A New Sub Class of Univalent Analytic Functions Associated with a Multiplier Linear Operator

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 04 Issue: 07 | July -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 2365
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 ),,(, BAm
k  which is a subclass of uniformly starlikefunctions involving a
multiplier linear operator
m
k , . 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 zazzf
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 nazazzf
Let f and g be analytic in ∆.Then g is said to be subordinate to f, written as
f(z)g(z)or  fg , if there exists a Schwartz function ω, which is analytic in ∆ with ω(0)=0 and )(1)(  zz such
that ).(z))(()(  zfzg  In particular, if the function f is univalent in ∆, we have the following equivalence([3],[7])
).f()g(andf(0)g(0))(z)()( zfzg 
Sharma and Raina ([4],[5],[6]) have introduced a multiplier linear operator
m
k , for
by0,1,  kZm 
(1.3)



































..],.........2,1{,)(
1
)(
,........}2,1{Zm,)(
1
)(
0,m),()(
1
,
1
11
2
,
-
0
1
,
2
11
1
,
,
Zmzfz
dt
d
z
k
zf
dttftz
k
zf
zfzf
m
k
kkm
k
z
m
k
kkm
k
m
k









The series representation of )(, zfm
k  for f(z) of the form (1.1) is given by












2
k, ,
1
)1(
1)((1.4)
n
n
n
m
m
za
nk
zzf


Also
[1]0,)( 0,  mDi m
k
m
k
[8]0,)( 0,1  mDii mm

[10]0,)( 1,1  mDiii mm
[2]0,)( ,1  mIiv mm

Involving the operator
m
k , , we define a class  BAT m
k ,,,  as follows:
Definition 1.1: For fixed A, B, ,,10,10,11  zAAB  a function Tf  is said to be in class
),,(, BAT m
k  if
  .,
1
)]()1{([1
)(
)(
)6.1(
,
,






z
Bz
zBAB
zf
zfz
m
k
m
k 



From the definition, it follows that ),,(, BATf m
k  if there exists a function w(z) analytic in ∆ and satisfies w(0)=0 and
1)( zw for ,z such that
  .,
)(1
)()]()1{([1
)(
)(
,
,







z
zBw
zwBAB
zf
zfz
m
k
m
k 


or equivalently
 
 
.,1
)(
)(
)()1{(
1
)(
)(
)7.1(
,
,
,
,









z
zf
zfz
BBAB
zf
zfz
m
k
m
k
m
k
m
k





The main object of this paper is to obtain necessary and sufficient conditions for the functions ),,()( , BATzf m
k  .
Furthermore we obtain extreme points, distortion bounds, Closure properties, radius of starlikeness and convexity
for ).,,()( , BATzf m
k 
2. COEFFICIENTS INEQUALITY
Theorem2.1: A necessary and sufficient condition for f(z) of the form (1.1) to be in the class ),,(, BAm
k  is that
   



2
, .)()1()(])()1()1)(1[()1.2(
n
n
m
k BAanBABn  
).10,10,11(
1
)1(
1)(, 







 

  AABfor
nk
nwhere
m
m
k
Proof: Assume that the inequality (2.1) holds true and |z|=1.Then we have
      


 )()()1()()()( ,,,, zfBzBABzfzfzfz m
k
m
k
m
k
m
k  
   



































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
zannzBBABzanz
zanzzannz





      









2
,
2
,
2
,
)()()()1()()1(
)()1(
n
n
n
m
k
n
n
n
m
k
n
n
n
m
k
zannBzanBABzBA
zann




   





2
,
2
, )(])()1()1([)()1()()1(
n
n
m
k
n
n
m
k anBAnBBAann  
    .0)()1()(])()1()1)(1[(
2
,  


BAanBABn
n
n
m
k   :Conversely,supposethatthefunction
f(z) defined by(1.1) be in the class ).,,(, BAT m
k 
Then from (2.1), we have
 
 
)(
)(
)()1{(
1
)(
)(
,
,
,
,
zf
zfz
BBAB
zf
zfz
m
k
m
k
m
k
m
k













 
 





)()()}]()1{([
)()(
,,
,,
zfBzzfBAB
zfzfz
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
k
zannzBzanzBAB
zann




Since zz )Re( for all, we have
1
)()}]()1{()1([)}()1{(
)()1(
Re)2.2(
2
,
2
,





















n
n
n
m
k
n
n
n
m
k
zanBAnBzBA
zann




We choose the value of z on the real axis so that
 
)(
)(
,
,
zf
zfz
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
Thus (2.1) proves the theorem.
The result is sharp. The extremal function being
 
 
.2,
)(])()1()1)(1[(
)()1(
)()3.2(
,



 nz
nBABn
BA
zzf n
m
k 


Corollary 2.2: Let the function f(z) defined by(1.1) be in the class ),,(, BAT m
k  .Then
(2.4)
 
.2,
)(])()1()1)(1[(
)}()1{(
,



 n
nBABn
BA
a m
k
n


3. DISTORTION THEOREMS
Theorem3.1: If f (z) ),,(, BAT m
k  , then for
 
Nnz
k
BAB
BA
zf m










 ,
1
1])()1()1[(
)}()1{(
(z))1.3(
2



and
 
.,
])()1()1[(
)}()1{(
)()2.3(
2m
k, Nnz
BAB
BA
zzf 






Proof: In view of inequality (2.1), it follows that
   



2
, .)()1()(])()1()1)(1[(
n
n
m
 
.2,
1
1])()1()1)(1[(
)}()1{(
or(3.3)
2













n
k
BABn
BA
a m
n
n



Therefore
)(
2
2
2






n
n
n
n
n azzzazzf
or
 
2
1
1])()1()1[(
)}()1{(
zf(z)(3.4) z
k
BAB
BA
m













and
or
)(
2
2
2






n
n
n
n
n azzzazzf
 
2
1
1])()1()1[(
)}()1{(
zf(z)(3.5) z
k
BAB
BA
m













From (3.4) and (3.5) inequality (3.1) follows.
Further for f (z) ),,(, BAT m
k  , the inequality (2.1) gives
   



2
, .)()1()(])()1()1[(
n
n
m
k BAanBAB  

 
.2,
])()1()1[(
)}()1{(
)(or(3.6)
2
, 





n
BAB
BA
an
n
n
m
k


 
Thus,






2
,
2
2
,, )()()(
n
n
m
k
n
n
n
m
k
m
k anzzzanzzf  
or
(3.7)
 
2
,
])()1()1[(
)}()1{(
z)( z
BAB
BA
zfm
k






And 





2
,
2
2
,, )()()(
n
n
m
k
n
n
n
m
k
m
k anzzzanzzf  
or
(3.8)
 
2
,
])()1()1[(
)}()1{(
z)( z
BAB
BA
zfm
k






On using (3.7) and (3.8) inequality (3.2) follows.
4. EXTREME POINTS
Theorem 4.1: Let
(4.1) zzf )(1 and
 
n
m
k
n z
nBABn
BA
zzf
)(])()1()1)(1[(
)}()1{(
)(
,




for n  2,then f (z) ),,(, BAT m
k  , if and only if it can be expressed in the form
(4.2) 



1
)()(
n
nn zfdzf , where 0nd and 



1
.1
n
nd
In particular the extreme points of ),,(, BAT m
k  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
nn z
nBABn
BA
zzfdzf






2n
n
nn zbdz
Where
  )(])()1()1)(1[(
)}()1{(
, nBABn
BA
b m
k
n





Now, since
  





2 2
, )}()1{()(])()1()1)(1[(
n
n
n
nn
m
k dBAbdnBABn  
)}.()1{()1)}(()1{( 1 BAdBA  
Therefore, f (z) ),,(, BAT m
k  .
Conversely, let f (z) ),,(, BAT m
k  , then (2.1) yields
 
n
m
k
n z
nBABn
BA
a
)(])()1()1)(1[(
)}()1{(
,



 for .2n

Setting
 
n
m
k
n a
BA
nBABn
d
)}()1{(
)(])()1()1)(1[( ,




 
for 2n
And 



2
1 1
n
ndd .
Then
 

 


2 , )(])()1()1)(1[(
)}()1{(
)(
n
n
nm
k
zd
nBABn
BA
zzf






2
)}({
n
nn zfzdz
  







2 2 1
).()()1(
n n n
nnnnn zfdzfddz
This completes the proof.
5. RADIUS OF STARLIKENESS
THEOREM 5.1: Let f (z) , then f(z) is starlike in whereBArz ),,,(
(5.1)
 
.2,
)}()1{(
)(])()1()1)(1[(
inf
1
1
,













n
BAn
nBABn
r
nm
k

 
Proof: It suffices to show that
11
)(
)(


zf
zfz
i.e. 1
1
)1(
1
)(
)(
2
1
1
2













n
n
n
n
n
n
za
zan
zf
zfz
(5.2) or 




2
1
.1
n
n
n zna
It is easily to see that (5.1) holds if
 
.
)}()1{(
)(])()1()1)(1[( ,1












BAn
nBABn
z
m
kn

 
This completes the proof.
6. RADIUS OF CONVEXITY
THEOREM 6.1: Let f (z) , then f(z) is convex in whereBArz ),,,(
(6.1)
 
.2,
)}()1{(
)(])()1()1)(1[(
inf
1
1
2
,













n
BAn
nBABn
r
nm
k

 
Proof: Upon noting the fact that f(z) is convex if and only if )(zfz  is starlike, the Theorem(6.1) follows.

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 subordinationresultsinvolvinga classofoperators”.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).
[9] Silverman, H., “Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51, (1975), 109-116.
[10]. Uralegaddi, B.A. and Somantha, C., “ Certain classes of univalent functions in H.M. Srivastava, S. Owa (Eds.) Currenttopics
in analytic function theory, World Scientific Publishing company, Singapore, (1992), 371-374.

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