Estimating coefficient bounds with respect to a generalized starlike functions on symmetric points

Abstract

The purpose of the present paper is to estimate the Certain Coefficient for generalized Starlike functions with respect to symmetric points defined on the open unit disk for which , () k R   of normalized analytic functions () fz lies in a region with respect to 1 and symmetric with respect to the real axis. 2010 AMS Subject Classification : Primary 30C45.

Full text

International Journal of Innovations in Engineering and Technology (IJIET)
http://dx.doi.org/10.21172/ijiet.91.02
Volume 9 Issue 1 Oct 2017 08 ISSN: 2319-1058
Estimating coefficient bounds with respect to
a generalized starlike functions on symmetric
points
R.Ambrose Prabhu
Department of Mathematics, Saveetha School of Engineering
Saveetha University, Thandalam, Chennai
Tamilnadu, India
M.Elumalai
Department of Mathematics, Saveetha School of Engineering
Saveetha University, Thandalam, Chennai
Tamilnadu, India
S. Chinthamani
Department of Mathematics, Saveetha School of Engineering
Saveetha University, Thandalam, Chennai
Tamilnadu, India
Abstract - The purpose of the present paper is to estimate the Certain Coefficient for generalized Starlike functions
with respect to symmetric points defined on the open unit disk for which
,()
k
R
 
of normalized analytic functions
()fz
lies in a region with respect to 1 and symmetric with respect to the real axis.
2010 AMS Subject Classification : Primary 30C45.
Key words and Phrases: Analytic function, Univalent function, Starlike function, Convex function, Subordination,
Hadamard product, Linear operator, Fekete-szegӧ Inequality.
1. INTRODUCTION
Let
A
denote the class of all analytic function
()fz
of the form
2
() n
n
n
f z z a z




(1.1)
which are analytic in the open unit disk
{ :| | 1}zz  U
and satisfy the condition
   
0 0, ' 0 1.ff
We
also denote by
S
the subclass of
A
consisting of all functions which are univalent in
.U
For functions
()fz
and
()gz
analytic in
U
, we say that the functions
()fz
is said to subordinate to
()gz
if there exist a schwarz
function
( ),z

analytic in
U
with
(0) 0 and ( ) 1 ( ),zz

  U
such that
( ) ( ( )) ( ).f z g z z

U
We denote this subordination by
or ( ) ( ) ( ).f g f z g z zU
In particular, if the function
()gz
is univalent in
U
, the above subordination is equivalent to
(0) (0) and ( ) ( ).f g f gUU
Let
()z

be an analytic function in
U
with
(0) 1


,
(0) 0


and
{ ( )} 0Re z


,
zU
which map the open
unit disk
U
onto a region starlike with respect to 1 and is symmetric with respect to the real axis. Then by
*()S

and
()C

, respectively, we denote the subclasses of the normalized analytic function class
A
, which satisfy the
following subordination relations:
() ( ),
()
zf z zz
fz

U
and

International Journal of Innovations in Engineering and Technology (IJIET)
http://dx.doi.org/10.21172/ijiet.91.02
Volume 9 Issue 1 Oct 2017 09 ISSN: 2319-1058
()
1 ( ), .
()
zf z zz
fz



U
These function were introduced and studied by Ma and Minda\cite {mamin}. In particular case, when
1 (1 2 )
( ) , ,0 1,
1z
zz
z



   
U
these function reduce respectively to the well-known classes
*( ),(0 1)S


of starlike functions of order

in
U
and
( ), (0 1)C


of convex functions of order

in
U
. Ma and Minda [9], the Fekete-Szegӧ
inequality for the functions in the class
()C

was derived and in view of the Alexander result relating the
function classes
*()S

and
()C

.
For a brief history of the Fekete-Szegӧ problems for the starlike, convex and other various subclasses of the
normalized analytic function in
A
, we refer the reader to the work done by Srivatsava et al [20] and
Ramachandran et al [14]. of course the main result shall refer back to Fekete and szegӧ [2] in the year 1933.
After 30 years or so, Keogh and Merkes [4] solved the problem for certain subclasses of univalent functions.
Koepf [6,7], gave excellent results for the class of close-to-convex functions. These articles [2,4,6,7] gave
valuable results which have to solve problems for other extended classes.
Recently Shanmugam et al [17] have studied the Fekete-Szegӧ problem for subclasses of starlike functions with
respect to symmetric points.
Motivated essentially by the aforementioned works, we prove the Fekete-Szegӧ inequality in Theorem 2.1
below for a more general class of normalized analytic functions.
For
,N


,
0
kN
the authors Darus [1] introduced the operator
,
k
D

defined by
,2
( ) [1 ( 1) ] ( , )
kkn
n
n
D f z z n C n a z
 


   

(1.2)
In the present paper, we obtain the Fekete-szegӧ inequality for the function
fA
in the class
,()
k
R
 
defined
as follows:
Definition: 1.1 Let
,:
k
D

AA
is a linear operator and
,
k
D

is analytic in
()fU
. Let
,2
( ) [1 ( 1) ] ( , )
k k n
n
n
D f z z n C n a z
 


   

where
()
( , ) ( ) ( 1)
n
Cn n



  
when
1


,
0


we get the sălăgean differential operator,
0k
or
0


gives Ruscheweyh operator,
0


gives Al-oboudi differential operator of order
k
01
1,0 1,0
( ) ( ), ( ) ( )D f z f z D f z zf z


Definition: 1.2 Let
()z

be a univalent starlike function with respect to 1 which map the unit disk
U
onto a
region in the right half plane which is symmetric with respect to the real axis
(0) 1


and
(0) 1


. A function
fA
is in the class
,()
k
R
 
if
,0
,,
( ) ( ) ( ), ( , , ).
[ ( )] [ ( )]
k
kk
s t z D f z zk
D f sz D f tz

  
 


 
  
In order to prove our main results, we need the following lemma.
Lemma: 1.3 [9] If
2
1 1 2
( ) 1 ...p z cz c z   
is an analytic function with positive real part in
U
, then
2
21
4 2, 0
2, 0 . 1
4 2, 1
v if v
c vc if v
v if v
  


   



∣∣

International Journal of Innovations in Engineering and Technology (IJIET)
http://dx.doi.org/10.21172/ijiet.91.02
Volume 9 Issue 1 Oct 2017 010 ISSN: 2319-1058
when
0v
or
1v
, the equality holds true if and only if
11
() 1z
pz z


or one of its rotations. If
01v
, then
the equality holds true if and only if
2
12
1
() 1z
pz z


or one of its rotation. If
0v
, then the equality holds true if
and only if
11 1 1 1 1 1
( ) (0 1),
2 2 1 2 2 1
zz
pz zz
  

   
     
   

   
or one of its rotations. If
1v
, the equality holds if and only if
1
p
is the reciprocal of one of the functions such
that the equality holds in the case of
0v
. Also the above upper bound is sharp, it can be improved as follows
when
01v
:
22
2 1 1 1
2, 0 2
c vc v c v    ∣ ∣ ∣ ∣
and
22
2 1 1 1
(1 ) 2, 1
2
c vc v c v     ∣ ∣ ∣ ∣
We also need the following result in our investigation.
Lemma: 1.4 [15] If
2
1 1 2
( ) 1 ...z c z c zp    
is a function with positive real part in
U
, then
 
2
21
2. 1, 2 1 .c vc max v  ∣ ∣ ∣ ∣
The result is sharp for the functions
1()pz
given by
2
12
1
() 1z
pz z


and
11
( ) .
1z
pz z


2. FEKETE-SZEGӧ PROBLEM FOR THE FUNCTION OF THE CLASS
,()
k
R
 
.
By making use of Lemma 1.4, we prove the Fekete-szegӧ Problem for the class
,()
k
R
 
.
Theorem: 2.1. Let
2
12
( ) 1z Bz Bz

   
. If
()fz
given by (1.1) belongs to the class
,()
k
R
 
, then
1
2
3 2 1 2
2
,
,,
.
aa

    




   

 

∣∣
where
2
22 1 1
12 2 2
1
2( )( 2) ( )
( 1)( 2)(1 ) ,
2( 2)(1 2 ) [( ) 3]
k
kB B s t B s t
sts st t B



    
   

     
2
221
22 2 2
1
2 ( 2) ( )
( 1)( 2)(1 ) ,
2( 2)(1 2 ) [( ) 3]
k
kB s t s t B
sts st t B



   
   

     
2
22 1 1
32 2 2
1
(2 )( 2) ( )
( 1)( 2)(1 ) ,
2( 2)(1 2 ) [( ) 3]
k
kB B s t B s t
sts st t B



    
   

     
22
2
1
222
21
21 22
4( 1)( 2)(1 2 ) [3 ( )]
2 (1 2 ) ( 2)[( ) 3]
( ) ,
22
( 1)(1 ) (2 )
k
k
k
s st t
B
Bs t s st t
BB st st
  
  

     



     
   


    


122
2.
(1 2 ) ( 1)( 2)(3 [ ])
kBs st t
  
     
Further, If
13
  

, then

International Journal of Innovations in Engineering and Technology (IJIET)
http://dx.doi.org/10.21172/ijiet.91.02
Volume 9 Issue 1 Oct 2017 011 ISSN: 2319-1058
2
2 2 2
3 2 1 2 4 1 2
2 2 2
1
( 1)( 2)(1 ) ( )( 2) .
( 2)(1 2 ) [( ) 3] {}
k
k
st
a a B B s t B a
s st t B

  

   
      
    
∣ ∣ ∣ ∣
If
32
  

, then
2
2 2 2
3 2 2 4 1 2
2 2 2
1
( 1)( 2)(1 ) ( 2) .
( 2)(1 2 ) [( ) 3] {}
k
k
st
a a B s t B a
s st t B

  

   
     
    
∣ ∣ ∣ ∣
Where
2 2 2
42
( )( 2)( 1)(1 ) ( 2)(1 2 ) [( ) 3] .
( 1)( 2)(1 )
kk
k
s t s t s st t
st
    


          

   

The result is sharp.
Proof: If
,()
k
fR
 

, then there exists a Schwarz function
()wz
, analytic in
U
with
(0) 0w
and
( ) 1wz ∣∣
in
U
such that
,
,,
( ) ( ) ( ( ))
[ ( )] [ ( )]
k
kk
s t z D f z wz
D f sz D f tz

  






Define a function
1()pz
by
11 ( )
( ) .
1 ( )
wz
pz wz


since
()wz
is a Schwarz function, we see that
1
{ ( )} 0Re p z 
and
1(0) 0p
. Define a function
()pz
by
,2
12
,,
( ) ( )
( ) ( ( )) 1
[ ( )] [ ( )]
k
kk
s t z D f z
p z wz bz b z
D f sz D f tz

  




     

(2.1)
From (2.1), we obtain
1
2(1 ) ( 1)(2 )
kb
ast

   
(2.2)
and
2 2 2
22
322
( )( 2)(1 ) ( 1)
2[3 ( )](1 2 ) ( 1)( 2)
k
k
b s t s t a
as st t

  
     
     
since
1
11
1 ( ( ))
() 1 ( ( ))
pz
pz pz






then
1
1
( ) 1
( ) ,
( ) 1
pz
pz pz






and
2
212
12 2
12
22
1 2 1
12
1 1 1
()
2 2 2
cz c z
bz bz cz c z
cz c c z




    
  


   





(2.3)
Equating the coefficients of
z
and
2
z
, we obtain
22
1 1 1 2 1 2 1 2 1
1 1 1 1
()
2 2 2 4
b Bc and b B c c Bc   
(2.4)
From (2.2) and (2.4), we get
11
22(1 ) ( 1)(2 )
kBc
ast

   
and
2
12
3 2 1 1
22 1
11
1 ( )
2 2 2
(1 2 ) ( 1)( 2)[3 ( )]
kBB
st
a c c B
B s t
s st t
  




   



      


.
Therefore we have

International Journal of Innovations in Engineering and Technology (IJIET)
http://dx.doi.org/10.21172/ijiet.91.02
Volume 9 Issue 1 Oct 2017 012 ISSN: 2319-1058
22
12
3 2 2 1 1
22 1
22
11
22
2
121
22
11
1 ( )
2 2 2
[3 ( )](1 2 ) ( 1)( 2)
( 1) (1 ) (2 )
{}
[3 ( )](1 2 ) ( 1)( 2)
k
k
k
BB
st
a a c c B
B s t
s st t
Bc st
Bc vc
s st t
  


  




    



      


   

     
∣∣
where
22
2 1 1 22
1
( 2)(1 2 ) [3 ( )]
1
1 ( )
2 2 2 ( 1)(1 ) (2 )
k
k
B B B s st t
st
vB s t st
  


    

   

    

If
1


, then by Lemma 1.3 and Lemma 1.4, we obtain
22
2
32 2
1
222
1
21 22
4( 1)( 2)(1 2 ) [3 ( )]
2 ( 2)(1 2 ) [( ) 3]
()
22
( 1)(1 ) (2 )
k
k
k
s st t
aa B
Bs t s st t
BB st st
  

 

     



     
  


    


∣∣
which is the first part of Theorem 2.1.
Similarly, if
2


, then by Lemma 1.3 and Lemma 1.4, we obtain
22
2
32 2
1
222
1
12 22
4( 1)( 2)(1 2 ) [3 ( )]
2 ( 2)(1 2 ) [( ) 3]
()
22
( 1)(1 ) (2 )
k
k
k
s st t
aa B
Bs t s st t
BB st st
  

 

     



     
  


    


∣∣
If
12
  

, we see that
22
1
3 2 2 1
22
122
2{}
2(1 2 ) ( 1)( 2)(3 [ ])
2
1( 2)(1 2 ) [3 ( )]
k
k
B
a a c vc
s st t
Bs st t
  
  
  
     
     
∣∣
Further, If
13
  

, then
22 1
3 2 1 2 22
2
() 1( 2)(1 2 ) [3 ( )]
k
B
a a a s st t
     
         
∣ ∣ ∣ ∣
Finally,we see that If
32
  

, then
22 1
3 2 2 2 22
2
( ) .
1( 2)(1 2 ) [3 ( )]
k
B
a a a s st t
     
         
∣ ∣ ∣ ∣
To show that the bounds are sharp, we define functions
( 2,3,...)
n
kn


by
,1
,
( ( )) ( ), (0) 0 ( (0)) 1,
()
knnnn
kn
z D k z z k k
D k z

 





   
and the function
F

and
(0 1)G


by
,
,
( ( )) ()
( ), (0) 0 ( (0)) 1
1
()
k
k
z D F z zz FF
z
D F z
  
 



   

and
,
,
( ( )) ()
( ), (0) 0 ( (0)) 1
1
()
k
k
z D G z zz GG
z
D G z
  
 



    

Clearly the functions
,
n
kF

and
,()
k
GR
   

. We also write
2
KK


.
If
1


or
2


, then the equality in Theorem 2.1 holds true if and only if
f
is
K

or one of its rotations.
When
12
  

, then the equality holds true if and only if
f
is
3
K

or one of its rotations. If
1


, then the

International Journal of Innovations in Engineering and Technology (IJIET)
http://dx.doi.org/10.21172/ijiet.91.02
Volume 9 Issue 1 Oct 2017 013 ISSN: 2319-1058
equality holds true if and only if
f
is
F

or one of its rotations. If
2


, then the equality holds true if and
only if
f
is
G

or one of its rotations.
By making use of Lemma 1.4, we can easily obtain the following theorem.
Theorem: 2.2. Let
2
12
( ) 1z Bz Bz

   
, where the coefficients
n
B
are real with
10B
and
20B
. If
()fz
given by (1.1) belongs to
,()
k
R
 
, then
21
32 22
22
2122
1
4
(1 2 ) ( 1)( 2)[3 ( )]
22 ( 2)(1 2 ) [3 ( )]
max 1,1 ,
2( 1)(2 ) (1 )
k
k
k
B
aa s st t
Bs t s st t
B
B s t st
  
 





     



     

   


    




∣∣
The result is Sharp.
Remark: 2.3. The coefficient bounds for
2
a∣∣
and
3
a∣∣
are special cases of those asserted by Theorem 2.1.
Remark: 2.4. In its special case when
1


,
0


and
0k
, we arrive at a known result due to Ma and Minda
[9]
Remark: 2.5. In its special case when
1


,
0


and
0k
,
1s
,
1t
, we arrive at a known result due to
T.N Shanmugam et al [17]
2. REFERENCE
[1] M.Darus and K.Al-shaqsi, “Differential Sandwich theorem with generalized derivative operator”, Proc.World Acad. Sci. Eng.
Tech., 28 (2008), 11-14.
[2] M.Fekete and G.szegӧ, “Eine Bermerkung ber ungerade schlichte Funktionen”, J.Lond.Math.Soc., 8 (1933), 85-89.
[3] A.W.Goodman, “Uniformly Convex Functions”, Ann. Polon. Math., 56 (1991), 87-92.
[4] F.R.Keogh and E.P.Merkes, “A Coefficient inequality for certain classes of analytic functions”, Proc.Amer.Math.Soc., 20 (1969),
8-12.
[5] A.A.Kilbas, H.M.Srivatsava and J.J.Trujillo, “Theory and Applications of Fractional Differential Equations, North Holland
Mathematics Studies”, Elsevier (North-Holland) Science Publishers, Amsterdam, London, New York, 204 (2006).
[6] W.Koepf, “On the Fekete-szegӧ problem for Close-to-convex functions II”, Arch.Math., 49 (1987), 490-533.
[7] W.Koepf, “On the Fekete-szegӧ problem for Close-to-convex functions II”, Proc.Amer.Math.Soc., 101 (1987), 89-95.
[8] S.Owa, “An Application of the Fractional Derivative”, Math. Japan., 29 (1984), 383-389.
[9] W. Ma and D. Minda, “A Unified treatment of some special classes of univalent functions”, in Proc.Conf. on Complex
Analysis(Tianjin,1992), 157--169, Conf. Proc. Lecture Notes Anal., Vol. I, Int. Press, Cambridge,MA .
[10] W. Ma and D. Minda, “Uniformly Convex Functions II”, Ann. Polon. Math., 58 (1993), 275-285.
[11] S.Owa, “On the Distortion Theorems”, I,Kyungpook Math.J., 18 (1978), 53-59.
[12] S.Owa and H.M.Srivatsava, “Univalent and Starlike Generalized Hypergeometric Functions”, Canad. J.Math., 39 (1987), 1057-
1077.
[13] I.Podlubny, “Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations,
Methods of their solutions and some of their Applications”, Mathematics in Science and Engineering, Academic Press, New
York, London and Toronto, 198 (1999).
[14] C.Ramachandran, S.Sivasubramanian, H.M.Srivatsava and A.Swaminathan, “Coefficient inequalities for certain subclasses of
analytic functions and their applications involving the Owa-Srivatsava operator of fractional calculus”, J.Math. Inequal. Appl. 12
(2) (2009), 351-363.
[15] V Ravichandran, Metin Bolcal, Yasar Polotoglu and A. Sen, “Certain Subclasses of Starlike and Convex functions of complex
order”, Hacettepe J. Math. and Stat., 34 (2005), 9-15.
[16] F.Ronning, “Uniformly Convex Functions and a corresponding class of starlike Functions”, Proc. Amer. Math. Soc. 118 (1993),
189-196.
[17] T.N.Shanmugam, C.Ramachandran and V.Ravichandran, “Fekete-szegӧ Problem for Subclass of Starlike Functions With
Respect to Symmetric Points”, Bull. Korean Math. Soc., 43 (3) (2006), 589-598.
[18] H.M.Srivatsava, “Some families of fractional derivative and other linear operators associated with analytic, univalent and
multivalent functions, in Analysis and its applications(Chennai 2000)”, Allied Publ. Ltd., New Delhi, Mumbai, Calcutta and
Chennai, (2001), 209-243.
[19] H.M. Srivatsava and Mishra H.M.Srivatsava and A.K.Mishra, “Applications of Fractional Calculus to Parabolic Starlike and
Uniformly Convex Functions”, Comput. Math Appl. 39 (3-4) (2000), 57-69.
[20] H.M.Srivatsava, A.K.Mishra and M.K.Das, the Fekete-szegӧ problem for a subclass of Close-to-convex functions, Complex
Variables Theory Appl., 44 (2001), 145-163.
[21] H.M.Srivatsava and S.Owa, “Univalent Functions, Fractional Calculus and their Applications”, Halsted Press(Ellis Horwood
Ltd., Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, (1989).