(1−B)+A−1 , then there exist ξ ∈ E such that

(f ∗g) (ξ)
ξ

= 0.

which is a contradiction, hence |λ| ≤ A−B
n(1−B)+A−1. �



STABILITY OF CONVOLUTION OF JANOWSKI FUNCTIONS 173

Corollary 3.3. Let −1 ≤ B < A ≤ 1 and let f (z) = z + λzn, n ≥ 2. Then f ∈ C [A,B] if and only if

|λ| ≤
A−B

n [n (1 −B) + A− 1]
.

Using the coefficient bounds of functions in set G, we now give alternate method to prove the
Theorem given in [1].

Corollary 3.4. Let −1 ≤ B < A ≤−1 and let f is of the form (1.1) and satisfy
∞∑
n=2

[n (1 −B) + A− 1] |an| ≤ A−B,

then f ∈ S∗ [A,B] .

Proof. Let f (z) = z +
∞∑
n=2

anz
n and g (z) = z +

∞∑
n=2

n(1−B)+A−1
A−B z

n, consider

(f ∗g) (z)
z

= 1 +

∞∑
n=2

n (1 −B) + A− 1
A−B

anz
n−1, z ∈ E.

it is known from Lemma 2.1 that f ∈ S∗ [A,B] if and only if (f∗g)(z)
z

6= 0. Now∣∣∣∣(f ∗g) (z)z
∣∣∣∣ ≥ 1 − ∞∑

n=2

n (1 −B) + A− 1
A−B

|an| |z|
n−1

> 0,

which gives us the required condition. �

We now consider two specific functions

Fα (z) =
f (z) + αz

1 + α
(3.2)

and

Fn,α (z) = f (z) +
α

n
zn (n ≥ 2) . (3.3)

Here α is a non zero complex number also we note that if f (z) ∈A, then both Fα (z) and Fn,α (z) ∈A.
The geometric properties of these functions are studied by various authors (see [3]). Using these two
functions, we study the geometric properties of Nδ (f) for classes of S

∗ [A,B] and C [A,B]. We first
discuss the relation between f (z) and Fα (z) in the following Lemma.

Lemma 3.1. Let −1 ≤ B < A ≤ 1, f ∈ A and δ > 0 and let for for all α ∈ C, Fα ∈ S∗ [A,B] (or
C [A,B]), then f ∈ S∗ [A,B] (or C [A,B]) furthermore for all g ∈ G (or H)∣∣∣∣(f ∗g) (z)z

∣∣∣∣ > δ,
where |α| < δ and z ∈ E.

Proof. Since Fα ∈ S∗ [A,B] then by Lemma 2.1, we know that for all g ∈ G,
(Fα ∗g) (z)

z
6= 0, z ∈ E.

Using (3.2) and simplifying, we obtain

(f ∗g) (z)
z

6= −α,

for all α. Thus we obtain ∣∣∣∣(f ∗g) (z)z
∣∣∣∣ > δ.

Using Lemma 2.1, we obtain that f ∈ S∗ [A,B] . This completes the proof. �

Applying the similar method, we have the following result.



174 NOOR AND SHAHID

Lemma 3.2. Let −1 ≤ B < A ≤ 1, f ∈ A and δ > 0 and let for for all α, Fn,α ∈ S∗ [A,B] , then for
all h ∈ G ∣∣∣∣(f ∗h) (z)zcn

∣∣∣∣ > δn,
where |α| < δ and z ∈ E.

Using Theorem 3.1 in Lemma 3.1, we obtain the following.

Corollary 3.5. Let −1 ≤ B < A ≤ 1, f ∈ A and δ > 0 and let for for all α, Fn,α ∈ S∗ [A,B] , then
for all h ∈ G ∣∣∣∣(f ∗h) (z)z

∣∣∣∣ > δn
∣∣∣∣n (B + 1) − (A + 1)A−B

∣∣∣∣ ,
where |α| < δ and z ∈ E.

We now prove the following

Theorem 3.2. Let −1 ≤ B < A ≤ 1 and δ > 0 if for all α, Fα ∈ S∗ [A,B] then Nδ1 (f) ⊂ S∗ [A,B]
where

δ1 =
δ (A−B)

1 −B
.

Proof. Let g ∈ Nδ1 (f) and g (z) = z +
∞∑
n=2

bnz
n. To prove that g ∈ S∗ [A,B] , it is enough to show that

(g ∗h) (z)
z

6= 0,

where h ∈ G and z ∈ E.
Consider ∣∣∣∣(g ∗h) (z)z

∣∣∣∣ =
∣∣∣∣(f ∗h) (z)z + ((g −f) ∗h) (z)z

∣∣∣∣
≥

∣∣∣∣(f ∗h) (z)z
∣∣∣∣−
∣∣∣∣((g −f) ∗h) (z)z

∣∣∣∣ .
Using Lemma 3.1 and series representations of f (z) , g (z) and h (z) , we obtain∣∣∣∣(g ∗h) (z)z

∣∣∣∣ > δ − ∞∑
n=2

(n (1 −B) − (1 −A)) |bn −an|
A−B

. (3.4)

Since
∞∑
n=2

(n (1 −B) − (1 −A)) |bn −an|
A−B

≤
1 −B
A−B

∞∑
n=2

n |bn −an| ≤
1 −B
A−B

δ1. (3.5)

Using (3.4) in (3.5), we obtain ∣∣∣∣(g ∗h) (z)z
∣∣∣∣ > δ − 1 −BA−Bδ1 > 0.

Hence

δ1 =
δ (A−B)

1 −B
.

This completes the proof. �

Theorem 3.3. Let −1 ≤ B < A ≤ 1. f ∈ C [A,B] , then Fα ∈ S∗ [A,B] for |α| < 14.

Proof. Let f (z) = z +
∞∑
n=2

anz
n. Then

Fα (z) =
f (z) + αz

(1 + α)

= (f (z) ∗ψ (z)) , z ∈ E.



STABILITY OF CONVOLUTION OF JANOWSKI FUNCTIONS 175

Here

ψ (z) =
z − α

1+α
z2

1 −z
.

Using the properties of convolution we obtain

f (z) ∗ψ (z) = zf′ (z) ∗
(
ψ (z) ∗ log

(
1

1 −z

))
.

Since f ∈ C [A,B], zf′ ∈ S∗ [A,B] , also if |α| < 1
4
, ψ ∈ S∗. Applying the convolution we obtain

ψ (z) ∗ log
(

1

1 −z

)
=

z∫
0

ψ (t)

t
dt. (3.6)

Using the Alexander relation in (3.6), we obtain ψ (z)∗log
(

1
1−z

)
∈ C. Using Lemma 2.3 one can prove

that C ∗S∗ [A,B] ⊂ S∗ [A,B] , hence

Fα (z) = zf
′ (z) ∗

(
ψ (z) ∗ log

(
1

1 −z

))
∈ S∗ [A,B] , |α| <

1

4
.

This completes the proof. �

We now prove the following.

Theorem 3.4. Let −1 ≤ B < A ≤−1. If f ∈ C [A,B] , then Nδ (f) ⊂ S∗ [A,B] where δ = A−B4(1−B).

Proof. If f ∈ C [A,B] , then by Theorem 3.3 Fα ∈ S∗ [A,B] for |α| < 14. choosing δ =
1
4

and applying
Theorem 3.2, we obtain our required result. �

For specific values of A and B we have the following

Corollary 3.6. [7]. If f ∈ C [1,−1] = C, then Nδ (f) ⊂ S∗ where δ = 14.

Corollary 3.7. If f ∈ C [1 − 2β,−1] = C (β) , then Nδ (f) ⊂ S∗ (β) where δ = 1−β4 and 0 ≤ β < 1.
We now prove the stability of convolution given in (1.5) for different classes of Nδ (f) . In the next

Theorem I represent the identity function I (z) = z.

Theorem 3.5. Let −1 ≤ B < A ≤ 1. The following relation holds

δ (I ∗ I,C [A,B]) ≥
√
A−B
1 −B

(3.7)

δ (I ∗ I,S∗ [A,B]) ≥
√

2(A−B)
1 −B

(3.8)

δ (C [A,B] ∗C,C [A,B]) = 0 (3.9)

δ (S∗ [A,B] ∗C,C [A,B]) = 0 (3.10)

δ (C [A,B] ∗C,S∗ [A,B]) ≥

√
4 +

(A−B)2

2 (1 −B)2
− 2 = δ0. (3.11)

Proof. Let f,g ∈ Nδ (I) , then applying definition of Nδ (f) , we obtain
∞∑
n=2

n |an| ≤ δ and
∞∑
n=2

n |bn| ≤ δ.

Consider
∞∑
n=2

n (n (1 −B) − 1 + A) |an| |bn|
A−B

≤
1 −B
A−B

∞∑
n=2

n2 |an| |bn| ≤
1 −B
A−B

δ2.

Now for h ∈ H,∣∣∣∣((f ∗g) ∗h) (z)z
∣∣∣∣ ≥ ∞∑

n=2

n (n (1 −B) − 1 + A) |an| |bn|
A−B

− 1 ≥
1 −B
A−B

δ2 − 1 > 0.



176 NOOR AND SHAHID

Using value of δ given in (3.7), we obtain our first inequality.
Similarly consider f, g ∈ Nδ (I) and consider

∞∑
n=2

(n (1 −B) − 1 + A) |an| |bn|
A−B

≤
1 −B
A−B

∞∑
n=2

n2 |an| |bn| ≤
1 −B

2 (A−B)
δ2,

Thus we obtain∣∣∣∣((f ∗g) ∗h) (z)z
∣∣∣∣ ≥ ∞∑

n=2

(n (1 −B) − 1 + A) |an| |bn|
A−B

− 1 ≥
1 −B

2(A−B)
δ2 − 1 > 0.

Which gives us inequality in (3.8).

To prove (3.9), consider f (z) = z +
(

A−B
2(2(1−B)−1+A)

)
z2 ∈ C [A,B]and g (z) = g0 (z) + δ2z

2 ∈ C,
where g0 (z) =

z
1−z . Taking the convolution of f and g, we get

(f ∗g) (z) = z +
(

A−B
2 (2 (1 −B) − 1 + A)

+
δ (A−B)

4 (2 (1 −B) − 1 + A)

)
z2,

Applying Corollary 3.3 with n = 2, (f ∗g) (z) ∈ C [A,B] if and only if, δ = 0.
To prove (3.10), we are applying the same method with f (z) = z +

(
A−B

(2(1−B)−1+A)

)
z2 and g (z) =

g0 (z) +
δ
2
z2.

For relation given in (3.11), consider f0 ∈ C [A,B] and g0 ∈ C and f ∈ Nδ (f0) and g ∈ Nδ (g0),
then for h ∈ G ∣∣∣∣(f ∗g ∗h) (z)z

∣∣∣∣ ≥
∣∣∣∣(f0 ∗g0 ∗h) (z)z

∣∣∣∣−
∣∣∣∣(f0 ∗ (g −g0) ∗h) (z)z

∣∣∣∣
−
∣∣∣∣(g0 ∗ (f −f0) ∗h) (z)z

∣∣∣∣ (3.12)
−
∣∣∣∣((f −f0) ∗ (g −g0) ∗h) (z)z

∣∣∣∣ .
Applying Lemma 2.3 one can prove f0 ∗g0 ∈ S∗ [A,B] and using Theorem 3.4, we obtain∣∣∣∣(f ∗g ∗h) (z)z

∣∣∣∣ > A−B4 (1 −B). (3.13)
If f0 (z) = z +

∞∑
n=2

a0nz
n and g0 (z) = z +

∞∑
n=2

b0nz
n and we know that f0 (z) ∈ C [A,B] ⊂ C therefore

|a0n| ≤ 1 and |b0n| ≤ 1. Now∣∣∣∣(f0 ∗ (g −g0) ∗h) (z)z
∣∣∣∣ ≤ ∞∑

n=2

|a0n| |bn − b0n| |n (1 −B) − 1 + A|
A−B

≤
1 −B
A−B

δ. (3.14)

Using definition of Nδ (f) , we know that n |an −a0n| ≤ δ or |an −a0n| ≤ δ2 as n ≥ 2. Now consider∣∣∣∣((f −f0) ∗ (g −g0) ∗h) (z)z
∣∣∣∣ ≤ ∞∑

n=2

|an −a0n| |bn − b0n| |n (1 −B) − 1 + A|
A−B

≤
(1 −B) δ2

2 (A−B)
. (3.15)

Using (3.13), (3.14) and (3.15) in (3.12), we obtain∣∣∣∣(f ∗g ∗h) (z)z
∣∣∣∣ ≥ A−B4 (1 −B) − 2 (1 −B)A−B δ − (1 −B) δ

2

2 (A−B)
> 0.

Solving for δ we obtain relation given in (3.11) which is non negative when δ ≤ δ0. This completes the
proof. �



STABILITY OF CONVOLUTION OF JANOWSKI FUNCTIONS 177

References

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Yokohama Math. J., 41(1993), 39-50.
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298-326.

[3] S. Kanas, Stability of convolution and dual sets for the class of k−uniformly convex and k−starlike functions,
Zeszyty Naukowe Politechniki Rzeszowskiej Matematyka, 22(1998), 51-64.

[4] S. Ponnusamy and V. Singh, Convolution properties of some classes of analytic functions, J. Math. Sci., 89(1998),

1008-1020.
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Comment. Math. Helv., 48(1973), 119-135.
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unit disc, Trans. Amer. Math. Soc., 210(1975), 63-74.

[7] S. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc., 81(1981), 521-527.
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37(1985), 48-61.

Department of Mathematics COMSATS Institute of Information Technology Park Road, Islamabad,

Pakistan

∗Corresponding author: shahid humayoun@yahoo.com


	1. Introduction
	2. Preliminaries
	3. Main Results
	References