16_914_cho.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 6, 2010, 1124-1136

ISSN 1307-5543 – www.ejpam.com

SPECIAL ISSUE ON COMPLEX ANALYSIS: THEORY AND APPLICATIONS

DEDICATED TO PROFESSOR HARI M. SRIVASTAVA,

ON THE OCCASION OF HIS 70TH
BIRTHDAY

Inclusion Properties for Certain Subclasses of Analytic

Functions Defined by a Multiplier Transformation

Oh Sang Kwon1, Nak Eun Cho2,∗

1 Department of Mathematics, Kyungsung University, Busan 608-736, Korea
2 Department of Applied Mathematics, Pukyong National University, Busan 608-737, Korea

Abstract. The purpose of the present paper is to investigate some inclusion properties of certain sub-

classes of analytic functions associated with a family of Multiplier transformations, which are defined

by means of the Hadamard product (or convolution).

2000 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Analytic function, Starlike function, Convex function, Hadamard product,

Choi-Saigo-Srivastava operator, Multiplier transformation, Inclusion relation, Convolution property,

Integral preserving property

1. Introduction

LetA denote the class of functions of the form

f (z) = z +

∞∑

k=2

akzk (1)

which are analytic in the open unit disk U = {z ∈ C : |z| < 1}. Let S ∗(α) and K (α) denote

the subclasses of A consisting of starlike and convex functions of order α (0 ≤ α < 1) and

let S ∗(0) = S ∗ and K (0) = K . If f and g are analytic in U, we say that f is subordinate

to g in U, written as f ≺ g or f (z) ≺ g(z), if there exists a Schwarz function w such that

f (z) = g(w(z)) for z ∈ U.

∗Corresponding author.

Email addresses: oskwon�ks.a
.kr (O. Kwon), ne
ho�pknu.a
.kr (N. Cho)

http://www.ejpam.com 1124 c© 2010 EJPAM All rights reserved.



O. Kwon, N. Cho / Eur. J. Pure Appl. Math, 3 (2010), 1124-1136 1125

A function f ∈A is said to be prestarlike of order α in U if

z

(1− z)2(1−α)
∗ f (z) ∈ S ∗(α) (0≤ α < 1),

where the symbol (∗) means the familiar Hadamard product (or convolution) of two analytic

functions in U. We denote this class by R(α) (see, for details, [9]). We note that a function

f ∈A is in the class R(0) if and only if f is convex univalent in U, and R(1/2) = S ∗(1/2).
Let N be the class of all analytic functions h which are univalent in U and for which h(U)

is convex with h(0) = 1 and and Re{h(z)} > 0 in U.

For any real number s, we define the multiplier transformations I s
λ

of functions f ∈ A by

I s
λ f (z) = z +

∞∑

k=2

�
k+λ

1+λ

�s
akzk (λ > −1).

Obviously, we observe that

I s
λ(I

t
λ f (z)) = I s+t

λ
f (z)

for all real numbers s and t. For λ = 1 and any integer s, the operator I s
λ

was studied by

Uralegaddi and Somanatha [13]. Also, for s = −1, the operator I s
λ

is the integral operator

studied by Owa and Srivastava [8]. Moreover, the operator I s
λ

is closely related to the mul-

tiplier transformation studied by Jung et al. [3] (also see [2]), and the differential operator

defined by Sălăgean [10].

Let

f s
λ(z) = z +

∞∑

k=2

�
k+λ

1+λ

�s
zk (s ∈ R; λ > −1)

and let f s
λ,µ

be defined such that

f s
λ(z) ∗ f s

λ,µ(z) =
z

(1− z)µ
(µ > 0; z ∈ U), (2)

where the symbol (∗) stands for the Hadamard product(or convolution). Then, motivated

essentially by the Choi-Saigo-Srivastava operator [1] (see also [5], [6] and [7]), we now

introduce the operator I s
λ,µ

:A →A , which are defined here by

I s
λ,µ f (z) =
�

f s
λ,µ ∗ f
�
(z) ( f ∈A ; s ∈ R; λ > −1; µ > 0), (3)

In particular, we note that I0
0,2 f (z) = z f ′(z) and I1

0,2 f (z) = f (z). In view of (2) and (3), we

obtain the following relations:

z
�

I s
λ,µ f (z)
�′
= µI s

λ,µ+1
f (z)− (µ− 1)I s

λ,µ f (z) ( f ∈A ; λ > −1; µ > 0) (4)

and

z
�

I s+1
λ,µ

f (z)
�′
= (λ+ 1)I s

λ,µ f (z)−λI s+1
λ,µ

f (z) ( f ∈ A ; λ > −1; µ > 0). (5)



O. Kwon, N. Cho / Eur. J. Pure Appl. Math, 3 (2010), 1124-1136 1126

We also define the function φ(a, c; z) by

φ(a, c; z) :=

∞∑

k=0

(a)k

(c)k
zk+1 (6)

(z ∈ U; a ∈ R; c ∈ R \Z−0 ;Z−0 := {−1,−2, · · · }),

where (ν)k is the Pochhammer symbol (or the shifted factorial) defined (in terms of the

Gamma function) by

(ν)k :=
Γ(ν + k)

Γ(ν)
=

(
1 if k = 0 and ν ∈ C \ {0} ,

ν(ν + 1) · · · (ν + k− 1) if k ∈ N := {1,2, · · · } and ν ∈ C.

By using the operator I s
λ,µ

, we introduce the following class of analytic functions for γ > 0,

λ > −1, s ∈ R, µ > 0 and h ∈ N :

T s
λ,µ(γ; h) :=

(
f ∈A : (1− γ)

I s
λ,µ

f (z)

z
+ γ(I s

λ,µ f (z))′ ≺ h(z)

)
.

In the present paper, we derive some inclusion relations, convolution properties and inte-

gral preserving properties for the class T s
λ,µ
(γ; h).

The following lemmas will be required in our investigation.

Lemma 1. [4] Let g be analytic in U and h be analytic and convex univalent in U with h(0) =

g(0). If

g(z) +
1

γ
zg′(z)≺ h(z) (Re{γ} ≥ 0;γ 6= 0), (7)

then

g(z) ≺eh(z) = γz−γ
∫ z

0

tγ−1h(t)d t ≺ h(z)

and eh is the best dominant of (7).

Lemma 2. [9] Let f ∈ S ∗(α) and g ∈ R(α). Then for any analytic function F in U,

g ∗ ( f F)

g ∗ f
(U)⊂ co(F(U))

where co(F(U)) denotes the convex hull of F(U).

Lemma 3. [12] Let 0< a ≤ c. Then

Re

�
φ(a, c; z)

z

�
>

1

2
(z ∈ U),

where φ is given by (1.6).



O. Kwon, N. Cho / Eur. J. Pure Appl. Math, 3 (2010), 1124-1136 1127

2. Inclusion Relations

Theorem 1. If 0≤ γ1 < γ2, then

T s
λ,µ(γ2; h)⊂ T s

λ,µ(γ1; h).

Proof. Let

g(z) =
I s
λ,µ

f (z)

z
( f ∈ T s

λ,µ(γ2; h) : z ∈ U). (8)

Then the function g is analytic in U with g(0) = 1. Differentiating both sides of (8), we have

(1− γ2)
I s
λ,µ

f (z)

z
+ γ2(I

s
λ,µ f (z))′ = g(z) + γ2zg′(z) ≺ h(z). (9)

Hence an application of Lemma 1 with µ = 1/γ2 yields

g(z) ≺ h(z). (10)

Since 0≤ γ1/γ2 < 1 and h is convex univalent in U, it follows from (8), (9) and (10) that

(1− γ1)
I s
λ,µ

f (z)

z
+ γ1(I

s
λ,µ f (z))′

=
γ1

γ2


(1− γ2)

I s
λ,µ

f (z)

z
+ γ2(I

s
λ,µ f (z))′


+
�

1−
γ1

γ2

�
g(z)

≺ h(z).

Therefore f ∈ T s
λ,µ
(γ1; h) and so we complete the proof of Theorem 1.

Theorem 2. If 0< µ1 ≤ µ2, then

T s
λ,µ2
(γ; h)⊂ T s

λ,µ1
(γ; h).

Proof. Let f ∈ T s
λ,µ2
(γ; h). Then

(1− γ)
I s
λ,µ1

f (z)

z
+ γ(I s

λ,µ1
f (z))′ (11)

=
φ(µ1,µ2; z)

z
∗


(1− γ)

I s
λ,µ2

f (z)

z
+ γ(I s

λ,µ2
f (z))′


 .

In view of Lemma 3, we see that the function φ(µ1,µ2; z)/z has the Herglotz representation

φ(µ1,µ2; z)

z
=

∫

|x |=1

dµ(x)

1− xz
(z ∈ U), (12)



O. Kwon, N. Cho / Eur. J. Pure Appl. Math, 3 (2010), 1124-1136 1128

where µ(x) is a probability measure defined on the unit circle |x |< 1 and
∫

|x |=1

dµ(x) = 1.

Since h is convex univalent in U, it follows from (11) and (12) that

(1− γ)
I s
λ,µ1

f (z)

z
+ γ(I s

λ,µ1
f (z))′ =

∫

|x |=1

h(xz)dµ(x)≺ h(z),

which completes the proof of Theorem 9.

Theorem 3. If µ > 0, then

T s
λ,µ+1

(γ; h)⊂ T s
λ,µ(γ;eh),

where

eh(z) = µz−µ
∫ z

0

tµ−1h(t)d t ≺ h(z).

Proof. Let

g(z) = (1− γ)
I s
λ,µ

f (z)

z
+ γ(I s

λ,µ f (z))′ ( f ∈A ; z ∈ U). (13)

Then from (4) and (13), we have

zg(z) = γµI s
λ,µ+1 f (z) + (1− γµ)I s

λ,µ f (z). (14)

Differentiating both sides of (13) and using (4) again, we obtain

z(zg′(z) + g(z))

= γµz(I s
λ,µ+1

f (z)) + (1− γµ)(µI s
λ,µ+1

f (z)− (µ− 1)I s
λ,µ

f (z)).
(15)

By a simple calculation with (14) and (15), we get

g(z) +
zg′(z)

µ
= (1− γ)

I s
λ,µ+1

f (z)

z
+ γ(I s

λ,µ+1
f (z))′. (16)

If f ∈ T s
λ,µ+1

(γ; h), then it follows from (16) that

g(z) +
zg′(z)

µ
≺ h(z) (µ > 0).

Hence an application of Lemma 1 yields

g(z)≺eh(z) = µz−µ
∫ z

0

tµ−1h(t)d t ≺ h(z),

which shows that

f ∈ T s
λ,µ+1

(γ;eh)⊂ T s
λ,µ(γ; h).



O. Kwon, N. Cho / Eur. J. Pure Appl. Math, 3 (2010), 1124-1136 1129

Theorem 4. If s ∈ R and λ > −1, then

T s
λ,µ(γ; h)⊂ T s+1

λ,µ
(γ;eh),

where

eh(z) = (λ+ 1)z−(λ+1)

∫ z

0

tλh(t)d t ≺ h(z).

Proof. By using the same techniques as in the proof of Theorem 3 and (5), we have

Theorem 11 and so we omit the detailed proof involved.

Theorem 5. Let γ > 0, β > 0 and f ∈ T s
λ,µ
(γ;βh+ 1− β). If β ≤ β0, where

β0 =
1

2


1−

1

γ

∫ 1

0

u
1

γ
−1

1+ u
du




−1

, (17)

then f ∈ T s
λ,µ
(0; h). The bound β0 is sharp for the function

h(z) =
1

1− z
(z ∈ U).

Proof. Let

g(z) =
I s
λ,µ

f (z)

z
( f ∈ T s

λ,µ(γ;βh+ 1− β);γ > 0;β > 0). (18)

Then we have

g(z) + γzg′(z) = (1− γ)
I s
λ,µ

f (z)

z
+ γ(I s

λ,µ f (z))′

≺ βh(z) + 1− β .

Hence an application of Lemma 1 yields

g(z) ≺
β

γ
z
− 1

γ

∫ z

0

t
1

γ
−1

h(t)d t + 1− β = (h ∗ψ)(z), (19)

where

ψ(z) =
β

γ
z
− 1

γ

∫ z

0

t
1

γ
−1

1− t
d t + 1− β . (20)

If 0< β ≤ β0, where β0 is given by (17), then from (20), we have



O. Kwon, N. Cho / Eur. J. Pure Appl. Math, 3 (2010), 1124-1136 1130

Re{ψ(z)} =
β

γ

∫ 1

0

u
1

γ
−1

Re

�
1

1− uz
du

�
+ 1− β

>
β

γ

∫ 1

0

u
1

γ
−1

1+ u
du+ 1− β

≥
1

2
.

By using the Herglotz representation for ψ, it follows from (18) and (19) that

I s
λ,µ

f (z)

z
≺ (h ∗ψ)(z)≺ h(z),

since h is convex univalent in U. This shows that f ∈ T s
λ,µ
(0; h).

For h(z) = 1/(1− z) and f ∈A defined by

I s
λ,µ

f (z)

z
=
β

γ
z
− 1

γ

∫ z

0

t
1

γ
−1

1− t
d t + 1− β ,

it is easy to verify that

(1− γ)
I s
λ,µ

f (z)

z
+ γ(I s

λ,µ f (z))′ = βh(z) + 1− β .

Thus f ∈ T s
λ,µ
(γ;βh+ 1− β) . Furthermore, for β > β0, we have

Re

(
I s
λ,µ

f (z)

z

)
to
β

γ

∫ 1

0

u
1

γ
−1

1+ u
du+ 1− β <

1

2
(z→−1),

which implies that f 6∈ T s
λ,µ
(0; h). Hence the bound β0 cannot be increased when

h(z) = 1/(1− z) (z ∈ U).

3. Convolution Properties

Theorem 6. If f ∈ T s
λ,µ
(γ; h) and

Re

�
g(z)

z

�
>

1

2
(g ∈A ; z ∈ U),

then

f ∗ g ∈ T s
λ,µ(γ; h).



O. Kwon, N. Cho / Eur. J. Pure Appl. Math, 3 (2010), 1124-1136 1131

Proof. Let f ∈ T s
λ,µ
(γ; h) and g ∈A . Then we have

(1− γ)
I s
λ,µ
( f ∗ g)(z)

z
+ γ(I s

λ,µ( f ∗ g)(z))′ =
g(z)

z
∗ψ(z),

where

ψ(z) = (1− γ)
I s
λ,µ

f (z)

z
+ γ(I s

λ,µ f (z))′ ≺ h(z).

The remaining part of the proof of Theorem 6 is similar to that of Theorem 2 and so we omit

the details involved.

Corollary 1. Let f ∈ T s
λ,µ
(γ; h) be given by (1). Then the function

σm(z) =

∫ 1

0

Sm(tz)

t
d t (z ∈ U),

where

Sm(z) = z +

m−1∑

n=1

an+1zn+1 m ∈ N \ {1}; z ∈ U),

is also in the class T s
λ,µ
(γ; h).

Proof. We have

σm(z) = z +

m−1∑

n=1

an+1

n+ 1
zn+1 = ( f ∗ gm)(z) (m ∈ N \ {1}), (21)

where

f (z) = z +

∞∑

n=1

an+1zn+1 ∈ T n
λ,µ(γ; h)

and

gm(z) = z +

m−1∑

n=1

zn+1

n+ 1
∈A ,

while, it is known [11] that

Re

�
gm(z)

z

�
= Re

(
1+

m−1∑

n=1

zn

n+ 1

)
>

1

2
(m ∈ N \ {1}; z ∈ U). (22)

In view of (21) and (22), an application of Theorem 6 leads to σm ∈ T s
λ,µ
(γ; h).

Theorem 7. If f ∈ T s
λ,µ
(γ; h) and

g(z) ∈ R(α) (g ∈A ; z ∈ U),

then

f ∗ g ∈ T s
λ,µ(γ; h).



O. Kwon, N. Cho / Eur. J. Pure Appl. Math, 3 (2010), 1124-1136 1132

Proof. By using a similar method as in the proof of Theorem 21, we have

(1− γ)
I s
λ,µ
( f ∗ g)(z)

z
+ γ(I s

λ,µ( f ∗ g)(z))′ =
g(z) ∗ (zψ(z))

g(z) ∗ z
(z ∈ U), (23)

where

ψ(z) = (1− γ)
I s
λ,µ

f (z)

z
+ γ(I s

λ,µ f (z))′ ≺ h(z).

Since h is convex univalent in U, it follows from (23) and Lemma 2 that Theorem 7 holds

true.

If we take α= 0 and α= 1/2 in Theorem 7, we have the following corollary.

Corollary 2. If f ∈ T s
λ,µ
(γ; h) and g ∈A satisfies one of the following conditions:

(i) g(z) is convex univalent in U

or

(ii) g(z) ∈ S∗(1

2
),

then f ∗ g ∈ T s
λ,µ
(γ; h).

4. Integral Operators

Theorem 8. If f ∈ T s
λ,µ
(γ; h), then the function F defined by

F(z) =
c + 1

zc

∫ z

0

t c−1 f (t)d t (Re{c} > −1) (24)

is in the class T s
λ,µ
(γ;eh), where

eh(z) = (c + 1)z−(c+1)

∫ z

0

t ch(t)d t ≺ h(z).

Proof. Let f ∈ T s
λ,µ
(γ; h). Then from (24), we obtain

(c + 1) f (z) = zF ′(z) + cF(z). (25)

Define the function G by

zG(z) = (1− γ)I s
λ,µF(z) + γz(I s

λ,µF(z))′ (z ∈ U). (26)

Differentiating both sides of (26) with respect to z, we get

G(z) + zG′(z) = (1− γ)
I s
λ,µ
(zF ′(z))

z
+ γ(I s

λ,µ(zF ′(z)))′. (27)



O. Kwon, N. Cho / Eur. J. Pure Appl. Math, 3 (2010), 1124-1136 1133

Furthermore, it follows from (25), (26) and (27) that

(1− γ)
I s
λ,µ

f (z)

z
+ γ(I s

λ,µ f (z))′

= (1− γ)z−1 I s
λ,µ

�
zF ′(z) + cF(z)

c + 1

�
+ γ

�
I s
λ,µ

�
zF ′(z) + cF(z)

c + 1

��′

= G(z) +
1

c + 1
zG′(z). (28)

Since f ∈ T s
λ,µ
(γ; h), from (28), we have

G(z) +
1

c + 1
zG′(z) ≺ h(z) (Re{c} > −1),

and so an application of Lemma 1 yields

G(z) ≺eh(z) =
c + 1

zc+1

∫ z

0

t ch(t)d t ≺ h(z).

Therefore we conclude that

F ∈ T s
λ,µ(γ;eh)⊂ T s

λ,µ(γ; h).

Theorem 9. If f ∈ A and F be defined as in Theorem 8. If

(1−α)
I s
λ,µ

F(z)

z
+α

I s
λ,µ

f (z)

z
≺ h(z) (α > 0), (29)

then F ∈ T s
λ,µ
(0;eh), where

eh(z) =
c + 1

α
z
− α

c+1

∫ z

0

t
c+1

α
−1h(t) ≺ h(z) (Re{c} > −1).

Proof. Let

G(z) =
I s
λ,µ

F(z)

z
(z ∈ U). (30)

Then G is analytic in U with G(0) = 1 and

zG′(z) = (I s
λ,µF(z))′ − G(z). (31)

It follows from (25), (29), (30), and (31) that

(1−α)
I s
λ,µ

F(z)

z
+α

I s
λ,µ

f (z)

z

= (1−α)
I s
λ,µ

F(z)

z
+

α

c + 1




cI s
λ,µ

F(z)

z
+ (I s

λ,µF(z))′






O. Kwon, N. Cho / Eur. J. Pure Appl. Math, 3 (2010), 1124-1136 1134

= G(z) +
α

c + 1
zG′(z) ≺ h(z) (Re{c} > 1;α > 0).

Therefore, by Lemma 1, we conclude that Theorem 9 holds true as stated.

Theorem 10. Let F ∈ T s
λ,µ
(γ; h). If the function f is defined by

F(z) =
c + 1

zc

∫ z

0

t c−1 f (t)d t (c > −1), (32)

then
f (σz)

σ
∈ T s

λ,µ(γ; h),

where

σ = σ(c) =

p
1+ (c + 1)2 − 1

c + 1
. (33)

The bound σ is sharp for the function

h(z) = β + (1− β)
1+ z

1− z
(β 6= 1; z ∈ U). (34)

Proof. We note that for F ∈A ,

F(z) = F(z) ∗
z

1− z
and zF ′(z) = F(z) ∗

z

(1− z)2
.

Then from (32), we have

f (z) =
cF(z) + zF ′(z)

c + 1
= (F ∗ g)(z) (c > −1; z ∈ U), (35)

where

g(z) =
1

c + 1

�
c

z

1− z
+

z

(1− z)2

�
∈A . (36)

Next, we show that

Re

�
g(z)

z

�
>

1

2
(|z| < σ), (37)

where σ = σ(c) is given by (4.10). Letting

1

1− z
= Reiθ (|z| = r < 1; R> 0),

we see that

cosθ =
1+R2(1− r2)

2R
and R≥

1

1+ r
. (38)



O. Kwon, N. Cho / Eur. J. Pure Appl. Math, 3 (2010), 1124-1136 1135

Then for (36) and (38), we have

2Re

�
g(z)

z

�
=

2

c + 1

�
cR cosθ + R2(2 cos2 θ − 1)

�

=
R2

c + 1

�
c(1− r2) +R2(1− r2)2 − 2

�
+ 1

≥
R2

c + 1

�
c + 1− 2r − (c + 1)r2

�
+ 1.

This evidently gives (37), which is equivalent to

Re

�
g(σz)

zσ

�
>

1

2
z ∈ U). (39)

Let F ∈ T s
λ,µ
(γ; h). Then, by using (35) and (39), an application of Theorem 6 yields

f (σz)

σ
= F(z) ∗

g(σz)

σ
∈ T s

λ,µ(γ; h).

For h given by (34), we consider the function F ∈A defined by

(1− γ)
I s
λ,µ

F(z)

z
+ γ(I s

λ,µF(z))′ = β + (1− β)
1+ z

1− z
(β 6= 1; z ∈ U). (40)

Then from (26), (28) and (40) , we find that

(1− γ)
I s
λ,µ

f (z)

z
+ γ(I s

λ,µ f (z))′

= β + (1− β)
1+ z

1− z
+

z

c + 1

�
β + (1− β)

1+ z

1− z

�′

= β +
(1− β)(c+ 1+ 2z− (c + 1)z2)

(c + 1)(1− z)2

= β (z = −σ).

Therefore we conclude that the bound σ = σ(c) cannot be increased for each c (c > −1).

ACKNOWLEDGEMENTS This research was supported by the Basic Science Research Pro-

gram through the National Research Foundation of Korea(NRF) funded by the Ministry of

Education, Science and Technology (No. 2010-0017111).



REFERENCES 1136

References

[1] J.H. Choi, M. Saigo and H.M. Srivastava. Some inclusion properties of a certain family

of integral operators. J. Math. Anal. Appl., 276:432–445, 2002.

[2] T.M. Flett. The dual of an inequality of Hardy and Littlewood and some related inequal-

ities. J. Math. Anal. Appl., 38:746–765, 2002.

[3] I.B. Jung, Y.C. Kim and H.M. Srivastava. The Hardy space of analytic functions associated

with certain one-parameter families of integral operators. J. Math. Anal. Appl., 176:138–

147, 1993.

[4] D.J. Hallenback and S. Ruscheweyh. Subordination by convex functions. Proc. Amer.

Math. Soc., 52:191–195, 1975.

[5] J.-L. Liu. The Noor integral and strongly starlike functions. J. Math. Anal. Appl.,

261:441–447, 2001.

[6] K.I. Noor. On new classes of integral operators. J. Natur. Geom., 16:71–80, 1999.

[7] K.I. Noor and M.A. Noor. On integral operators. J. Math. Anal. Appl., 238:341–352,

1999.

[8] S.Owa and H.M. Srivastava. Some applications of the generalized Libera integral oper-

ator. Proc. Japan Acad. Ser. A Math. Sci., 62:125–128, 1986.

[9] St.Ruscheweyh. Convolution in Geometric Function Theory. Les Presses de L’Université de

Montréal, 1982.

[10] G.S. Sǎlǎgean. Subclasses of univalent functions. Lecture Notes in Math., Springer-

Verlag, 1013:362–372, 1983.

[11] R. Singh and S. Singh. Convolution properties of a class of starlike functions. Proc. Amer.

Math. Soc., 106:145–152, 1989.

[12] Lucyna Trojnar-Spelina. On certain applications of Hadamard product. Applied Math.

Comput., 199:653–662, 2008.

[13] B.A. Uralegaddi and C. Somanatha. Certain classes of univalent functions. Current Top-

ics in Analytic Function Theory (Eds. H. M. Srivastava and S. Owa), World Scientific

Publishing Company Singapore, New Jersey, London, and Hong Kong, pages 371–374,

1992.