6_murugusundaramoorthy.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 2, No. 2, 2009, (239-249)

ISSN 1307-5543 – www.ejpam.com

Subordination Results For Spirallike Functions

G. Murugusundaramoorthy1∗ and N. Magesh2

1 School of Science and Humanities, VIT University,

Vellore - 632014, India

2 Department of Mathematics, Adhiyamaan College of Engineering,

Hosur - 635109, India

Abstract. In this paper, we introduce a new class of functions which is defined by Dziok-

Srivastava operator and obtain the subordination results for this class of functions. Some

known and new results, which follow as special cases of our results, have also been mentioned.

AMS subject classifications: 30C45

Key words: Univalent functions, starlike functions, convex functions, Spirallike functions,

subordinating factor sequence, Hadamard product, generalized hypergeometric functions.

1. Introduction

Let A denote the class of functions of the form

f (z) = z+

∞
∑

n=2

anzn (1.1)

∗Corresponding author.

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

http://www.ejpam.com 239 c© 2009 EJPAM All rights reserved.



G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 2 (2009), (239-249) 240

which are analytic and univalent in the open disc U = {z : |z| < 1}. For functions

f ∈ A given by (1.1) and g ∈ A given by g(z) = z+
∞
∑

n=2

bnzn, we define the Hadamard

product (or Convolution ) of f and g by

( f ∗ g)(z) = z +

∞
∑

n=2

anbnzn, z ∈ U . (1.2)

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!
(1.3)

(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 .
(1.4)

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 (1.5)

where

Γn =
(α1)n−1 . . . (αl)n−1

(n− 1)!(β1)n−1 . . . (βm)n−1

. (1.6)

For notational simplicity, we can use a shorter notation

H l
m
[α1,β1] for H(α1, . . .αl;β1, . . . ,βm) in the sequel. The linear operator H l

m
[α1,β1] is

called Dziok-Srivastava operator (see [3]), includes (as its special cases) various other



G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 2 (2009), (239-249) 241

linear operators introduced and studied by Bernardi [1], Carlson and Shaffer [2],

Libera [4], Livingston [6], Ruscheweyh [7] and Srivastava-Owa [11].

For 0 ≤ λ < 1, 0 ≤ γ < 1 and −π
2
< η < π

2
, we let Rl

m
(η,γ,λ) be the subclass of A

consisting of functions of the form (1.1) and satisfying the analytic criterion

Re

¨

eiη
z(H l

m
[α1,β1] f (z))

′

(1−λ)H l
m
[α1,β1] f (z) +λz(H l

m
[α1,β1] f (z))

′

«

> γ cosη, z ∈ U , (1.7)

where H l
m
[α1,β1] f (z) is given by (1.5).

Several known and new subclasses can be obtained from the class Rl
m
(η,γ,λ), by

suitably specializing the values of l, m, α1,α2, . . . ,αl, β1,β2, . . . ,βm, λ, γ and η. We

present below some of these subclasses of Rl
m
(η,γ,λ) consisting of functions of the

form (1.1). We observe that

Example 1.1. If l = 2 and m= 1 with α1 = 1, α2 = 1, β1 = 1 then

R
2
1
(η,γ,λ) ≡ S(η,γ,λ)

:=

¨

f ∈ A : Re

¨

eiη
z f ′(z)

(1−λ) f (z) +λz f ′(z)

«

> γ cosη, |η|<
π

2
, 0≤ γ < 1, z ∈ U

«

.

Also R2
1
(η,γ, 0)≡ S(η,γ) denotes the η−spirallike functions of order γ studied by Libera

[5]. Further R2
1
(η, 0, 0) ≡ S(η), |η|< π

2
. Spacek [10] proved that the members of S(η),

known as η−spirallike functions, are univalent in U.

Example 1.2. If l = 2 and m= 1 with α1 = δ+ 1 (δ >−1), α2 = 1, β1 = 1, then

R
2
1
(η,γ,λ)≡ Dδ(η,γ,λ)

:=

�

f ∈ A : Re

¨

eiη
z(Dδ f (z))′

(1−λ)Dδ f (z) +λz(Dδ f (z))′

«

> γ cosη,

|η|<
π

2
, 0≤ γ < 1, z ∈ U

�

,

where Dδ f (z) is called Ruscheweyh derivative operator [7] defined by

Dδ f (z) :=
z

(1− z)δ+1
∗ f (z)≡ H2

1
(δ+ 1, 1; 1) f (z).



G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 2 (2009), (239-249) 242

Example 1.3. If l = 2 and m= 1 with α1 = µ+ 1(µ > −1), α2 = 1, β1 = µ+ 2, then

R
2
1
(η,γ,λ)≡ Bµ(η,γ,λ)

:=

¨

f ∈ A : Re

�

eiη
z(Jµ f (z))′

(1−λ)Jµ f (z) +λz(Jµ f (z))′

«

> γ cosη, |η|<
π

2
,

0 ≤ γ < 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 [4] and Livingston [6].

Example 1.4. If l = 2 and m= 1 with α1 = a (a > 0), α2 = 1, β1 = c (c > 0), then

R
2
1(η,γ,λ) ≡ La

c (η,γ,λ)

:=

�

f ∈ A : Re

¨

eiη
z(L(a, c) f (z))′

(1−λ)L(a, c) f (z) +λz(L(a, c) f (z))′

«

> γ cosη, |η| <
π

2
,

0≤ γ < 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 coefficient estimates and sub-

ordination properties for the class of functions Rl
m
(η,γ,λ). Some interesting conse-

quences of the results are also pointed out.

2. Main Results

To prove our results we need the following definitions and lemmas.



G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 2 (2009), (239-249) 243

Definition 2.1. For analytic functions g and h with g(0) = h(0), g is said to be subor-

dinate to h, denoted by g ≺ h, if there exists an analytic function w such that w(0) = 0,

|w(z)| < 1 and g(z) = h(w(z)), for all z ∈ U .

Definition 2.2. A sequence {bn}
∞
n=1

of complex numbers is said to be a subordinating

sequence if, whenever f (z) =
∞
∑

n=1

anzn, a1 = 1 is regular, univalent and convex in U , we

have
∞
∑

n=1

bnanzn ≺ f (z), z ∈ U . (2.1)

In 1961, Wilf [12] proved the following subordinating factor sequence.

Lemma 2.1. The sequence {bn}
∞
n=1

is a subordinating factor sequence if and only if

Re

(

1+ 2

∞
∑

n=1

bnzn

)

> 0, z ∈ U . (2.2)

Next we obtain the coefficient inequality theorem for the class Rl
m
(η,γ,λ).

Theorem 2.1. A function f (z) of the form (1.1) is in Rl
m
(η,γ,λ) if

∞
∑

n=2

[(1−λ)(n− 1) secη+ (1− γ)(1+ nλ−λ)]Γn |an| ≤ 1− γ, (2.3)

where |η|< π

2
, 0≤ λ < 1, 0≤ γ < 1 and Γn is given by (1.6).

Proof. Suppose the inequality (2.3) holds true. Then we get,

�

�z(H l
m
[α1,β1] f (z))

′− [(1−λ)H l
m
[α1,β1] f (z) +λz(H l

m
[α1,β1] f (z))

′]
�

�

− (1− γ) cosη
�

�[(1−λ)H l
m
[α1,β1] f (z) +λz(H l

m
[α1,β1] f (z))

′]
�

�

≤

�

�

�

�

�

∞
∑

n=2

[(n− 1)(1−λ)anΓnzn]

�

�

�

�

�

− (1− γ) cosη

�

�

�

�

�

z +

∞
∑

n=2

(1+ nλ−λ)anΓnzn]

�

�

�

�

�

≤
∞
∑

n=2

(n− 1)(1−λ)|an|Γn|z|
n− (1− γ) cosη|z|+

∞
∑

n=2

(1− γ) cosη(1+ nλ− λ)|an|Γn|z|
n



G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 2 (2009), (239-249) 244

By taking z→ 1 on the real axis we obtain

≤
∞
∑

n=2

[(n− 1)(1−λ) + (1− γ) cosη(1+ nλ−λ)]|an|Γn− (1− γ) cosη

≤ 0.

This completes the proof of the Theorem 2.1.

In the view of Examples 1.1 to 1.4, we state the following corollaries.

Corollary 2.1. A function f (z) of the form (1.1) is in S(η,γ,λ) if

∞
∑

n=2

[(1− λ)(n− 1) secη+ (1− γ)(1+ nλ−λ)] |an| ≤ 1− γ,

where |η|< π

2
, 0≤ λ < 1 and 0≤ γ < 1.

Remark 2.1. We observe that Corollary 2.1, yields the result of Silverman [8] for the

special values of η, λ and γ.

Corollary 2.2. A function f (z) of the form (1.1) is in Dδ(η,γ,λ) if

∞
∑

n=2

[(1−λ)(n− 1) secη+ (1− γ)(1+ nλ−λ)]
(δ+ 1) . . . (δ+ n− 1)

(n− 1)!
|an| ≤ 1− γ,

where |η|< π

2
, 0≤ λ < 1, 0≤ γ < 1 and δ > −1.

Corollary 2.3. A function f (z) of the form (1.1) is in Bµ(η,γ,λ) if

∞
∑

n=2

[(1−λ)(n− 1) secη+ (1− γ)(1+ nλ−λ)]
�

µ+ 1

µ+ n

�

|an| ≤ 1− γ,

where |η|< π

2
, 0≤ λ < 1, 0≤ γ < 1 and µ >−1.

Corollary 2.4. A function f (z) of the form (1.1) is in La
c
(η,γ,λ) if

∞
∑

n=2

[(1−λ)(n− 1) secη+ (1− γ)(1+ nλ−λ)]
(a)n−1

(c)n−1

|an| ≤ 1− γ,

where |η|< π

2
, 0≤ λ < 1, 0≤ γ < 1 and a > 0, c > 0.



G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 2 (2009), (239-249) 245

Next we obtain the subordination result for the class Rl
m
(η,γ,λ).

Theorem 2.2. Let f ∈ Rl
m
(η,γ,λ) and g(z) be any function in the usual class of convex

functions C , then

((1−λ) secη+ (1− γ)(1+λ))Γ2

2[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]
( f ∗ g)(z) ≺ g(z) (2.4)

where |η|< π

2
, 0≤ γ < 1; 0 ≤ λ < 1, with

Γ2 =
α1 . . .αl

β1 . . .βm

(2.5)

and

Re
�

f (z)
	

>−
[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]

((1−λ) secη+ (1− γ)(1+λ))Γ2

, z ∈ U . (2.6)

The constant factor
((1−λ) secη+(1−γ)(1+λ))Γ2

2[1−γ+((1−λ) sec η+(1−γ)(1+λ))Γ2]
in (2.4) cannot be replaced by a larger

number.

Proof. Let f ∈ Rl
m
(η,γ,λ) and suppose that g(z) = z +

∞
∑

n=2

cnzn ∈ C . Then

((1− λ) secη+ (1− γ)(1+λ))Γ2

2[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]
( f ∗ g)(z)

=
((1−λ) secη+ (1− γ)(1+λ))Γ2

2[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]

 

z+

∞
∑

n=2

cnanzn

!

. (2.7)

Thus, by Definition 2.2, the subordination result holds true if

�

((1−λ) secη+ (1− γ)(1+λ))Γ2

2[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]
an

�∞

n=1

is a subordinating factor sequence, with a1 = 1. In view of Lemma 2.1, this is equiva-

lent to the following inequality

Re

(

1+

∞
∑

n=1

((1−λ) secη+ (1− γ)(1+λ))Γ2

[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]
anzn

)

> 0, z ∈ U . (2.8)



G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 2 (2009), (239-249) 246

By noting the fact that
((1−λ)(n−1) sec η+(1−γ)(1+nλ−λ))Γn

(1−γ)
is increasing function for n ≥ 2

and in particular

((1−λ) secη+ (1− γ)(1+λ))Γ2

(1− γ)
≤
((1−λ)(n− 1) secη+ (1− γ)(1+ nλ−λ))Γn

(1− γ)
,

n≥ 2, |η|<
π

2
,

therefore, for |z|= r < 1, we have

Re

(

1+
((1−λ) secη+ (1− γ)(1+ λ))Γ2

[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]

∞
∑

n=1

anzn

)

= Re

�

1+
((1−λ) secη+ (1− γ)(1+λ))Γ2

[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]
z+

∞
∑

n=2

((1−λ) secη+ (1− γ)(1+λ))Γ2anzn

[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]

�

≥ 1−
((1−λ) secη+ (1− γ)(1+ λ))Γ2

[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]
r

−
1

[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]
×

∞
∑

n=2

((1−λ)(n− 1) secη+ (1− γ)(1+ nλ−λ))Γn|an|r
n

≥ 1−
((1−λ) secη+ (1− γ)(1+ λ))Γ2

[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]
r −

1− γ

[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]
r

> 0, |z|= r < 1,

where we have also made use of the assertion (2.3) of Theorem 2.1. This evidently

proves the inequality (2.8) and hence also the subordination result (2.4) asserted by

Theorem 2.2. The inequality (2.6) follows from (2.4) by taking

g(z) =
z

1− z
= z +

∞
∑

n=2

zn ∈ C .



G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 2 (2009), (239-249) 247

Next we consider the function

F(z) := z −
1− γ

((1−λ) secη+ (1− γ)(1+λ))Γ2

z2

where |η|< π

2
, 0 ≤ γ < 1, 0 ≤ λ < 1 and Γ2 is given by (2.5). Clearly F ∈ Rl

m
(η,γ,λ).

For this function (2.4)becomes

((1−λ) secη+ (1− γ)(1+λ))Γ2

2[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]
F(z)≺

z

1− z
.

It is easily verified that

min

�

Re

�

((1−λ) secη+ (1− γ)(1+ λ))Γ2

2[1− γ+ ((1−λ) secη+ (1− γ)(1+λ))Γ2]
F(z)

��

=−
1

2
, z ∈ U .

This shows that the constant
((1−λ) secη+(1−γ)(1+λ))Γ2

2[1−γ+((1−λ) secη+(1−γ)(1+λ))Γ2]
cannot be replaced by any

larger one.

By taking different choices of l, m, α1,α2, . . . ,αl , β1,β2, . . . ,βm, λ, γ and η in the

above theorem and in view of the Examples 1 to 4 in Section 1, we state the following

corollaries for the subclasses defined in those examples.

Corollary 2.5. If f ∈ S(η,γ,λ), then

(1−λ) secη+ (1− γ)(1+λ)

2[1− γ+ (1−λ) secη+ (1− γ)(1+λ)]
( f ∗ g)(z) ≺ g(z) (2.9)

where |η|< π

2
, 0≤ γ < 1; 0 ≤ λ < 1, g ∈ C and

Re
�

f (z)
	

> −
[1− γ+ (1−λ) secη+ (1− γ)(1+λ)]

(1−λ) secη+ (1− γ)(1+λ)
, z ∈ U .

The constant factor
(1−λ) secη+(1−γ)(1+λ)

2[1−γ+(1−λ) secη+(1−γ)(1+λ)]
in (2.9) cannot be replaced by a larger one.

Corollary 2.6. If f ∈ Dδ(η,γ,λ), then

(δ+ 1)[(1−λ) secη+ (1− γ)(1+λ)]

2[1− γ+ (δ+ 1){(1−λ) secη+ (1− γ)(1+λ)}]
( f ∗ g)(z) ≺ g(z), (2.10)



G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 2 (2009), (239-249) 248

where |η|< π

2
, 0≤ γ < 1; 0 ≤ λ < 1, δ > −1, g ∈ C and

Re
�

f (z)
	

>−
[1− γ+ (δ+ 1){(1−λ) secη+ (1− γ)(1+λ)}]

(δ+ 1)[(1−λ) secη+ (1− γ)(1+λ)]
, z ∈ U .

The constant factor
(δ+1)[(1−λ) sec η+(1−γ)(1+λ)]

2[1−γ+(δ+1){(1−λ) sec η+(1−γ)(1+λ)}]
in (2.10) cannot be replaced by a

larger one.

Corollary 2.7. If f ∈ B∗
µ
(η,γ,λ), then

(µ+ 1)[(1−λ) secη+ (1− γ)(1+λ)]

2[(µ+ 2)(1− γ) + (µ+ 1){(1−λ) secη+ (1− γ)(1+λ)}]
( f ∗ g)(z) ≺ g(z), (2.11)

where |η|< π

2
, 0≤ γ < 1; 0 ≤ λ < 1, µ > −1, g ∈ C and

Re
�

f (z)
	

>−
[(µ+ 2)(1− γ) + (µ+ 1){(1−λ) secη+ (1− γ)(1+ λ)}]

(µ+ 1)[(1−λ) secη+ (1− γ)(1+λ)]
, z ∈ U .

The constant factor
(µ+1)[(1−λ) secη+(1−γ)(1+λ)]

2[(µ+2)(1−γ)+(µ+1){(1−λ) sec η+(1−γ)(1+λ)}]
in (2.11) cannot be replaced by

a larger one.

Corollary 2.8. If f ∈ L∗a
c
(η,γ,λ), then

a[(1−λ) secη+ (1− γ)(1+λ)]

2[c(1− γ) + a{(1−λ) secη+ (1− γ)(1+λ)}]
( f ∗ g)(z) ≺ g(z), (2.12)

where |η|< π

2
, 0≤ γ < 1; 0 ≤ λ < 1, a > 0, c > 0, g ∈ C and

Re{ f (z)}> −
[c(1− γ) + a{(1−λ) secη+ (1− γ)(1+λ)}]

a[(1−λ) secη+ (1− γ)(1+λ)]
, z ∈ U .

The constant factor

2[c(1− γ) + a{(1−λ) secη+ (1− γ)(1+λ)}]

in (2.12) cannot be replaced by a larger one.

Remark 2.2. We observe that Corollary 2.5, yields the results obtained by Singh [9] for

the special values of λ,γ and η.



REFERENCES 249

ACKNOWLEDGEMENTS The authors would like to thank the referee for his valu-

able comments and suggestions.

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