Vol 3, No 12 (2012)

Available online throug

ISSN 2229 – 5046

AN INEQUALITY FOR CERTAIN ANALYTIC FUNCTIONS DEFINED BY A NEW

MULTIPLIER TRANSFORMATION

S R SWAMY

1

*, ROHAN R

1

AND NIRMALA J

2

1

Department of Computer Science and Engineering, R V College of Engineering,

Mysore Road, Bangalore - 560 059, India

2

Department of Mathematics, Maharani’s Science College for women,

Palace road, Bangalore-560 001, India

(Received on: 16-11-12; Revised & Accepted on: 19-12-12)

ABSTRACT

I

n this paper, the authors investigate an interesting differential inequality of certain analytic functions, defined by a new multiplier transformation in the open unit disc

U

=

{

z

:

z

<

1

}

.

2000 Mathematical Subject Classification: 30C45.

Key words and Phrases: Multivalent function, Starlike function, Convex function, Multiplier transformation.

1. INTRODUCTION

Let

A

_p

(

n

)

denote the class of functions of the form

...}).

3

,

2

,

1

{

,

(

,

)

(

=

+

∑

∈

=

∞

+ =

N

n

p

z

a

z

z

f

n p k

k k p

(1.1)

which are analytic in the open unit disc

U

=

{

z

:

z

<

1

}.

In particular, we set

A

_p

(

1

)

=

A

_p,

A

₁

(

n

)

=

A

(

n

)

and

A

A

A

A

1

(

1

)

=

1

=

(

1

)

=

,a well-known class of normalized analytic functions in

U

.

For

0

≤

ρ

<

p

,

p

,

n

∈

N

,

we denote by

S

_p*

(

n

,

ρ

)

and

K

p

(

n

,

ρ

)

the subclasses of

A

p

(

n

)

consisting of all analytic functions which are, respectively, n-p-valent starlike of order

ρ

and n-p-valent convex of order

ρ

. Note that

)

(

)

,

1

(

*

*

1

ρ

S

ρ

S

=

and

K

₁

(

1

,

ρ

)

=

K

(

ρ

)

are, respectively, the usual classes of univalent starlike functions of order

ρ

and univalent convex functions of order

ρ

,

0

≤

ρ

<

1

.

We also write

S

*

(

0

)

=

S

* and

K

(

0

)

=

K

, which are the classes of univalent starlike ( w.r.t origin) and univalent convex functions in

U

.

Definition 1.1. ([18, 19]). Let

p

,

n

∈

N

,

m

∈

N

₀

=

N



{

0

}

,

β

≥

0

and

α

a real number with

α

+

p

β

>

0

.

Then we define the operator

I

_pm_,α,β

:

A

_p

(

n

)

→

A

_p

(

n

),

by

)),

(

(

),

(

)

(

0

,

,

f

z

f

z

f

A

n

I

_pαβ

=

∈

_p

,

)

(

)

(

)

(

' 1

,

,

_α

_β

β

α

β α

p

z

zf

z

f

z

f

I

_p

+

+

=

…,

))

(

(

)

(

_{, , ,}1_,

,

,

f

z

I

I

f

z

I

m_pαβ

=

_{pαβ p}m_α−_β .

Remark 1.2. We observe that

I

m_p,α,β is a linear operator and for

f

(

z

)

given by (1.1), we have

MULTIPLIER TRANSFORMATION /IJMA- 3(12), Dec.-2012.

k k m

n p k p m

p

a

z

p

k

z

z

f

I

∑

∞

+

=













+

+

+

=

β

α

β

α

β

α,

(

)

, . (1.2)

It follows from (1.2) that

),

(

)

(

0 ,

,

f

z

f

z

I

m_pα

=

(1.3)

,

))

(

(

)

(

)

(

)

(

α

+

p

β

I

_pm_,α+1_,β

f

z

=

α

I

m_p,α,β

f

z

+

β

z

I

m_p,α,β

f

z

'

(1.4)

We note that

•

I

₁m_,α,β

f

(

z

)

=

I

_αm_,β

f

(

z

)

(See [17]).

•

I

m_p,α,1

f

(

z

)

=

I

_pm

(

α

)

f

(

z

),

α

>

−

p

(See [1], [14] and [16]).

•

I

m_{p,l+p−pβ,β}

f

(

z

)

=

I

_pm

(

β

,

l

)

f

(

z

),

l

>

−

p

,

β

≥

0

(See Catas [6]).

•

I

_pm_,0,β

f

(

z

)

=

D

m_p

f

(

z

)

(See [4], [9] and [12]).

Remark 1.3 i)

I

m_p

(

α

)

f

(

z

)

was considered in [1],[14] and [16], for

α

≥

0

and

I

m_p

(

β

,

l

)

f

(

z

)

was defined in [6]

for

l

≥

0

,

β

≥

0

, ii)

I

l

f

z

I

mp

l

f

z

l

p

m

p

(

)

(

)

=

(

1

,

)

(

),

>

−

, iii)

I

(

,

0

)

f

(

z

)

D

(

)

f

(

z

)

m

p m

p

β

=

β

,

β

≥

0

,

was mentioned in Aouf et.al. [3], iv) 1

(

β

),

β

≥

0

,

m

D

was introduced by Al-Oboudi [2],v)

D

1

(

1

)

f

(

z

)

D

f

(

z

)

m

m

=

was

defined by Salagean [13] and was considered for

m

≥

0 in [5] , vi)

I

1

(

α

)

f

(

z

),

α

≥

0

,

m

was investigated in [7] and

[8] and vii) 1

(

1

)

m

I

was due to Uralegaddi and Somanatha[20].

Definition 1.4. A function

f

∈

A

_p

(

n

)

is said to be in the class

S

_pm

(

n

,

α

,

β

,

ρ

)

for all

z

in

U

if it satisfies

p

z

f

I

z

f

I

m p m

p

ρ

β α

β

α

_>

















+

)

(

)

(

Re

, ,

1 , ,

, (1.3)

where

0

≤

ρ

<

p

,

p

,

n

∈

N

,

m

∈

N

0

=

N



{

0

}

,

β

≥

0

and

α

a real number with

α

+

p

β

>

0

.

We note that

S

_pm

(

1

,

α

,

1

,

ρ

)

is the class considered in [15] for

α

≥

0

.

we also note that

)

,

(

)

,

1

,

0

,

(

*

0

ρ

ρ

n

S

n

S

_p

=

_p and

S

1_p

(

n

,

0

,

1

,

ρ

)

=

K

_p

(

n

,

ρ

).

The basic tool in proving our result is the following lemma.

Lemma 1.5 [10, 11]). Let

Ω

be a set in the complex plane C. Suppose that the function

Ψ

:

C

2

×

U

→

C

satisfies the condition

Ψ

(

ix

₂

,

y

₁

;

z

)

∉

Ω

for all

z

∈

U

and for all real x₂and y₁ such that

).

1

(

2

2 2

1

x

n

y

≤

+

If p (z) = 1 + c_nzn+ … is analytic in E and for z

∈

E,

ψ

(

p

(

z

),

zp

'

(

z

);

z

)

⊂

Ω

,

then

Re(

p

(

z

))

>

0

in

U

.

2. MAIN RESULTS

Theorem 2.1. Let

λ

≥

0

,

γ

≤

1

,

0

≤

ρ

<

p

be real numbers such that

γ

≤

λ

and

2

1

1

_

<











−

p

ρ

γ

. Let

β

≥

0

,

α

a

real number such that

α

+

p

β

>

0

and let

))

(

1

(

1

)

(

2

))

(

1

(

)

/

(

)

/

)(

1

(

)

,

,

,

,

,

,

(

2

p

p

p

n

p

p

n

p

M

ρ

γ

β

α

ρ

λβ

ρ

λ

ρ

λ

ρ

β

α

γ

λ

−

−

+

−

−

+

−

If

f

∈

A

p

(

n

)

satisfies the condition

),

,

,

,

,

,

,

(

)

(

)

(

)

1

(

)

(

)

(

)

1

(

Re

₁ , , , , 2 , , 1 ,

,

λ

γ

α

β

ρ

γ

γ

λ

λ

β α β α β α β α

n

p

M

z

f

I

z

f

I

z

f

I

z

f

I

m p m p m p m p

_>

















+

−

+

−

+ + + (2.1) then

p

z

f

I

z

f

I

m p m p

ρ

β α β α

_>

















+

)

(

)

(

Re

, , 1 , ,

i.e.,

f

(

z

)

∈

S

m_p

(

n

,

α

,

β

,

ρ

).

Proof. Since

,

0

≤

ρ

<

p

, let us write

p

ρ

τ =

.Thus, we have

,

0

≤

τ

<

1

.

Now we define

.

),

(

)

1

(

)

(

)

(

, , 1 , ,

U

z

z

p

z

f

I

z

f

I

m p m p

∈

−

+

=

+

τ

τ

β α β α (2.2)

Therefore

p

(

z

)

is analytic in

U

and

p

(

0

)

=

1

.

Differentiating (2.2) logarithmically and using (1.4), we obtain

.

)]

(

)

1

(

)[

(

)

(

'

)

1

(

)]

(

)

1

(

[

)

(

)

(

1 , , 2 , ,

z

p

p

z

zp

z

p

z

f

I

z

f

I

m p m p

τ

τ

β

α

β

τ

τ

τ

β α β α

−

+

+

−

+

−

+

=

+ + (2.3)

A simple calculation yields

),

);

(

'

),

(

(

)

(

)

(

)

1

(

)

(

)

(

)

1

(

1 , , , , 2 , , 1 , ,

z

z

zp

z

p

z

f

I

z

f

I

z

f

I

z

f

I

m p m p m p m p

_ψ

γ

γ

λ

λ

β α β α β α β α

₌

















+

−

+

−

+ + + where

)]

(

)

1

(

[

)

1

(

)

(

'

)

(

)

1

(

)]

(

)

1

(

[

)]

(

)

1

(

)[

1

(

)

);

(

'

),

(

(

2

z

p

z

zp

p

z

p

z

p

z

z

zp

z

p

τ

τ

γ

γ

β

α

λβ

τ

τ

τ

λ

τ

τ

λ

ψ

−

+

+

−

+

−

+

−

+

+

−

+

−

=

. (2.4)

Let

p

(

z

)

=

u

₁

+

iu

₂and

zp

'

(

z

)

=

v

₁

+

iv

₂, where

u

₁

,

u

₂

,

v

₁

,

v

₂are real numbers with

(

1

)

2

2 2

1

u

n

v

≤

−

+

. Then

we have

),

(

max

)

(

)

1

(

)

1

(

))

;

,

(

Re(

2 2 2

2 2 2 2 2 2 1

2

u

u

u

Ru

S

z

v

iu

φ

φ

τ

γ

γτ

γ

ψ

=

≤

−

+

+

−

+

≤

(2.5)

where,

)

1

(

)

(

2

)

1

(

)

1

(

2

γ

γτ

β

α

τ

λβ

λτ

τ

λ

_

−

+











+

−

−

+

−

=

p

n

S

and

.

)

(

2

)

1

(

)

1

(

)

(

)

1

(

2

β

α

τ

λβ

γτ

γ

λγτ

λ

γ

τ

p

n

R

+

−

+

−

−

+

−

−

=

It can be easily verified that

φ

'

(

u

₂

)

=

0

implies

u

₂

=

0

and

φ

''

(

0

)

<

0

,

under the given conditions. Therefore,

max

φ

(

u

₂

)

=

φ

(

0

)

=

M

(

p

,

n

,

λ

,

γ

,

α

,

β

,

ρ

).

(2.6) Let

)}.

,

,

,

,

,

,

(

)

Re(

:

{

w

w

>

M

p

n

λ

γ

α

β

ρ

=

Ω

MULTIPLIER TRANSFORMATION /IJMA- 3(12), Dec.-2012.

p

z

f

I

z

f

I

m p m p

ρ

β α β α

_>

















+

)

(

)

(

Re

, , 1 , , .

Remark 2.2. Taking

β

=

n

=

1

in Theorem 2.1, we obtain Theorem 2.2 of Singh et.al. [15] (Considered for

α

≥

0

).

Our result hold true for

α

>

−

p

.

0

=

α

in Theorem 2.1 yields

Corollary 2.3. Let

λ

≥

0

,

γ

≤

1

,

0

≤

ρ

<

p

be real numbers such that

γ

≤

λ

and

.

2

1

1

_

<











−

p

ρ

γ

Let

.

))

(

1

(

1

2

))

(

1

(

)

/

(

)

/

)(

1

(

)

,

,

,

,

(

2

p

p

n

p

p

n

p

T

ρ

γ

ρ

λ

ρ

λ

ρ

λ

ρ

γ

λ

−

−

−

−

+

−

=

If

f

∈

A

_p

(

n

)

satisfies the condition

),

,

,

,

,

(

)

(

)

(

)

1

(

)

(

)

(

)

1

(

Re

₁ 2 1

ρ

γ

λ

γ

γ

λ

λ

n

p

T

z

f

D

z

f

D

z

f

D

z

f

D

m p m p m p m p

>

















+

−

+

−

+ + + then

.

)

(

)

(

Re

1

p

z

f

D

z

f

D

m p m p

ρ

>

















+

Remark 2.4. In the case when

p

=

n

=

1

, Corollary 2.3 reduces to Corollary 3.1 of Singh et. al [15].

Taking

α

=

l

+

p

−

p

β

,

l

>

−

p

,

in Theorem 2.1, we obtain

Corollary 2.5. Let

λ

≥

0

,

γ

≤

1

,

0

≤

ρ

<

p

be real numbers such that

γ

≤

λ

and

2

1

1

_

<











₋

p

ρ

γ

. Let

β

≥

0

and

p

l

>

−

and let

))

(

1

(

1

)

(

2

))

(

1

(

)

/

(

)

/

)(

1

(

)

,

,

,

,

,

,

(

2

p

p

l

p

n

p

p

l

n

p

ρ

γ

ρ

λβ

ρ

λ

ρ

λ

ρ

β

γ

λ

−

−

+

−

−

+

−

=

Τ

.

If

f

∈

A

_p

(

n

)

satisfies the condition

(

,

,

,

,

,

,

),

)

(

)

,

(

)

(

)

,

(

)

1

(

)

(

)

,

(

)

(

)

,

(

)

1

(

Re

₁ 2 1

ρ

β

γ

λ

β

γ

β

γ

β

λ

β

λ

l

n

p

z

f

l

I

z

f

l

I

z

f

l

I

z

f

l

I

m p m p m p m p

Τ

>

















+

−

+

−

+ + + then

.

)

(

)

,

(

)

(

)

,

(

Re

1

p

z

f

l

I

z

f

l

I

m p m p

ρ

β

β

>

















+ REFERENCES

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