Vol 3, No 7 (2012)

(1)

Available online throug

ISSN 2229 – 5046

ON A NEW SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS DEFINED

BY GENERALIZED SALAGEAN OPERATOR

Saurabh Porwal

a

, K. K. Dixit

b

, A. L. Pathak

c

and R. Tripathi

c*

a

Department of Mathematics, U.I.E.T. Campus, C.S.J.M. University, Kanpur-208024

b

Department of Mathematics, Gwalior Institute of Information Technology, Gwalior-474015 (M.P.), India

c

Department of Mathematics, Brahmanand College, The Mall, Kanpur(U.P.), India

(Received on: 15-03-12; Accepted on: 18-05-12)

ABSTRACT

T

he purpose of the present paper is to study some results involving coefficient conditions, extreme points, distortion bounds, convolution conditions and convex combination for a new class of generalized Salagean-Type harmonic univalent functions in the open unit disc. Relevant connections of the results presented here with various known results are briefly indicated.

AMS 2010 Mathematics Subject Classification: 30C45, 30C50, 30C55.

Keywords and Phrases: Harmonic, Univalent functions, Convex and Starlike functions.

1. INTRODUCTION

A continuous complex-valued function

f

=

u

+

iv

defined in a simply connected domain

D

is said to be harmonic in

D

if both

u

and

v

are real harmonic in

D

. In any simply connected domain we can write

f

=

h

+

g

, where

h

and

g

are analytic in

D

. We call

h

the analytic part and

g

the co-analytic part of

f

.

A necessary and sufficient condition for

f

to be locally univalent and sense preserving in

D

is that

|

h z

′

( ) |>|

g z

′

( ) |, .

z

∈

D

Let

S

_H denote the class of functions

f

=

h

+

g

which are harmonic univalent and sense preserving in the open unit disc

U

= { :| |< 1}

z

for which

f

(0) =

f

_z

(0) 1 = 0

−

. Then for

f

=

h

+ ∈

g

S

H we may express the analytic functions

h

and

g

as

1

=2 =1

( ) =

_k k

, ( ) =

_k k

,

< 1 .

k k

h z

z

a z g z

b z

b

∞ ∞

+

∑

(1.1)

Clunie and Sheil-Small [3] investigated the class

S

_H as well as its geometric subclasses and established some coefficient bounds. Since then, there have been several related papers on

S

_H and its subclasses.

For

f

=

h

+

g

given by (1.1), we defined the modified generalized Salagean operator of

f

as

0 0

( ) =

( ) ( 1)

( ), (

,

=

{0}, 0

1)

m m m m

D f z

_λ

D h z

_λ

+ −

D g z

_λ

m

∈

N N

N

∪

≤ ≤

λ

(1.2)

where

=2

( ) =

{(

1)

1}

m m k

k k

D h z

_λ

z

k

λ

a z

∞

+

∑

−

+

Corresponding author: R. Tripathic*

c

(2)

Generalized Salagean Operator / IJMA- 3(7), July-2012, Page: 2827-2835

and

=1

( ) =

{(

1)

1} ,

m m k

k k

D g z

_λ

k

λ

b z

∞

−

+

∑

where

D

_λm stands for the generalized Salagean operator introduced by Al-Obaudi[8]. For

λ

= 1

it reduces to Salagean operator introduced by Salagean [10].

Now for

0 ≤

γ

< 1

,

β

≥

0

,

m

∈

N

,

n

∈

N

₀ ,

m

>

n

,

α

∈

R

, 0

≤ ≤

λ

1

and

z U

∈

, suppose that

( , ; ; ; , )

H

RS

m n

β γ λ

t

denote the family of harmonic functions

f

of the form (1.1) such that

( )

Re (1

)

,

( )

m

i i

n t

D f z

e

D f z

α α

β

γ





+

−

≥









(1.3)

where

D f

m is defined by (1.2) and

f z

_t

( ) = (1

−

t z

)

+

(

h z

( )

+

g z t

( ) , 0

)

≤ ≤

t

1

.

Further, let the subclass

RS

_H

( , ; ; ; , )

m n

β γ λ

t

consist of harmonic univalent functions

f

_m

=

h

+

g

_m in

( , ; ; ; , )

H

RS

m n

β γ λ

t

so that

h

and

g

_m are of the form

1

=2 =1

( ) =

|

_k

|

k

, ( ) = ( 1 )

_m m

|

_k

|

k

.

k k

h z

z

a

z

g

z

b

z

∞ ∞

−

∑

−

∑

(1.4)

By specializing the parameters in subclass

RS

_H

( , ; ; ; , )

m n

β γ λ

t

we obtain the following known subclasses studied earlier by various authors.

1.

RS

_H

(

n

+

q n

, ; ;1; ,1)

β γ

≡

R

_H

( , , , )

n

γ β

q

studied by Dixit et al. [4]. 2.

RS

_H

(

n

+

1, ;1;1; ,1)

n

γ

≡

RS

_H

( , )

n

γ

studied by Yalcin et al. [14]. 3.

RS

_H

( , ; 0;1; ,1)

m n

γ

≡

S

_H

( , ; )

m n

γ

studied by Yalcin [13]. 4.

RS

_H

(

n

+

1, ; 0;1; ,1)

n

γ

≡

H n

( , )

γ

studied by Jahangiri et al. [6]. 5.

RS

_H

(2,1; ;1; ,1)

β γ

≡

HCV

( , )

β γ

studied by Kim et al. [7]. 6.

RS

_H

(1, 0; ; ; ,1)

β γ

t

≡

G

_H

( , , )

β γ

t

studied by Ahuja et al. [1]. 7.

RS

_H

(1, 0;1;1; ,1)

γ

≡

G

_H

( )

γ

studied by Rosy et al. [9]. 8.

RS

_H

(1, 0; 0;1; ,1)

γ

≡

S

_H*

( )

γ

studied by Jahangiri [5].

9.

RS

_H

(1, 0; 0;1; 0,1)

≡

S

*_H studied by Avci and Zlotkiewicz [2], Silverman [11] and Silverman and Silvia [12]. In the present paper, results involving the coefficient bounds, extreme points, distortion bounds, convex combinations for the above classes

RS

_H

( , ; ; ; , )

m n

β γ λ

t

and

RS

_H

( , ; ; ; , )

m n

β γ λ

t

of harmonic univalent functions have been investigated.

2. MAIN RESULTS

In our first theorem, we introduce a sufficient condition for function in

RS

_H

( , ; ; ; , )

m n

β γ λ

t

.

Theorem 2.1: Let

f

=

h

+

g

be such that

h

and

g

are given by (1.1). Furthermore, let

=2

{(

1)

1} (1

) {(

1)

1} (

)

|

1

m n

k k

k

t

a

λ

β

λ

γ β

γ

∞

₋

₊

₋

₊

−

∑

=1

{(

1)

1} (1

) ( 1)

{(

1)

1} (

)

|

| 1

1

m m n n

k k

k

t

b

λ

β

λ

γ β

γ

−

∞

₋

₊

_{− −}

₋

₊

+

≤

−

∑

(2.1)

(3)

in

U

and

f

∈

RS

_H

( , ; ; ; , )

m n

β γ λ

t

.

Proof: If

z

₁

≠

z

₂ then

1 2 1 2

( )

1 ( )

( )

f z

g z

h z

−

≥ −

−

(

)

(

)

(

)

1 2 =1

1 2 1 2

=2

= 1

k k

b

z

a

z

∞

−

+

−

∑

=1

=2

{(

1)

1} |

|

> 1

1 {(

1)

1} |

|

k k

k

b

k

a

λ

∞

−

+

−

+

∑

=1

=2

{(

1)

1} (1

) ( 1)

{(

1)

1} (

)

|

1

1 {(

1)

1} (1

) {(

1)

1} (

)

1 |

|

1

m m n n

k

m n

k

t

b

k

t

a

λ

β

λ

γ β

γ

λ

β

λ

γ β

γ

∞

−

∞



₋

₊

_{− −}

₋

₊











−













≥ −

−

+

−

+

−

∑

0, (Using (2.1))

≥

which proves the univalence.

Also, we have

1 =2

( )

1

_k k

k

h z

k a

z

∞ ₋

′

≥ −

∑

=2

> 1

_k

k

k a

∞

−

∑

=2

{(

1)

1} (1

) {(

1)

1} (

)

1 |

|

1

m n

k

t

a

λ

β

λ

γ β

γ

∞

−

+

−

+

≥ −

−

∑

=1

{(

1)

1} (1

) ( 1)

{(

1)

1} (

)

|

1

m m n n

k

t

b

λ

β

λ

γ β

γ

∞

−

+

− −

−

+

≥

−

∑

1 =1

{(

1)

1} (1

) ( 1)

{(

1)

1} (

)

>

|

|| |

1

m m n n

k k

k

t

b

z

λ

β

λ

γ β

γ

∞

−

+

− −

−

+

−

∑

1 =1

|

_k

|| |

k

k b

z

∞

−

≥

∑

≥

|

g z

′

( ) | .

(4)

Using the fact Re

( )

w

≥

γ

if and only if

|1

− + ≥ + −

γ

w

| |1

γ

w

|

, it suffices to show that

| (1

−

γ

)

D f z

_λn _t

( ) (1

+ +

β

e

iα

)

D f z

_λm

( )

−

β

e D f z

iα _λn _t

( ) |

| (1

γ

)

D f z

_λn _t

( ) (1

β

e

iα

)

D f z

_λm

( )

β

e D f z

iα _λn _t

( ) | 0 .

−

+

− +

+

≥

(2.2)

Substituting for m

( )

D f z

_λ and n

( )

t

D f z

_λ in L.H.S. of (2.2) we have

=2

=| (2

)

[(1

){(

1)

1}

n

(1

i

){(

1)

1}

m i

k

z

k

t

e

α

k

e

α

γ

∞

γ

λ

β

λ

β

−

+

∑

−

+

+ +

−

+

−

=1

{(

1)

1} ]

n _k k

( 1)

n

[(1

){(

1)

1}

n

( 1)

m n

(1

i

)

k

λ

t a z

γ

k

λ

t

β

e

α

∞

−

+

+ −

∑

−

+

+ −

+

=2

{(

1)

1}

m i

{(

1)

1} ]

n k

|

[(1

i

){(

1)

1}

m

k

λ

β

e

α

k

λ

t b z

γ

z

β

e

α

k

λ

∞

−

+

−

+

−

+

∑

+

−

+

=1

(1

){(

1)

1}

n i

{(

1)

1} ]

n _k k

( 1)

n

[( 1)

m n

k

t

e

α

k

t a z

γ

λ

β

λ

∞ −

− +

−

+

−

+

+ −

∑

−

(1

){(

1)

1}

(1

){(

1)

1}

{(

1)

1} ]

|

i m n i n k

k

e

α

k

t

e

α

k

t b z

β

λ

γ

λ

β

λ

+

−

+

− +

−

+

−

+

=2

2(1

) | | 2

[{(

1)

1} (1

m

) (

) {(

1)

1} ] |

n _k

|| |

k

z

k

t k

a

z

γ

∞

λ

β

γ β

λ

≥

−

∑

−

+

− +

−

+

=1

| [(1

){(

1)

1}

n

( 1)

m n

(1

i

){(

1)

1}

m i

{(

1)

1} ] ||

n _k

|| |

k

t

e

α

k

e

α

k

t

b

z

γ

λ

β

λ

β

λ

∞

−

∑

−

+

+ −

+

−

+

−

+

=1

| [( 1)

m n

(1

i

){(

1)

1}

m

(1

){(

1)

1}

n i

{(

1)

1} ] ||

n

|| |

k k k

e

α

k

t

e

α

k

t

b

z

β

λ

γ

λ

β

λ

∞

−

∑

−

+

−

+

− +

−

+

−

+

=2

=1

=2

=1

2(1

) | | 2

[{(

1)

1} (1

) (

) {(

1)

1} ] |

|| |

2 [{(

1)

1} (1

) (

) {(

1)

1} ] |

|| |

if

is odd

=

2(1

) | | 2

[{(

1)

1} (1

) (

) {(

1)

1} ] |

|| |

2 [{(

1)

m n k

k k

m n k

k k

m n k

k k

k

z

k

t k

a

z

k

t k

b

z

m n

z

k

t k

a

z

k

γ

λ

β

γ β

λ

β

γ β

λ

γ

λ

β

γ β

λ

∞

−

+

− +

−

+

−

+

−

+

−

+

− +

−

+

−

∑

1} (1

) (

) {(

1)

1} ] |

|| |

if

is even

m n k

k

t k

b

z

m n

λ

β

γ β

λ









₊

_{− +}

₋

₊



−



1 =2

{(

1)

1} (1

) (

) {(

1)

1}

= 2 (

1 ) | | {1

|

|| |

1

n n

k k k

k

t k

z

λ

β

γ β

λ

a

z

γ

∞

−

+

− +

−

+

−

∑

1 =1

{(

1)

1} (1

) ( 1)

(

) {(

1)

1}

|

|| |

}

1

m m n n

k k k

k

t k

b

z

λ

β

γ β

λ

γ

− ∞

−

+

− −

+

−

+

−

∑

=2

{(

1)

1} (1

) (

) {(

1)

1}

> 2(1

){1 (

|

1

m n

k k

k

t k

a

λ

β

γ β

λ

γ

∞

₋

₊

_{− +}

₋

₊

−

(5)

=1

{(

1)

1} (1

) ( 1)

(

) {(

1)

1}

|

|)}.

1

m m n n

k k

k

t k

b

λ

β

γ β

λ

γ

−

∞

₋

₊

_{− −}

₊

₋

₊

−

∑

The last expression in non-negative by (2.1) and so that proof is complete.

The harmonic univalent functions

=2

1 ( ) =

{(

1)

1} (1

) (

) {(

1)

1}

k k

m n

k

f z

z

x z

k

t k

γ

λ

β

γ β

λ

∞

₋

+

−

+

− +

−

+

∑

=1

1 .

{(

1)

1} (1

) ( 1)

(

) {(

1)

1}

k k

m m n n

k

y z

k

t k

γ

λ

β

γ β

λ

∞

−

+

−

+

− −

+

−

+

∑

(2.3)

where

2 1

|

_k

|

_k

| 1

k k

x

y

∞ ∞

= =

+

=

∑

, show that the coefficient bound given by (2.1) is sharp.

This completes the proof of Theorem 2.1.

In the following theorem, it is proved that the condition (2.1) is also necessary for function

f

_m

=

h

+

g

_m, where

h

and

g

_m are of the form (1.4).

f

_m

=

h

+

g

_m be given by (1.4). Then

f

_m

∈

RS

_H

( , ; ; ; , )

m n

β γ λ

t

if and only if

=1

({(

1)

1} (1

m

) {(

1)

1} (

n

) |

_k

| {(

1)

1}

m

k

λ

β

k

λ

t

γ β

a

k

λ

∞

−

+

−

+

−

+

∑

(1

+

β

) ( 1)

− −

m n−

{(

k

−

1)

λ

+

1} (

n

t

γ β

+

) |

b

_k

|)

≤

2(1

−

γ

).

(2.4)

Proof: Since

RS

_H

( , ; ; ; , )

m n

β γ λ

t

⊆

RS

_H

( , ; ; ; , )

m n

β γ λ

t

, we only need to prove the “only if” part of the theorem. To this end, for functions

f

_m of the form (1.4), we notice that condition

( )

(1

)

( )

m

i i

n t

D f z

Re

e

D f z

α λ α

λ

β

γ





+

−

≥









is equivalent to

=2 2 1

=1

1

=2 =1

(1

)

[{(

1)

1} (1

) (

){(

1)

1} ] |

|

( 1)

[{(

1)

1} (1

) ( 1)

(

){(

1)

1} ] |

|

{(

1)

1} |

|

( 1)

{(

1)

1} |

|

m i i n k

k k

m m i m n i n k

k k

n k m n n k

k k

z

k

e

k

t

a

z

k

e

k

t b

z

Re

z

k

t a

z

k

t b

z

α α

γ

λ

β

γ

λ

β

γ

λ

∞

− −

∞ ∞

+ −



₋

₊

₋

₊

₋

₊



+ −

−

+

− −

+

−

+







₋

₊

_{+ −}

₋

₊





∑

0. 





_≥





_

(2.5)

(6)

1 =2

1 =1

1 1

=2 =1

(1

)

[{(

1)

1}

{(

1)

1} ] |

|

[{(

1)

1}

( 1)

{(

1)

1} ] |

|

1 {(

1)

1} |

|

( 1)

{(

1)

1} |

|

m n k

k k

m m n n k

k k

n k m n n k

k k

k

t k

a

r

k

t k

b

r

Re

k

t a

r

k

t b

r

γ

λ

γ

λ

γ

λ

∞

−

∞

− −

∞ ∞

− − −



_{− −}

₋

₊

₋

₊



₋

₊

_{− −}

₋

₊





 −

−

+

− −

−

+





∑

1 =2

1 =1

1 1

=2 =1

({(

1)

1}

{(

1)

1} ) |

|

({(

1)

1}

( 1)

{(

1)

1} ) |

|

0.

1 {(

1)

1} |

|

( 1)

{(

1)

1} |

|

m n k

k k

m m n n k

k

i k

n k m n n k

k k

k

t k

a

r

k

t k

b

r

e

k

t a

r

k

t b

r

α

β

λ

β

λ

∞

−

∞

− −

∞ ∞

− − −



−

+

−

+

_



−

+

− −

−

+

_

−

_

≥



−

+

− −

−

+





∑

Since

Re

(

−

e

iα

)

≥ −

|

e

iα

|= 1

−

, the above inequality reduces to

1 =2

1 =1

1 1

=2 =1

(1

)

[{(

1)

1} (1

) (

){(

1)

1} ] |

|

[{(

1)

1} (1

) ( 1)

(

){(

1)

1} ] |

|

0.

1 {(

1)

1} |

|

( 1)

{(

1)

1} |

|

m n k

k k

m m n n k

k k

n k m n n k

k k

k

t

a

r

k

t b

r

k

t a

r

k

t b

r

γ

λ

β

β γ

λ

β

β γ

λ

∞

−

∞

− −

∞ ∞

− − −

− −

−

+

−

+

−

+

−

+

− −

+

−

+

≥

−

+

− −

−

+

∑

(2.6)

If the condition (2.4) does not hold, then the numerator in (2.6) is negative for

r

sufficiently close to 1. Hence there exists a

z

₀

=

r

₀ in (0,1) for which the quotient in (2.6) is negative. This contradicts the required condition for

( , ; ; ; , )

m H

f

∈

RS

m n

β γ λ

t

and so the proof is complete.

We employ the techniques of Dixit et al. [1.4] in the proof of Theorem 2.3 and 2.4.

f

_m be given by (1.4). Then

f

m

∈

RS

H

( , ; ; ; , )

m n

β γ λ

t

if and only if

1 =1

( ) =

(

( )

( )), wh e

re

( )

,

m k k k mk

k

f

z

x h z

y g

z

h z

z

∞

+

=

∑

1 ( ) =

, ( = 2, 3, 4...),

{(

1)

1} (1

) (

) {(

1)

1}

k

k m n

h z

z

k

t k

γ

λ

β

γ β

λ

−

+

− +

−

+

1

1 ( ) =

( 1)

, ( = 1, 2, 3...),

{(

1)

1} (1

) ( 1)

(

) {(

1)

1}

m k

mk m m n n

g

z

k

t k

γ

λ

β

γ β

λ

−

+ −

(7)

0

k

x

≥

,

y

_k

≥

0

,

1

(

k k

) = 1

k

x

y

∞

=

+

∑

. In particular the extreme points of

RS

_H

( , ; ; ; , )

m n

β γ λ

t

are

{ }

h

_k and

{

g

mk

}

.

f

m

∈

RS

H

( , ; ; ; , )

m n

β γ λ

t

. Then for

| |=

z

r

< 1

we have

2

1 1

1

1 (1

) ( 1)

(

)

|

( ) | (1 |

|)

|

(

1)

(

1)

(1

)

(

)

(

1)

(1

)

(

)

m n

m n m n m n

t

f

z

b

r

b

r

t

γ

β

γ β

λ

β

γ β

λ

β

γ β

−

− −



−

+

− −

+



≤ +

+

_

−

_

+

_

+

−

+

−

+

_

and

2

1 1

1

1 (1

) ( 1)

(

)

|

( ) | (1 |

|)

|

.

(

1)

(

1)

(1

)

(

)

(

1)

(1

)

(

)

m n

m n m n m n

t

f

z

b

r

b

r

t

γ

β

γ β

λ

β

γ β

λ

β

γ β

−

− −



−

+

− −

+



≥ +

−

_

−

_

+

_

+

−

+

−

+

_

The following covering result follows from the left hand inequality in Theorem 2.4.

Corollary 2.5: Let

f

_m of the form (1.4) be so that

f

m

∈

RS

H

( , ; ; ; , )

m n

β γ λ

t

. Then

(

1) (1

) (

1) (

)

1 { :|

|<

(

1) (1

) (

1) (

)

m n

t

w w

t

λ

β

λ

γ β

γ

λ

β

λ

γ β

+

−

+

− +

+

−

+

((

1)

1)(1

)

(

)((

1)

( 1)

)

|

₁

|}.

(

1) (1

) ((

1)) (

)

m n m n

m n

t

b

t

λ

β

γ β λ

λ

β

λ

γ β

−

+

−

+

−

+

− −

−

+

−

+

For our next theorem, we need to define the convolution of two harmonic functions. For harmonic functions of the form

1

=2 =1

( ) =

|

k

( 1)

m

|

k

m k k

k k

f

z

a

z

b

z

∞ ∞

−

∑

+ −

∑

and

1

=2 =1

( ) =

|

k

( 1

m

)

|

k

m k k

k k

F z

z

A

z

B

z

∞ ∞

−

∑

+ −

∑

we define the convolution of two harmonic functions

f

and

F

as

1

=2 =1

(

_m

*

_m

)( ) =

_m

( ) *

_m

( ) =

|

_k _k

|

k

( 1 )

m

|

_k _k

|

k

.

k k

f

F

z

f

z

F z

z

a A

z

b B

z

∞ ∞

−

∑

+ −

∑

(2.7)

Using the definition, we show that the class

S

_H

( , ; ; )

m n

α λ

is closed under convolution.

Theorem 2.6: For

0 ≤ ≤

γ

₁

γ

₂

< 1

let

f

_m

∈

RS

_H

( , ; ; ; , )

m n

β γ λ

t

₁ and

F

_m

∈

RS

_H

( , ; ; ;

m n

β γ λ

t

₂

, )

. Then

2 1

*

( , ; ; ;

, )

( , ; ; ; , )

m m H H

f

F

∈

RS

m n

β γ λ

t

⊆

RS

m n

β γ λ

t

.

Proof: Let 1

2 1

( ) =

k

( 1)

m k

m k k

k k

f

z

a z

b z

∞ ∞

−

= =

−

∑

+ −

∑

be in

RS

_H

( , ; ; ; , )

m n

β γ λ

t

₁ and

1

=2 =1

( ) =

k

( 1)

m k

m k k

k k

F z

z

A z

B z

∞ ∞

−

∑

+ −

∑

(8)

1

k

A

≤

and

B

_k

≤

1

. Now, for the convolution function

f

_m

*

F

_m we get

2

=2 2

{(

1)

1} (1

) {(

1)

1} (

)

|

1

m n

k k k

k

t

a A

λ

β

λ

γ

β

γ

∞

₋

₊

₋

₊

−

∑

2

=1 2

{(

1)

1} (1

) ( 1)

{(

1)

1} (

)

|

1

m m n n

k k k

k

t

b B

λ

β

λ

γ

β

γ

−

∞

₋

₊

_{− −}

₋

₊

+

−

∑

1

=2 1

{(

1)

1} (1

) {(

1)

1} (

)

|

1

m n

k k

k

t

a

λ

β

λ

γ

β

γ

∞

₋

₊

₋

₊

≤

−

∑

1

=1 1

{(

1)

1} (1

) ( 1)

{(

1)

1} (

)

|

1

m m n n

k k

k

t

b

λ

β

λ

γ

β

γ

−

∞

₋

₊

_{− −}

₋

₊

+

−

∑

≤

1. (Since

f

m

∈

RS

H

( , ; ; ; , ))

m n

β γ λ

t

1

Therefore

f

_m

*

F

_m

∈

RS

_H

( , ; ; ;

m n

β γ λ

t

₂

, )

⊆

RS

_H

( , ; ; ; , )

m n

β γ λ

t

₁ for

0 ≤ ≤

β α

< 1

. Now we show that

RS

_H

( , ; ; ; , )

m n

β γ λ

t

is closed under convex combinations.

Theorem 2.7: The class

RS

_H

( , ; ; ; , )

m n

β γ λ

t

is closed under convex combination.

Proof: For

i

= 1, 2, 3...

let _m

( )

_H

( , ; ; ; , ),

i

f

z

∈

RS

m n

β γ λ

t

where _m

( )

i

f

z

is given by

1

2 1

( ) =

k

( 1)

m k

m_i k_i k_i

k k

f

z

a z

b z

∞ ∞

−

= =

−

∑

+ −

∑

.

Then by Theorem 2.2

=1

{(

1)

1} (1

) {(

1)

1} (

)

|

1

m n

k_i k

k

t

a

λ

β

λ

γ β

γ

∞

₋

₊

₋

₊

−

∑

{(

1)

1} (1

) ( 1)

{(

1)

1} (

)

|

| 2.

1

m m n n

k_i

k

t

b

λ

β

λ

γ β

γ

−

+

− −

−

+

≤

−

For

1

= 1, 0

1

i i

i

t

∞

=

≤ ≤

∑

, the convex combination of _m i

f

may be written as

1

=1 =2 =1 =1 =1

( ) =

k

( 1)

m k

.

i m_i i k_i i k_i

i k i k i

t f

z

t a

z

t b

z

∞ ∞ ∞ ∞ ∞

−









−

_

_

+ −

_

_









∑

∑ ∑

Then by 2.2,

=1 =1

{(

1)

1} (1

) {(

1)

1} (

)

[

|

1

m n

i k_i

k i

k

t

t a

λ

β

λ

γ β

γ

∞

₋

₊

₋

₊

∞

−

∑

=1

{(

1)

1} (1

) ( 1)

{(

1)

1} (

)

|

|]

2

1

m m n n

i k_i i

k

t

t b

λ

β

λ

γ β

γ

− ∞

−

+

− −

−

+

≤

−

∑

=1 =1

{(

1)

1} (1

) {(

1)

1} (

)

=

{

|

1

m n

i k_i

i k

k

t

λ

β

λ

γ β

a

γ

∞ ∞

₋

₊

₋

₊

−

∑ ∑

{(

1)

1} (1

) ( 1)

{(

1)

1} (

)

|

1

n m n n

k_i

k

t

b

λ

β

λ

γ β

γ

−

+

− −

−

+

(9)

=1

2

i

= 2.

i

t

∞

≤

∑

This is the condition required by Theorem 2.2 and so

1

( )

( , ; ; ; , )

i m_i H

i

t f

z

RS

m n

β γ λ

t

∞

=

∈

∑

.

1_{ACKNOWLEDGEMENT}

The author is thankful to University Grants Commission, New Delhi, Govt. of India for financial support under grant serial no. F.No. 8-2 (223)/2011 (MRP/NRCB).

REFERENCES

[1] Ahuja, O.P., Aghalary, R. and Joshi, S.B. (2005). ``Harmonic univalent functions associate with

k

-Uniformly starlike functions'', J. Math. Res. Sci, 9(1), 9-17.

[2] Avci, Y. and Zlotkiewicz, E. (1990). ``On harmonic univalent mappings''. Ann. Univ. Mariae Curie-Sklodowska, Sect.A 44, 1-7.

[3] Clunie, J. and Sheil-Small, T. (1984). ``Harmonic univalent functions'', Ann. Acad. Sci. Fenn. Ser. AI Math 9, 3-25.

[4] Dixit, K.K., Pathak, A.L., Porwal, S. and Agarwal, R. (2009), ``A new subclass of harmonic univalent functions defined by Salagean operator'', Int. J. Contemp. Math. Sci. 4(8), 371-383.

[5] Jahangiri, J.M., (1999). ``Harmonic functions starlike in the unit disc'', J. Math. Anal. Appl. 235, 470-477. [6] Jahangiri, J.M., Murugusundaramoorthy, G. and Vijay, K. (2002), ``Salagean-type harmonic univalent

functions'', South J. Pure Appl. Math. 2, 77-82.

[7] Kim, Y.C., Jahangiri, J.M. and Choi, J.H. (2002), ``Certain convex harmonic functions'', Int. J. Math. Math. Sci., 29 (8), 459-465.

[8] Oboudi, Al. (2004), On Univalent function defined by Generalized Salagean operator, International Journal of Mathematics and Mathematical Sciences, 27, 1429-1436.

[9] Rosy, T, Stephen, B.A., Subramanian, K.G. and Jahangiri, J.M. (2001), ``Goodman-Ronning-type harmonic univalent functions'', Kyungpook Math. J. 41 (1), 45-54.

[10] Salagean, G.S. (1983) ``Subclasses of univalent functions'', Complex Analysis Fifth Romanian Finish Seminar, Bucharest 1, 362-372.

[11] Silverman, H. (1998) ``Harmonic univalent functions with negative coefficients'' J. Math. Anal. Appl. 220, 283-289.

[12] Silverman, H. and Silvia, E.M. (1999), ``Subclass of harmonic univalent functions'', New Zealand J. Math. 28, 275-284.

[13] Yalcin, S. (2005), ``A new class of Salagean-type harmonic univalent functions'', Appl. Maths. Letters 18, 191-198.

[14] Yalcin, S., Öztürk and Yamankaradeniz, M. (2007), ``On the subclass of Salagean-type harmonic univalent functions'', J. Inequl. Pure Appl. Maths. 8(2), Art 54, 1-17.