doi:10.4236/apm.2011.14034 Published Online July 2011 (http://www.SciRP.org/journal/apm)
Inclusion and Argument Properties for Certain
Subclasses of Analytic Functions Defined by
Using on Extended Multiplier Transformations
Oh Sang Kwon
Department of Mathematics, Kyungsung University, Busan, Korea E-mail: [email protected]
Received March 28, 2011; revised April 27, 2011; accepted May 5, 2011
Abstract
Making use of a multiplier transformation, which is defined by means of the Hadamard product (or convolu-tion), we introduce some new subclasses of analytic functions and investigate their inclusion relationships and argument properties.
Keywords:Subordination, Starlike Functions, Convex Functions, Closed-to-Convex Functions, Multiplier Transformation, Multivalent Functions, Argument Principle
1. Introduction
Let Ap denote the class of functions f normalized by
=1
= p k p ( : {1, 2,3, })
k p k
f z z a z p
(1.1)which are analytic and p-valent in the open unit disk
= : and < 1 U z z z
If f and g are analytic in U , we say that f is subordinate to g, and write
or ( )
f g f z g z zU
if there exists a Schwarz function , analytic in
with and
z
U
0 = 0
z < 1 in zU , such that
=
f z g z for z U .
We denote by *
p
S and Cp
the subclasses ofp
A consisting of all analytic functions which are, respectively, p-valent starlike of order (0< p) in U and -valent convex of order p (0< p) in U.
Let M be the class of analytic functions with , which are convex and univalent in U and
0 = 1
satisfy the following inequality:
Re z > 0 (zU)
Making use of the aforementioned principle of subordination between analytic functions, we define each of the following subclasses of Ap:
* ;
1
: Re : and
(0 < ; ; )
p
p
S
zf z
f f A z
p f z
p z U M
(1.2)
;
1
: Re : and 1
(0 < ; ; )
p
p
K
zf z
f f A z
p f z
p z U M
(1.3) For m0: {0,1, 2, } , we define the multiplier transformation Jm
p, , l
of functionsp
f A by
, ; ,
: Re : and *
;
. . 1
(0 , < ; ; , )
p p p
zf z
C f f A g S s t
p g z p z U M
z
(1.4)
=1
, , =
( > 0; 0; )
m
m p
k p k
l k k p
J p l f z z a z l
l z
U
(1.5)
Put
, ,
=1
=
( ; > 0; 0; )
m
m p
p l
k
l k
z z z
l
m l z U
k p
(1.6)
The operators mp, ,l and p lm,1, , are the multiplier transformations introduced and studied earlier by Sarangi and Uralegaddi [16] and Uralegaddi and Somanatha ([1] and [2]), respectively. Correspending to the function
, ,
m pl z
defined by (1.6), we introduce a function
, , ,
m pl z
given by the Hadamard product (or convolu- tion):
,
, , * , , = ( >
1 p
m m
p l p l p
z
z z
z
p)
Then, analogous to Jm
p, , l , we have define anew multiplier transformation
, , :
mp p
I p l A A
as follows:
,
, ,
, , = *
m m
p l
I p l f z z f z (1.7) We note that
0 1
2
,1,1 = and 1,1, 2 =
p
I p f z f z I f z zf z
It is easily verifed from the above definition of the operator Im
p, ,l
, that
1
, ,
= , , , ,
m
m m
z I p l f z
p I p l f z I p l f z
(1.8)and
1
, ,
= , , , ,
m
m m
z I p l f z
lI p l f z p l I p l f z
(1.9)
The definition (1.6) of the multiplier transformation , ,
m pl
is motivated essentially by the Choi-Saigo- Srivastava operator [3] for analytic functions, which includes a simpler integral operator studied earlier by Noor [7] and others (cf. [4-6]).
Next, by using the operator Im
p, ,l
defined by (1.7), we introduce the following subclasses of analytic functions:
*
= : and , , ;
( ; , , > 0; ;0 < 1
m
p p
f f A I p l f z S
M l m
, , , ;
m p l
S
)
(1.10)
, , , ;
= : and , , ;
( ; , , > 0; ;0 < 1
m p l
m
p p
K
f f A I p l f z K
M l m
)
(1.11)
and
, ,
, ; ,
= : and , , , ; ,
( , ; , , > 0; ;0 , < 1)
p l
m p
f f A I p l f z C
M l m
,
m
C
(1.12) We also note that
pm, ,, l
;
zf z
Smp, ,,l
;
f z K (1.13)
In particular, we set
, ,
, , , ,
1
m ; Az= m ; , ( 1 < < 1)
1
p l p l
S S A B B A
Bz
(1.14) and
, ,
, , , ,
1
; = ; , ( 1 < <
1
m m
p l p l
Az
K K A B B
Bz
A 1)
(1.15) In the present paper, we investigate some inc re
lusion lationships and argument properties associated with such multivalent functions in the class Ap as those be- longing to the subclasses ,
, , ;
m p l
S , ,
, ; ,m
Kp l and
,
, , , ; ,
m p l
C defined by (1.10), .11) and (1(1 .12), respectively.
2. Inclusion Properties
emma 2.1: Let
L be convex univalent in U with
0 1 and Re
z
> 0 ( , ). I p is n Uf analytic i with p
0 = 1, then
zp z
( )p z z z U
z
implies that p z
z2.2: Let
(z U ).
Theorem M with
Re
> max ,min
z U
l p z
p p
then , 1
,
1,
;
, , ; , , ; , ,
m m m
p l p l p l
S S S . Proof. First of all, we show that
, 1 ,
, , ; , , ;
m m
p l p l
S S . Let , ,, 1
;
m p l
fS and set
1 z I
m
p, ,l f z
=
, ,
m
p z
p I p f z
l
(2.1)
where the function p z
obtain
is analytic in with .
ng (2.1), we
U
0 = 1p
Applyi
1
, ,m
p p p z
I p l f
, ,
= m
I p l f z z
(2.2)
tiating both sides of (2.2) and multiplying the reseulting equation by , we have
By logarithmically differen
z
, , 1= ( )
m
z I p l f z
zp z
p z z U
p p z
Since
, by appma 2.1 to (2.3), it follows that in , that is, that
, ,
m
p I p l f z
(2.3)
Re p z > 0 lying Lem-
p z z U
,
, , ;
m p l
f z S . prove the second
To part of Theore 2.1, letm
z S ,
, , ;m p l
f and put
1
1=
, ,
m
q z
p I p l f z
, ,
1 z Im p l f z
where the function is analytic in with .
precisely the sa nner, we can ult that in , that is, that
q z
me ma
U
find the res
0 = 1q
In
q z z U
1,
, , ;
m p l
f z S under the hypothesis
Re p z p > 0
t
z M withl
Theorem 2.3: Le
> max ,min
z U
l p Re z
p p
then , 1
,
1,
;
, , ; , , ; , ,
m m m
p l p l p l
K K K
, 1 , 1
, , , ,
, ,
, , , ,
; ;
; ;
m m
p l p l
m m
p l p l
f z K zf z S zf z S f z K
and
,
,
, , ; , , ;
m m
p l p l
f z K zf z S
1,
1,
, , ; , , ;
m m
p l p l
zf z S f z K
which evidently prove Theorem 2.3. By setting
=1 ( 1 < < 1;1
Az
z B A
Bz
z U )
in Theorems 2.2 and 2.3, we deduce the fo corollary.
Corollary 2.4: Suppose that
llowing
. Proof. Applying (1.11) and Theorem 2.2, we observe that
1
> max ,
1
l p A
B p p
Then, for the function classes defined by (1.12) and (1.13),
1,
, , , , m, , ; ,
pl pl Spl A B
and
, 1 ; , , ; ,
m m
S A B S A B
, 1 , 1,
, , ; , , , ; , , , ; ,
m m m
p l p l p l
K A B K A B K A B
Theorem 2.5: Let , M with
> max ,min
z U
l p
Re z
p p
then
, 1 ,
, , , ,
, ,
, ; , , ; ,
; ,
m m
p l p l
p l
C C
C
1, , m
Proof. We begin by proving that
, 1 ,
, , , ; , , , , ; ,
m m
p l p l
C C , which is t first inclusion relationship asserted by Theorem 2.5.
he
Let
, 1
, , , ; ,
m p l
f z C . Then there exists a function k z
Sp*
;
such that
1 , ,
1
( )
m
z I p l f z
z z U p k z
Choose the function
g z such that
1 , , =
m
p
Then
,
, ,
; Sp l ;
, 1
, ,
m m
p l
g z S , and
1
1
, , 1
( )
, ,
m
m
z I p l f z
z z U p I p l g z
(2.4) Now let
=
, ,
, ,m
m
z I p l f z I p l g z
p z
(2.5)
where the function p z
is analytic in U with
0 = 1p .
nd th Using (1.9), we fi at
1
1
2
, , , ,
, ,
1 = 1
, , , , , ,
1 , , , ,
1 =
, , , ,
1 =
m m
m m
m m
m m
m m
z z I p l f z I p l f z z I p l f z
p I p l g z p z I p l g z I p l g z
z I p l f z z I p l f z p z I p l g z I p l g z
p
2
, , , ,
1
, , , ,
, , , ,
m m
m m
m m
z I p l f z z I p l f z I p l g z I p l g z
z I p l g z I p l g z
Since
,
, , ;m p l
g z S , then we set
= 1
, ,
, ,m
m
z I p l f z q z
p I p l g z
, , , ,
= 1
m
m
z I p l f z I p l f z
p zp z p q z
p p z
(2.8)
Hence (2.6)
where q
z
z in U with the assumption thatM
. By (2.5),
2 , ,
, ,
= 1
m
m
z I p l f z I p l g z
p zp z p q z p p z
, ,
= , ,
m m
z I p l f z
p p z I p l g z
and mu , we obtain
(2.7)
Differentiating both side of (2.7) with respect to z
ltiplying by z Computing the above equations, we can obtain
1
1
, , 1
, ,
=
m m
z I p l f z p I p l g z
p z
p p q z
zp z p z
p q z
1 1
= p p z p z p q z 1 p p z
Since , applying Lemma
2.1 wit
Re p z > 0
h w z
= 1p z , we can show that p z
z in U, so that
,
, , , ; ,
m p l
f z C .
3. Argument Properties
emma 3.1: Let
L be convex univalent in U and be analytic in U with Re
z
0. If p z
is analytic in U and p
0 =
0 , then)
p z
z zp z
z (z Uimplies that p z
z (zU). Lemma 3.2: Let p be analytic in U with
0 = 1p and p z
= 0 for o points z z, U such thatall z U . If there exist
tw 1 2
1= arg 1 < arg 2 2
2 p p z
< arg =
z p z
2
(3.1) for some 1 and 2 (1,2> 0) and f r all o z
1 2
(z < z = z ).
1 1 1 2 2 2 1 2
1 2
= and =
2 2
z p z z p z
i m i
p z p z
m
(3.2)
where 1
1
b m
b
and
2 1
1 2
= tan 4
b i
.
Theorem 3.3: Let f Ap. 0 < , 1 21. 0 < < p. If
1
1 2
1
, ,
< arg <
2 , , 2
m
m
z I p l f z I p l g z
, 1
, , , ; ,
m p l
g S p AB , then
for some
1< arg <
z I
2
, ,
2 , , 2
m
m
p l f z I p l g z
where
, 2 1
are the solutions for the following equations:
1 1
1
( 2
= tan
2) 1 b c
1
1
1 2 1
os 2 1
2 1 1 sin
1 2
t
p A
b b t
B
and
1 1
1
2 2
1 2 1
2 2
= tan
1
2 1 1 sin
1 2
p A
b b t
B
is given by (3.2), and
2 1 b cos t
( )
b
1 1= 1
t t = cos2 2
1 1
p A B
p AB B
(3.3)
Proof. Let
, , 1
=
, ,
m
m
z I p l f z p z
p I p l g z
.
in with . By
using (1 we obtain
(3.4)Differentiating both sides of the above equation and multiplying the resulting equation by , we find that
Then p z
is analytic U p
0 = 1.9),
1
, ,
= , , , ,
m
m m
p p z I p l g z
p I p l f z I p l f z
z
1
, ,
( ) , ,
= , , , ,
m
m
m m
p p z I p l g z p p z I p l g z
p I p l f z I p l f z
Since
, 1
, , , ; ,
m p l
g z S p A B hat
, by Corollary 2.4, it
follows t
,
, , , ; ,
m p l
g z S p A B .
Next we let
, , 1
=
m
m
z I p l g z q z
g z
, ,
p I p l
Then, using (1.9), we have
, ,
=
, ,
m m
I p l g z
p p q
I p l g z
z
(3.5)
From (3.4) and (3.5), we obtain
1
1 , ,
=
m
p I p l g z
zp z p z
p q z
, ,
1 z Im p l f z
Furthermore, by using a known result, we have
2 21 <
1 1
AB A B
q z
B B
(3.6)
Thus, from (3.6), we obtain
= exp2
i p q z
where, in terms of t1 given by (3.3).
1 < < 1
t t
1 1
< <
1 1
p A p
B B
A
We note that is analytic in with
p U p
0 = 1.Let =h z
lar domain
be function whic aps
angu
the h m U onto the
1 2
: < < with 0
2 arg 2 h
= 1
Applying Lemma 3.1 for this function with
h
z = 1 p q z
we see that Re
p z
> 0 ( z U ), and hence
= 0p z (z U ). By using 1 2
z z U
Lemma 3.2, if there exist two points , such th
we obtain (3.2
at the h
condition (3 ) is ) under t e constraint (3.2). A
.1 satisfied, then
nd we obtain
1 1
1
1
1 1
1 2 1
1 2
1 2 1
1 1
arg
= arg 1 exp
2 2 2
2 1
1
cos 1
2 tan
2 2 cos 1
2
1 cos
2 tan
2 1
2 1
z p z p z
p q z
i
i m
m
b t
p A
B
m
1 2
1 1= 2
1 1 cos
2
b b t
and
1 2 1
2 2 1
2 2 2
2
1 2 1
1 cos
π 2
arg tan =
2 1 2
2 1 1 cos
1 2
b t z p z
p z
p q z p A
b b t
B
which would obviously contradict the assertion of Theorem 3.3. We thus complete the proof of Theorem 3.3.
If we let 1=2 onseque
in Theorem 3.5, we easily obtain the following c nce.
rollary 3.4: Let
Co f Ap. 0 < 1. 0 < < p. If
, ,
arg <
2 , ,
m m
z I p l f z I p l g z
, 1
, , , ; ,
m p l
g S p AB , then for some
, , arg
, ,
< 2
m
m
z I p l f z I p l g z
1 1
1
1 cos
2 2
= tan
1
1 1 cos
1 2
b t
p A
b b t
B
b is given by (3.2), and
1
, ,
< arg <
2
m
m
z I p l f z I p
2
2 , ,l g z
(3.7)
Theorem 3.5: Let
p
f A. 0 < , 1 2 1. 0 < < p. If
1 2
, ,
< arg <
2 , , 2
m
m
z I p l f z I p l g z
for some ,
, , , ; ,
m p l
g S p AB , then
1
1 1
2
, , < arg
2 , ,
< 2
m m
z I p l f z I p l g z
where 1, 2 are the solutions for the following equations:
1 2 1
1
1 1
1 2 1
1 cos
2 2
= tan
1
2 1 1
1 2
b t
p A p l b b
B
cos t
and
1 2 1
1
2 2
1 2 1
1 cos
2 2
= tan
1
2 1 1 cos
b t
p A l b b t
B
is given by (3.2), and
1 p 2
b
1 1
1
2
=
( )
2 = cos
1 1
t t
p A B l
p AB p
B
(3.8) we let
If 1=2 in Theorem 3.5, we easily obtain onsequence.
Corollary 3.6: Let the following c
p
f A
, , arg
m
< 2 , ,
m
z I p l f z I p l g z
for some
,
, , , ; ,
m p l
g S p A B , then
1 , ,
arg <
2 , ,
m m
z I p l f z I p l g z
. 0 < 1. 0 < < p. If where is the solutions for the following equation:
1 1
1
1 cos
2 2
= tan
1
1 1 cos
1 2
b t
p A p l b b
B
t
is given by (1.17), and b
1 1
1
2 =
2 = cos
1 t t
p A B
l
p AB p B
1
(3.9)
4. Acknow
The research was supported by Kyungsung University Research Grants in 2011.
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