A New General Multivalent Integral Operator
G. Thirupathi Assistant Professor, Department of Mathematics, Ayya Nadar Janaki Ammal College
Sivakasi - 626 124, Tamilnadu, India.
Abstract — In this paper, we define a new multivalent integral operator for certain subclass of analytic functions in the open unit disc
U
. We obtain some interesting properties for this integral operator.Keywords — analytic function, multivalent function, integral operator, starlike and convex functions.
I. INTRODUCTION, DEFINITIONS AND PRELIMINARIES Let
A
p be the class of functionsf z
( )
, of the form
1
,
p n
n n p
f z
z
a z
p
(1.1) which are analytic in the unit discU
z
:
z
1
. And letA
A
1.We denote by
S
*, ,
C K
andC
* the familiar subclasses ofA
consisting of functions which are respectively starlike, convex, close-to-convex and quasi-convex inU
.A function
f z
( )
A
p is said to bep
valently starlike of order
0
p
andz
U
denoted byS
*p( )
, if and only if( )
.
( )
zf z
Re
f z
A function
f z
( )
A
pis said to bep
valently convex of order
0
p
andz
U
denoted byC
p( )
, if and only if( )
1
.
( )
zf
z
Re
f z
It is easy to see that
( )
p( )
( )
p*( ).
zf z
f z
p
C
S
Furthermore,
S
p*(0)
S
p*,
C
p(0)
C
p are respectively, the classes ofp
valently starlike, convex functions inU
. Also, letp
1
, the above classes reduced toS
1*
S
*,
C
1(0)
C
.A function
f z
( )
A
p is said to be in the classk
US
p( , )
ofk
- uniformlyp
valent starlike of order
0
p
inz
U
and satisfies( )
( )
.
( )
( )
zf z
zf z
Re
k
p
f z
f z
Further, a function
f z
( )
A
p is said to be in the classk
UC
p( , )
ofk
- uniformlyp
valent convex( )
( )
1
1
.
( )
( )
zf
z
zf
z
Re
k
p
f z
f z
In particular, when
p
1
, we obtaink UST
( )
andk UCV
( )
, the classes ofk
uniformly starlike andk
uniformly convex functions of order
, 1
1
, respectively which were studied by various authors, example see[9].A function
f z
( )
A
p is said to be in the classS
p*( , )
b
ofp
valently starlike of complex order
0
b b
and type
0
p
, if it satisfies the following inequality
1
( )
,
(
).
( )
zf z
Re p
p
z
b
f z
U
(1.2)A function
f z
( )
A
p is said to be in the classC
p( , )
b
ofp
valently convex of complex order
0
b b
and type
0
p
, if it satisfies the following inequality1
( )
,
(
).
( )
zf
z
Re p
z
b f z
U
(1.3)For
p
1
and
0
, the above classes reduced to the following classes:
*
1
( )
( )
1
1
0,
0
(
)
( )
pzf z
b
Re
b
z
b
f z
U
S
which is defined by Nasr and Aouf [8] and
1
( )
( )
1
,
0
(
) .
( )
p
zf
z
b
Re
b
z
b f z
U
C
defined by Wiatrowski [14].
The Hadamard product of two functions
2
( )
n nn
f z
z
a z
and2
( )
n nn
g z
z
b z
is given by
2
*
( )
n n n.
n
f
g
z
z
a b z
For a function
f
A
p, we define the following operator
0
1
1
0
( )
( )
1
( )
( )
( )
( )
,
.
k k
D f z
f z
D f z
zf z
p
D f z
D D
f z
k
z
U
(1.4)
The differential operator
D
k was introduced by Shenan et al.[12]. Whenp
1
, we get a familiar Salagean derivative [10].By using the above operator, we define the following new classes:
Definition 1.1: A function
f
A
p is said to be in the classS
p k*,( , )
b
ofp
valently starlike of complex orderb b
0
and type
0
p
, if it satisfies the following inequality
( )
1
,
(
).
( )
kk
z D f z
Re p
p
z
b
D f z
Definition 1.2: A function
f
A
p is said to be in the classC
p k,( , )
b
ofp
valently convex of complex orderb b
0
and type
0
p
, if it satisfies the following inequality
( )
1
,
(
).
( )
k
k
z D f z
Re p
z
b
D f z
U
(1.6)Definition 1.3: A function
f
A
p is said to be in the classCV
p k,( , )
and
0
p
, if it satisfies the following inequality
( )
( )
1
1
,
(
)
( )
( )
k k
k k
z D f z
z D f z
Re
p
z
D f z
D f z
U
(1.7)for some
0
and
(0
1)
.The class
CV
p,0( , )
introduced and studied by Yang and Owa [15]. Forp
1,
1
, we have the class( )
UC
considered by Owa [9]. Specializing the values of the parametersp k
, ,
andb
, the above classes *,
( , )
p k
b
S
andC
p k,( , )
b
reduces to the several well-known subclasses, which subclasses are introduced and investigated by various authors (see [4], [14], [10] and [6]).Definition 1.4: Let
n
,
i,
i
0 ,
m
(
m m
1,
2,
,
m
n)
n0,
(
1,
2,
,
n),
1 2
(
,
,
,
n)
and
withRe
( )
0
. Forf g
i,
i
A
p for alli
1,2,3,
,
n
, we introduced anew general integral operator , ,i, i
( , ) :
p n
f g
i i p p
I
A
A
by
1
, 1
, , 1
1 0
( )
( )
( )
.
i i i
i i i
m
z n m
i
p i
p n p p
i
D g t
D f t
z
pt
dt
t
pt
I
(1.8)Remark 1.1: This integral operator , ,i, i
p n
I
generalizes the following several well-known operators introduced and studied by various authors: If
i
0,
m
i
0
and
1
, then this integral operator reduced to the operatorF z
p( )
which was studied by Frasin [5]. If
i
0,
1
and
i
i, then this integral operator reduced to the operator, , , 1 0
1
( )
( )
i i
l n
z p
i
p n l p
i
D f t
F
z
pt
dt
t
(1.9) which was studied by Saltik et al. [11]. For
p
1,
1,
m
i
0
and
i
i, then we obtain the operator
0 1
( )
( )
( )
i
i
n z
i
n i
i
f t
G z
g t
dt
t
(1.10) introduced and studied by Stanciu and Breaz [13]. For
p
1,
1,
m
i
0
and
i
0
, then we obtain the operator
0 1
( )
( )
i
n z
i n
i
f t
F z
dt
t
For
p
1,
1,
m
i
0,
i
i and
i
0
, then we obtain the operator
1
2
1, 2, ,
( )
0 1( )
2( )
( )
n n
z
n
F
z
g t
g t
g t
dt
(1.12) introduced and studied by Breaz et al. [3]. For
p
n
1,
1,
m
i
0,
i
and
i
0
, then we obtain the operator0
( )
( )
zf t
F z
dt
t
(1.13) introduced and studied in [7]. In particular, for
1
, we obtain Alexander integral operator
0
( )
( )
zf t
I z
dt
t
introduced in [1].I. MAIN RESULTS
In this section, we obtain the sufficient condition for the integral operator , ,i, i
( )
p n
z
I
.Theorem 2.1: Let
i,
i be positive real numbers(
i
1,2,3,
, )
n
. Iff
i
S
*p k,( , ) (0
b
1)
andg
i
C
p k,( , ), (
b
i
1,2,3,
, )
n
then the integral operator, , ,
i i
p n
I
defined in (1.8) is in the classC
p( )
, where
2 2
1
(
)
(
)
(
1)
(
1)
.
n
i i i i i
i
Re b
Re b
p
p
p
p
p
b
b
Proof: From (1.8), it is easy to see that
, , 1
1
1
( )
( )
( )
.
i i i
i m
m n
i
p i
p n p p
i
D g z
D f z
z
pz
z
pz
I
(2.1) Differentiating (2.1) logarithmically with respect toz
and multiply byz
, we get
, ,
1 1
, ,
( )
( )
( )
(
1)
(
1) ,
( )
( )
( )
i i
i i
m m
n n
i i
p n
i m i
m
i i
p n i i
z D f z
z D g z
z
z
p
p
p
z
D f z
D g z
I
I
which implies
, ,
1 , ,
1 1 1 1
( )
( )
1
(
1)
1
( )
( )
( )
1
1
.
( )
i i i
i
m n
i p n
i m
i
p n i
m
n n n n
i
i i i i
m
i i i i
i
z D f z
z
z
p
p
p
p
b
z
b
b
D f z
z D g z
p
p
p
p
p
b
D g z
b
I
I
(2.2)
We calculate the real part of (2.2), we get
, ,
1 , ,
1 1 1 1
( )
( )
1
1
1
( )
( )
( )
1
1
.
( )
i i i
i
m n
i p n
i m
i
p n i
m
n n n n
i
i i i i
m
i i i i
i
z D f z
z
z
p
Re p
p
Re
Re p
p
b
z
b
b
D f z
z D g z
p
Re p
Re
p
p
b
D g z
b
I
I
Since
f
i
S
*p k,( , ) (0
b
1)
andg
i
C
p k,( , ), (
b
i
1,2,3,
, )
n
, we obtain
, ,
1 1
, ,
1 1 1
2 2
1
( )
1
1
( )
1
( )
( )
(
1)
(
1)
.
n n
p n
i i i i
i i
p n
n n n
i i i
i i i
n
i i i i i
i
z
z
p
Re p
p
Re
b
z
b
p
Re
p
p
b
Re b
Re b
p
p
p
p
p
b
b
I
I
(2.4)
Hence , ,i, i
( )
p n p
I
C
, where
21
( )
(
1)
n
i i i i i
i
Re b
p
p
p
p
b
2( )
(
p
1)
Re b
b
.Remark 2.1: For the choices of the parameters
,
and
, we get the following results for the various authors: Letting
p
1,
1,
m
i
0
and
i
i, in Theorem 2.1, we obtain Theorem 2.1 in [13]. If
i
0,
1
and
i
i, in Theorem 2.1, we obtain the result in Theorem 2.1 Saltik et al. [11].Theorem 2.2: Let
i,
ibe positive real numbers(
i
1,2,3,
, )
n
. We suppose that the functionsf
i arestarlike functions by order
1
i
, that is* ,
1
1,
i p k
i
f
S
and
g
i
CV
p,i(
i), 0
i
1,
i
1,2,
,
n
.If
1
,
ni i i
i
p
p
n
p
then the integral operator , ,i, ip n
I
defined in (1.8}) is in the classC
( )
, where
1
.
ni i i
i
p
n
p
p
Proof: Using a similar argument in Theorem 2.1, we have
, ,
1 , ,
1
( )
( )
( )
(
1)
(
1)
( )
( )
( )
( )
( )
(
1)
(
1)
( )
( )
i i
i i
i i
i i
m m
n
i i
p n
i m i m
i
p n i i
m m
i i
i m i i
m
i i
i
z D f z
z D g z
z
z
p
p
p
z
D f z
D g z
z D f z
z D g z
p
p
p
D f z
D g z
I
I
.
n
(2.5)
From (2.5), we have
, ,
1 , ,
( )
( )
( )
1
(
1)
( )
( )
( )
i i
i i
m m
n
i i
p n
i m i i
m i
p n i i
z D f z
z D g z
z
z
p
p
p
z
D f z
D g z
I
I
(2.6)
, , 1 1 , , 1 1 1 1 1( )
( )
1
( )
( )
( )
1
( )
( )
( )
( )
i i i i i i i m n n i p ni m i
i i
p n i
m n n i i i m i i i m n n i
i m i
i i i
m n i i m i
z D f z
z
z
Re
p
Re
p
z
D f z
z D g z
Re
p
D g z
z D f z
p
Re
p
D f z
z D g z
Re
D
I
I
11
.
( )
i n i i ip
g z
(2.7)But *,
1
1,
i p k
f
i
S
, so
( )
1
(
)
( )
i i m i m i iz D f z
Re
D f z
and sinceg
i
CV
p,i(
i)
for
i
0
and0
i1,
i
1,2,
,
n
, from (2.7),
, ,
1 1 1
, ,
1 1 1
( )
( )
1
1
( )
( )
( )
1
( )
i i i i mn n n
i p n
i i i i i
m
i i i
p n i
m
n n n
i
i i i i i
m
i i i
i
z D g z
z
z
Re
p
n
p
p
p
z
D g z
z D g z
p
n
p
p
p
D g z
I
I
(2.8)Since
( )
1
0
( )
i i m i i i m iz D g z
p
D g z
, we obtain
, , 1 1 , , 1( )
1
( )
.
n n p ni i i
i i
p n
n
i i i
i
z
z
Re
p
n
p
p
z
p
n
p
p
I
I
(2.9)Using the hypothesis
1n
i i i
i
p
p
n
p
in (2.9), we obtain that the integral operator , ,i, ip n
I
is in the classC
( )
, where
1
.
ni i i
i
p
n
p
p
Taking
n
1
in Theorem 2.2, we obtain the following corollary:Corollary 2.1: Let
,
be positive real numbers. We suppose that the functionsf
is a starlike functions by order1
, that is* ,
1
1,
p kf
p
p
1
p
,
then the integral operator
I
p ,, defined in (1.8}) is in the classC
( )
, where
1
.
p
p
p
Letting
p
1
, for the choices of
and
the above Corollary 2.1 reduce to the following result, which was proved earlier by [13].Corollary 2.2: Let
,
be positive real numbers. We suppose that the functionsf
is a starlike functions by order1
, that is* 1,
1
1,
kf
S
andg
i
CV
1,( )
. If
1
2,
then the integral operator
I
1, , defined in (1.8}) is in the classC
( )
, where
2
1
.
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