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A NEW SUB CLASS OF UNIVALENT ANALYTIC FUNCTIONS ASSOCIATED
WITH A MULTIPLIER LINEAR OPERATOR
DR. JITENDRA AWASTHI
DEPARTMENT OF MATHEMATICS
,
S.J.N.P.G.COLLEGE, LUCKNOW-226001
---***---ABSTRACT: This paper deals with a new class
km,(
,
A
,
B
)
which is a subclass of uniformly starlikefunctions involving a multiplier linear operator mk,
. Coefficients inequality, Distortion theorem, Extreme points, Radius of starlikeness and radius of convexity for functions belonging to this class are obtained.Key words and phrases: Univalent, starlike, subordination, convex, analytic, linear operator. 2010 Mathematics Subject Classification: 30C45. 1. INTRODUCTION
Let S denote the class of functions of the form
2
,
)
(
)
1
.
1
(
n n n
z
a
z
z
f
which are analytic and univalent in the unit open disk ∆={z: |z|<1}.
Silverman[9] had introduced and studied a subclass T of S consisting of functions of the form
2
)
2
,
0
(
,
)
(
)
2
.
1
(
n
n n
n
z
a
n
a
z
z
f
Let f and g be analytic in ∆.Then g is said to be subordinate to f, written as
f(z)
g(z)
or
f
g
, if there exists a Schwartz function ω, which is analytic in ∆ with ω(0)=0 and
(
z
)
1
(
z
)
such thatg
(
z
)
f
(
(
z
))
(z
).
In particular, if the function f is univalent in ∆, we have the following equivalence([3],[7])).
f(
)
g(
and
f(0)
g(0)
)
(z
)
(
)
(
z
f
z
g
Sharma and Raina ([4],[5],[6]) have introduced a multiplier linear operator m k,
forby
0
,
1
,
Z
k
m
(1.3)
..]
,...
2
,
1
{
,
)
(
1
)
(
,...}
2
,
1
{
Z
m
,
)
(
1
)
(
0,
m
),
(
)
(
1 , 1 1 1
2 ,
-0
1 , 2 1 1 1 ,
,
Z
m
z
f
z
dt
d
z
k
z
f
dt
t
f
t
z
k
z
f
z
f
z
f
m k k k
m k
z
m k k k m
k m k
The series representation of
mk,f
(
z
)
for f(z) of the form (1.1) is given by
2
k,
,
1
)
1
(
1
)
(
(1.4)
n
n n m m
z
a
n
k
z
z
f
Also
[1]
0
,
)
(
i
mk,0
D
kmm
[8]
0
,
)
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| Page 2366
[10]
0
,
)
(
iii
1m,1
D
mm
[2]
0
,
)
(
iv
1m,
I
mm
Involving the operator m k,
, we define a classT
km,
,
A
,
B
as follows:Definition 1.1: For fixed A, B,
1
B
A
1
,
0
A
1
,
0
1
,
z
,
a functionf
T
is said to be in class)
,
,
(
,
A
B
T
km
if
.
,
1
)]
(
)
1
{(
[
1
)
(
)
(
)
6
.
1
(
,
,
z
Bz
z
B
A
B
z
f
z
f
z
m k m
k
From the definition, it follows that
f
T
km,(
,
A
,
B
)
if there exists a function w(z) analytic in ∆ and satisfies w(0)=0 and1
)
(
z
w
forz
,
such that
.
,
)
(
1
)
(
)]
(
)
1
{(
[
1
)
(
)
(
,
,
z
z
Bw
z
w
B
A
B
z
f
z
f
z
m k m
k
or equivalently
1
,
.
)
(
)
(
)
(
)
1
{(
1
)
(
)
(
)
7
.
1
(
, , ,
,
z
z
f
z
f
z
B
B
A
B
z
f
z
f
z
m k m k m
k m k
The main object of this paper is to obtain necessary and sufficient conditions for the functions
f
(
z
)
T
km,(
,
A
,
B
)
. Furthermore we obtain extreme points, distortion bounds, Closure properties, radius of starlikeness and convexity forf
(
z
)
T
km,(
,
A
,
B
).
2. COEFFICIENTS INEQUALITY
Theorem2.1: A necessary and sufficient condition for f(z) of the form (1.1) to be in the class
km,(
,
A
,
B
)
is that
2
,
(
)
(
1
)
(
)
.
]
)
(
)
1
(
)
1
)(
1
[(
)
1
.
2
(
n
n m
k
n
a
A
B
B
A
B
n
).
1
0
,
1
0
,
1
1
(
1
)
1
(
1
)
(
,
n
k
n
for
B
A
A
where
m m
k
Proof: Assume that the inequality (2.1) holds true and |z|=1.Then we have
,f
(
z
)
,f
(
z
)
,f
(
z
)
B
(
1
)
(
A
B
)
Bz
,f
(
z
)
z
mk mk mk
mk
2 , 2
,
2 , 2
,
)
(
)
(
)
1
(
)
(
)
(
)
(
n
n n m k n
n n m k
n
n n m k n
n n m k
z
a
n
n
z
B
B
A
B
z
a
n
z
z
a
n
z
z
a
n
n
z
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| Page 2367
2 , 2 , 2 ,)
(
)
(
)
(
)
1
(
)
(
)
1
(
)
(
)
1
(
n n n m k n n n m k n n n m kz
a
n
n
B
z
a
n
B
A
B
z
B
A
z
a
n
n
2 , 2,
(
)
(
1
)
(
)
[
(
1
)
(
1
)
(
)
]
(
)
)
1
(
n n m k n n mk
n
a
A
B
B
n
A
B
n
a
n
(
1
)
(
)
]
(
)
(
1
)
(
)
0
.
)
1
)(
1
[(
2 ,
B
A
a
n
B
A
B
n
n n mk
: Conversely, suppose that the functionf(z) defined by(1.1) be in the class
T
km,(
,
A
,
B
).
Then from (2.1), we have
)
(
)
(
)
(
)
1
{(
1
)
(
)
(
, , , ,z
f
z
f
z
B
B
A
B
z
f
z
f
z
m k m k m k m k
)
(
)
(
)}]
(
)
1
{(
[
)
(
)
(
, , , ,z
f
Bz
z
f
B
A
B
z
f
z
f
z
m k m k m k m k
.
1
)
(
)
(
)}]
(
)
1
{(
[
)
(
)
1
(
2 , 2 , 2 ,
n n n m k n n n m k n n n m kz
a
n
n
z
B
z
a
n
z
B
A
B
z
a
n
n
Since
Re(
z
)
z
for all, we have1
)
(
)}]
(
)
1
{(
)
1
(
[
)}
(
)
1
{(
)
(
)
1
(
Re
)
2
.
2
(
2 , 2 ,
n n n m k n n n m kz
a
n
B
A
n
B
z
B
A
z
a
n
n
We choose the value of z on the real axis so that
)
(
)
(
, ,z
f
z
f
z
m k m k
is real. Upon clearingthe denominator of (2. 2) and letting
z
1
, we can write (2.2) as
2,
(
)
(
1
)
(
)
..
]
)
(
)
1
(
)
1
)(
1
[(
n n m
k
n
a
A
B
B
A
B
n
Thus (2.1) proves the theorem.
The result is sharp. The extremal function being
(
1
)
(
)
]
(
)
,
2
.
)
1
)(
1
[(
)
(
)
1
(
)
(
)
3
.
2
(
,
z
n
n
B
A
B
n
B
A
z
z
f
m n© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal
| Page 2368
Corollary 2.2: Let the function f(z) defined by(1.1) be in the class
T
km,(
,
A
,
B
)
.Then (2.4)
(
1
)
(
)
]
(
)
,
2
.
)
1
)(
1
[(
)}
(
)
1
{(
,
n
n
B
A
B
n
B
A
a
mk n
3. DISTORTION THEOREMS
Theorem3.1: If f (z)
T
km,(
,
A
,
B
)
, then for
N
n
z
k
B
A
B
B
A
z
f
m
,
1
1
]
)
(
)
1
(
)
1
[(
)}
(
)
1
{(
(z)
)
1
.
3
(
2
and
(
1
)
(
)
]
,
.
)
1
[(
)}
(
)
1
{(
)
(
)
2
.
3
(
mk,z
2n
N
B
A
B
B
A
z
z
f
Proof: In view of inequality (2.1), it follows that
2
,
(
)
(
1
)
(
)
.
]
)
(
)
1
(
)
1
)(
1
[(
n
n m
k
n
a
A
B
B
A
B
n
.
2
,
1
1
]
)
(
)
1
(
)
1
)(
1
[(
)}
(
)
1
{(
or
(3.3)
2
n
k
B
A
B
n
B
A
a
mn n
Therefore
)
(
2 2
2
n n n
n
n
z
z
z
a
a
z
z
f
or
2
1
1
]
)
(
)
1
(
)
1
[(
)}
(
)
1
{(
z
f(z)
(3.4)
z
k
B
A
B
B
A
m
and
or
)
(
2 2
2
n n n
n
n
z
z
z
a
a
z
z
f
2
1
1
]
)
(
)
1
(
)
1
[(
)}
(
)
1
{(
z
f(z)
(3.5)
z
k
B
A
B
B
A
m
From (3.4) and (3.5) inequality (3.1) follows.
Further for f (z)
T
km,(
,
A
,
B
)
, the inequality (2.1) gives
2
,
(
)
(
1
)
(
)
.
]
)
(
)
1
(
)
1
[(
n
n m
k
n
a
A
B
B
A
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| Page 2369
,
2
.
]
)
(
)
1
(
)
1
[(
)}
(
)
1
{(
)
(
or
(3.6)
2
,
n
B
A
B
B
A
a
n
n
n m
k
Thus,
2 , 2
2 ,
,
(
)
(
)
(
)
n
n m k n
n n m k m
k
f
z
z
n
a
z
z
z
n
a
or (3.7)
2,
]
)
(
)
1
(
)
1
[(
)}
(
)
1
{(
z
)
(
z
B
A
B
B
A
z
f
m
k
And
2 , 2
2 ,
,
(
)
(
)
(
)
n
n m k n
n n m k m
k
f
z
z
n
a
z
z
z
n
a
or
(3.8) ,
2]
)
(
)
1
(
)
1
[(
)}
(
)
1
{(
z
)
(
z
B
A
B
B
A
z
f
m
k
On using (3.7) and (3.8) inequality (3.2) follows. 4. EXTREME POINTS
Theorem 4.1: Let
(4.1)
f
1(
z
)
z
and
n m k
n
z
n
B
A
B
n
B
A
z
z
f
)
(
]
)
(
)
1
(
)
1
)(
1
[(
)}
(
)
1
{(
)
(
,
for n
2,then f (z)T
km,(
,
A
,
B
)
, if and only if it can be expressed in the form(4.2)
1
)
(
)
(
n n n
f
z
d
z
f
, whered
n
0
and
1
.
1
n n
d
In particular the extreme points of
T
km,(
,
A
,
B
)
are the functions given by (4.1). Proof: Let f(z) be expressed in the form (4.1),then
1 2
[(
1
)(
1
)
(
1
)
(
)
]
,(
)
)}
(
)
1
{(
)
(
)
(
n n
n m k n
n
z
n
B
A
B
n
B
A
z
z
f
d
z
f
2
n
n n n
b
z
d
z
Where
(
1
)
(
)
]
(
)
)
1
)(
1
[(
)}
(
)
1
{(
,
n
B
A
B
n
B
A
b
mk n
Now, since
2 2
,
(
)
{(
1
)
(
)}
]
)
(
)
1
(
)
1
)(
1
[(
n
n n
n n m
k
n
d
b
A
B
d
B
A
B
n
{(
1
)
(
A
B
)}(
1
d
1)
{(
1
)
(
A
B
)}.
Therefore, f (z)T
km,(
,
A
,
B
)
.Conversely, let f (z)
T
km,(
,
A
,
B
)
, then (2.1) yields
m nk
n
z
n
B
A
B
n
B
A
a
)
(
]
)
(
)
1
(
)
1
)(
1
[(
)}
(
)
1
{(
,
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| Page 2370
Setting
nm k
n
a
B
A
n
B
A
B
n
d
)}
(
)
1
{(
)
(
]
)
(
)
1
(
)
1
)(
1
[(
,
for
n
2
And
2 1
1
n n
d
d
.Then
2
[(
1
)(
1
)
(
1
)
(
)
]
,(
)
)}
(
)
1
{(
)
(
n
n n m k
z
d
n
B
A
B
n
B
A
z
z
f
2
)}
(
{
n
n n
z
f
z
d
z
2 2 1
).
(
)
(
)
1
(
n n n
n n n
n
n
d
f
z
d
f
z
d
z
This completes the proof. 5. RADIUS OF STARLIKENESS
THEOREM 5.1: Let f (z) , then f(z) is starlike in
z
r
(
,
A
,
B
),
where
(5.1)
,
2
.
)}
(
)
1
{(
)
(
]
)
(
)
1
(
)
1
)(
1
[(
inf
1 1
,
n
B
A
n
n
B
A
B
n
r
n m k
Proof: It suffices to show that
1
1
)
(
)
(
z
f
z
f
z
i.e.
1
1
)
1
(
1
)
(
)
(
2
1 1
2
n
n n
n
n
n
z
a
z
a
n
z
f
z
f
z
(5.2) or
2
1
.
1
n
n n
z
na
It is easily to see that (5.1) holds if
.
)}
(
)
1
{(
)
(
]
)
(
)
1
(
)
1
)(
1
[(
,1
B
A
n
n
B
A
B
n
z
m k n
This completes the proof. 6. RADIUS OF CONVEXITY
THEOREM 6.1: Let f (z) , then f(z) is convex in
z
r
(
,
A
,
B
),
where
(6.1)
,
2
.
)}
(
)
1
{(
)
(
]
)
(
)
1
(
)
1
)(
1
[(
inf
1 1
2
,
n
B
A
n
n
B
A
B
n
r
n m k
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| Page 2371
REFERANCES
[1]. Al-Oboudi, F.M., “On univalent functions defined by a generalized Salageon operator”. Indian J. Math, Math. Sci. 25-28(2004), 1429-1436.
[2]. Cho, N.E. and Kim, T.H., “Multiplier transformations and strongly close-to-convex functions”, Bull. Korean Math. Soc. 40(2003), 399-410.
[3].Owa,S. and Srivastava,H.M., “Univalent and starlike generalized hypergeometric functions”. Canad.J.Math.,39(5)(1987),1057-1077.
[4].Raina, R.K. and Sharma, P., “Subordination properties of univalent functions involving a new class of operators”,Electr. J. Math. Anal. App. 2(1), (2014),37-52.
[5].Raina, R.K. and Sharma, P., “Subordination preserving properties associated with a class of operators”, Le Mathematiche 68(1),(2013),217-228.
[6].Raina, R.K., Sharma, P. and Sokol,J. “Certain subordination results involving a class of operators”. Analele Univ. Oradea Fasc. Matematica 21(2),(2014),89-99.
[7]. RUSCHEWEYH S., “Neighborhoods of Univalent Functions”, Proc. Amer. Math., Soc., 81(1981), 521-527.
[8]. Salageon, G.S., “Subclasses of univalent function in complex analysis” v Romanian-Finnish seminar, Part- I (Bucharest, 1981), Lecture notes in Mathematics 1013, Springer Verlag, Berlin and New York (1983).
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