9_520_murugus.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 4, No. 1, 2011, 76-82

ISSN 1307-5543 – www.ejpam.com

On Certain Sufficient Conditions for Analytic Univalent

Functions

G. Murugusundaramoorthy1,∗and N. Magesh2

1 School of Advanced Sciences, VIT University, Vellore - 632014, Tamilnadu, India.
2 Department of Mathematics, Government Arts College(Men), Krishnagiri-635001, Tamilnadu, India.

Abstract. In this paper, we introduce a new class Bl
m
(α,δ) of functions which is defined by hyperge-

ometric function and obtain its relations with some well-known subclasses of analytic univalent func-

tions. Furthermore, as a special case, we show that convex functions of order 1/2 are also members of

the family Bl
m
(α,δ).

2000 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Univalent functions, starlike functions, convex functions, Hadamard product,

generalized hypergeometric functions.

1. Introduction

LetA denote the class of functions of the form

f (z) = z +

∞
∑

n=2

anzn (1)

which are analytic and univalent in the open disc U = {z : |z| < 1} and normalized by

f (0) = 0 = f ′(0)− 1. We denote by S∗(α) and K(α) the subclasses of A consisting of all

functions which are, respectively starlike and convex of order α. Thus,

S∗(α) =

�

f ∈A : Re

�

z f ′(z)

f (z)

�

> α, 0≤ α < 1, z ∈ U

�

and

K(α) =

�

f ∈ A : Re

�

1+
z f ′′(z)

f ′(z)

�

> α, 0≤ α < 1, z ∈ U

�

.

∗Corresponding author.

Email addresses: gmsmoorthy�yahoo.
om (G. Murugusundaramoorthy), nmagi_2000�yahoo.
o.in
(N.Magesh)

http://www.ejpam.com 76 c© 2010 EJPAM All rights reserved.



G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 4 (2011), 76-82 77

We notice that K(α)⊂ S∗(α)⊂A . Further,

R(α) =
�

f ∈ A : Re
�

f ′(z)
�

> α, 0≤ α < 1, z ∈ U
	

.

If f and g are analytic functions in U , we say that f is subordinate to g, written f ≺ g,

if there is a function w analytic in U , with w(0) = 0, |w(z)| < 1, for all z ∈ U such that

f (z) = g(w(z)) for all z ∈ U . If g is univalent, then f ≺ g if and only if f (0) = g(0) and

f (U)⊆ g(U).

For functionsΦ ∈A given by Φ(z) = z+
∞
∑

n=2

φnzn andΨ ∈A given byΨ(z) = z+
∞
∑

n=2

ψnzn,

we define the Hadamard product (or Convolution ) of Φ and Ψ by

(Φ ∗Ψ)(z) = z +

∞
∑

n=2

φnψnzn, z ∈ U . (2)

For complex parameters α1, . . . ,αl and β1, . . . ,βm (β j 6= 0,−1, . . . ; j = 1,2, . . . , m) the

generalized hypergeometric function l Fm(z) is defined by

l Fm(z) ≡ l Fm(α1, . . .αl ;β1, . . . ,βm; z) :=

∞
∑

n=0

(α1)n . . . (αl)n

(β1)n . . . (βm)n

zn

n!
(3)

(l ≤ m+ 1; l, m ∈ N0 := N ∪ {0}; z ∈ U)

where N denotes the set of all positive integers and (α)n is the Pochhammer symbol defined

by

(α)n =

¨

1, n= 0

α(α+ 1)(α+ 2) . . . (α+ n− 1), n ∈ N .
(4)

Let H(α1, . . .αl ;β1, . . . ,βm) :A →A be a linear operator defined by

[(H(α1, . . .αl ;β1, . . . ,βm))( f )](z) := z l Fm(α1,α2, . . .αl ;β1,β2 . . . ,βm; z) ∗ f (z)

= z +

∞
∑

n=2

Γn anzn (5)

where

Γn =
(α1)n−1 . . . (αl)n−1

(n− 1)!(β1)n−1 . . . (βm)n−1

. (6)

For notational simplicity, we can use a shorter notation H l
m[α1] for H(α1, . . .αl ;β1, . . . ,βm)

in the sequel. The linear operator H l
m[α1] is called Dziok-Srivastava operator (see [3]), in-

cludes (as its special cases) various other linear operators introduced and studied by Bernardi

[1], Carlson and Shaffer [2], Libera [6], Livingston [7], Ruscheweyh [8] and Srivastava-Owa

[9].

For 0≤ α < 1 and δ ≥ 0, let Bl
m(α,δ) consisting of functions of the form (1) and satisfying

the condition
�

�

�

�

�

H l
m[α1 + 1] f (z)

z

�

z

H l
m[α1] f (z)

�δ

− 1

�

�

�

�

�

< 1−α, z ∈ U . (7)



G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 4 (2011), 76-82 78

The class Bl
m(α,δ) is a unified class of analytic functions which includes various new

subclasses of analytic univalent functions. We observe that

Example 1. If l = 2 and m= 1 with α1 = 1, α2 = 1, β1 = 1 then

B2
1(α,δ) :=

(

f ∈A :

�

�

�

�

�

f ′(z)
�

z

f (z)

�δ

− 1

�

�

�

�

�

< 1−α, δ ≥ 0, 0≤ α < 1, z ∈ U .

)

The class B2
1(α,δ) has been studied by Frasin and Jahangiri [5]. Further B2

1(α, 2) has been

studied by Frasin and Darus [4]. Also we note that B2
1(α, 1)≡ S∗(α) and B2

1(α, 0)≡ R(α).

Example 2. If l = 2 and m= 1 with α1 = η+ 1 (η > −1), α2 = 1, β1 = 1, then

B(η,α,δ) :=

(

f ∈A :

�

�

�

�

�

Dη+1 f (z)

z

�

z

Dη f (z)

�δ

− 1

�

�

�

�

�

< 1−α, η > −1, δ ≥ 0, 0≤ α < 1, z ∈ U .

)

,

where Dη f (z) is called Ruscheweyh derivative operator [8] defined by

Dη f (z) :=
z

(1− z)η+1
∗ f (z) ≡ H2

1(η+ 1,1; 1) f (z).

Also we observe that B(0,α, 1) ≡ K(α).

Example 3. If l = 2 and m= 1 with α1 = µ+ 1(µ > −1), α2 = 1, β1 = µ+ 2, then

B(µ,α,δ) :=

(

f ∈A :

�

�

�

�

�

Jµ+1 f (z)

z

�

z

Jµ f (z)

�δ

− 1

�

�

�

�

�

< 1−α, µ > −1, δ ≥ 0, 0≤ α < 1, z ∈ U

)

,

where Jµ is a Bernardi operator [1] defined by

Jµ f (z) :=
µ+ 1

zµ

∫ z

0

tµ−1 f (t)d t ≡ H2
1(µ+ 1,1;µ+ 2) f (z).

Note that the operator J1 was studied earlier by Libera [6] and Livingston [7].

Example 4. If l = 2 and m= 1 with α1 = a (a > 0), α2 = 1, β1 = c (c > 0), then

B(a, c,α,δ) :=

(

f ∈A :

�

�

�

�

�

L(a+ 1, c) f (z)

z

�

z

L(a, c) f (z)

�δ

− 1

�

�

�

�

�

< 1−α, δ ≥ 0, 0≤ α < 1, z ∈ U

)

,

where L(a, c) is a well-known Carlson-Shaffer linear operator [2] defined by

L(a, c) f (z) :=

 

∞
∑

k=0

(a)k

(c)k
zk+1

!

∗ f (z) ≡ H2
1(a, 1; c) f (z).

The object of the present paper is to investigate the sufficient condition for functions to be

in the class Bl
m(α,δ). Furthermore, as a special case, we show that convex functions of order

1/2 are also members of the family Bl
m(α,δ).



G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 4 (2011), 76-82 79

2. Main Results

To prove our results we need the following lemma.

Lemma 1. [5] Let p be analytic in U with p(0) = 1 and suppose that

Re

�

1+
zp′(z)

p(z)

�

>
3α− 1

2α
. (8)

Then Re {p(z)} > α for z ∈ U and 1

2
≤ α < 1.

Using Lemma 1, we first prove the following theorem.

Theorem 1. Let f (z) be the functions of the form (1), δ ≥ 0 and 1

2
≤ α < 1. If

(α1 + 1)
H l

m[α1 + 2] f (z)

H l
m[α1 + 1] f (z)

−δα1

H l
m[α1 + 1] f (z)

H l
m[α1] f (z)

+α1(δ− 1)≺ 1+ βz, (9)

where β = 3α−1

2α
, then f (z) ∈ Bl

m(α,δ).

Proof. Define the function p(z) by

p(z) :=
H l

m[α1 + 1] f (z)

z

�

z

H l
m[α1] f (z)

�δ

(10)

Then the function p(z) is analytic in U and p(0) = 1. Therefore, differentiating (10) logarith-

mically and the simple computation yields

zp′(z)

p(z)
= (α1+ 1)

H l
m[α1 + 2] f (z)

H l
m[α1 + 1] f (z)

− δα1

H l
m[α1 + 1] f (z)

H l
m[α1] f (z)

+α1(δ− 1)− 1.

By the hypothesis of the theorem, we have

Re

�

1+
zp′(z)

p(z)

�

>
3α− 1

2α
.

Hence by Lemma 1, we have

Re

(

H l
m[α1 + 1] f (z)

z

�

z

H l
m[α1] f (z)

�δ
)

> α, z ∈ U .

Therefore in view of definition f (z) ∈ Bl
m(α,δ).

For l = 2 and m = 1 with α1 = a (a > 0), α2 = 1, β1 = c (c > 0), we obtain the following

corollary.



G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 4 (2011), 76-82 80

Corollary 1. Let 1

2
≤ α < 1. If f ∈A and

Re

�

(a+ 1)
L[a+ 2, c] f (z)

L[a+ 1, c] f (z)
− δa

L[a+ 1, c] f (z)

L[a, c] f (z)
+ a(δ− 1)

�

>
3α− 1

2α
, z ∈ U ,

then

Re

¨

L[a+ 1, c] f (z)

z

�

z

L[a, c] f (z)

�δ
«

> α, z ∈ U .

Therefore f (z) ∈ B(a, c,α,δ).

Taking l = 2 and m = 1 with α1 = µ+ 1(µ > −1), α2 = 1, β1 = µ+ 2, we get

Corollary 2. Let 1

2
≤ α < 1. If f ∈A and

Re

¨

(µ+ 2)
Jµ+2 f (z)

Jµ+1 f (z)
− δ(µ+ 1)

Jµ+1 f (z)

Jµ f (z)
+ (µ+ 1)(δ− 1)

«

>
3α− 1

2α
, z ∈ U ,

then

Re

(

Jµ+1 f (z)

z

�

z

Jµ f (z)

�δ
)

> α, z ∈ U .

Therefore f (z) ∈ B(µ,α,δ).

Choosing l = 2 and m= 1 with α1 = η+ 1 (η > −1), α2 = 1, β1 = 1, we have

Corollary 3. Let 1

2
≤ α < 1. If f ∈A and

Re

¨

(η+ 2)
Dη+2 f (z)

Dη+1 f (z)
− δ(η+ 1)

Dη+1 f (z)

Dη f (z)
+ (η+ 1)(δ− 1)

«

>
3α− 1

2α
, z ∈ U ,

then

Re

¨

Dη+1 f (z)

z

�

z

Dη f (z)

�δ
«

> α, z ∈ U .

Therefore f (z) ∈ B(η,α,δ).

Choosing l = 2 and m= 1 with α1 = 1, α2 = 1 and β1 = 1, we have

Corollary 4. [5] If f ∈A and

Re

�

1+
z f ′′(z)

f ′(z)
+ δ

�

1−
z f ′(z)

f (z)

��

>
3α− 1

2α
, z ∈ U ,

then

Re

¨

f ′(z)
�

z

f (z)

�δ
«

> α, z ∈ U .

Therefore f (z) ∈ B2
1(α,δ).



G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 4 (2011), 76-82 81

Choosing l = 2 and m= 1 with α1 = 2, α2 = 1, β1 = 1, δ = 1 and α = 1

2
we have

Corollary 5. If f ∈A and

Re

¨

z2 f ′′′+ 6z f ′′(z) + 6 f ′(z)

2 f ′(z) + z f ′′
−

z f ′′(z)

f ′(z)

«

>
3

2
, z ∈ U ,

then

Re

�

1+
z f ′′(z)

f ′(z)

�

> 0, z ∈ U .

That is, f (z) ∈ K .

Choosing l = 2 and m= 1 with α1 = 2, α2 = 1, β1 = 1, δ = 0 and α = 1

2
we have

Corollary 6. If f ∈A and

Re

¨

z2 f ′′′+ 4z f ′′(z) + 2 f ′(z)

2 f ′(z) + z f ′′

«

>
1

2
, z ∈ U ,

then

Re

�

f ′(z) +
z f ′′(z)

2

�

>
1

2
, z ∈ U .

Choosing l = 2 and m= 1 with α1 = 1, α2 = 1, β1 = 1, δ = 1 and α = 1

2
we have

Corollary 7. If f ∈A and

Re

�

z f ′′(z)

f ′(z)
−

z f ′(z)

f (z)

�

>
−3

2
, z ∈ U ,

then

Re

�

z f ′(z)

f (z)

�

>
1

2
, z ∈ U .

That is, f (z) is starlike of order 1/2 .

Choosing l = 2 and m= 1 with α1 = 1, α2 = 1, β1 = 1, δ = 0 and α = 1

2
we have

Corollary 8. If f ∈A and

Re

�

1+
z f ′′(z)

f ′(z)

�

>
1

2
, z ∈ U

then

Re
�

f ′(z)
	

>
1

2
, z ∈ U .

That is f (z) ∈ B(0,1/2) = R1/2.

ACKNOWLEDGEMENTS The authors would like to thank the referee(s) for their insightful

comments and suggestions.



REFERENCES 82

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