A coefficient inequality for the starlike univalent functions in the unit disc with two fixed points of complex order

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
99
A Coefficient Inequality for the Starlike Univalent Functions in
the Unit Disc with Two Fixed Points of Complex Order
Dr. R. K. Sharma(Assistant Professor)
Dept. of Mathematics, BUIT, Barkatullah University Bhopal
Email:rksharma178@rediffmail.com
Abstract
The aim of this paper is to obtained a coefficient inequality for the class of starlike function S*
(A, B, b) in the
unit disc U= {z: │z│< 1}, where A and B are two fixed points and b is non zero complex number.
1. Introduction
Let An denote the class of functions which is the following form
1
( ) n
n
n
w z b z
∞
=
= ∑ , { }{1,2,3.....}n N∈ = (1.1)
which are analytic in the open unit disc U={ z: │z│< 1} and satisfying the condition w(0)=0 and │w(z)│< 1.
Further, let Sn denote the subclass of functions in An which are univalent in U. Also Sα
*
(n) represents the
subclasses of An which is starlike functions of order α (0 ≤ α < 1). The starlike function is defined as follows
'( )
( ) : ,0 1,
( )
n
zf z
S n f A R z U
f z
α α α∗   
= ∈ > ≤ < ∈   
  
(1.2)
The present paper is devoted to a unified study of various subclasses of univalent functions. For this purpose, we
introduce the new class of analytic functions S*
(A, B, b)
of the form
2
( ) n
n
n
f z z a z
∞
=
= + ∑ , (1.3)
analytic in the open unit disc U and satisfying the conditions
'( ) 1
1
( ) { '( ) 1}
f z
b A B B f z
−
<
− − −
(1.4)
and
{ }
1 1 ( )
1 '( ) 1
1 ( )
Aw z
f z
b Bw z
+
+ − =
+
(1.5)
where A and B are arbitrary fixed numbers such that -1 ≤ B < A ≤ 1 and b is non-zero complex number.
2. Preliminaries Lemmas:
Before stating and proving our main results, we need the following lemmas due to Keogh and Merkes [1] and
Silverman [3].
Lemma 1: Let the function w(z) defined by
1
( ) n
n
n
w z b z
∞
=
= ∑ ,
be analytic with w(0)=0 and │w(z)│< 1 in U. If s is any complex number then
2
2 1 max(1, )b sb s− ≤ (1.6)
Lemma 2: Let the function f(z) defined by
2
( ) n
n
n
f z z a z
∞
=
= + ∑ ,
then,

100
1
1
1
k
k n
k
a
α
α
∞
= +
− 
≤ 
− 
∑ (1.7)
( ) ( )nf z S α∗
⇒ ∈ { }: : 0 1z U n N α∈ ∈ ≤ <
3. Main Results: (Coefficient inequalities)
Theorem 1: If the function f (z) defined by
2
( ) n
n
n
f z z a z
∞
=
= + ∑ ,
belongs to the class S*
(A, B, b). If δ is any complex number, then
2
3 2
( )
max{1, },
( ,3)
A B b
a a d
µ
δ
α λ
−
− ≤ (1.8)
where
2
2
{ ( , 2)} ( ) { ( , 3)}
{ ( , 2)}
B A B b
d
α λ µ δ α λ
α λ
+ −
=
The inequality (1.8) is sharp.
Proof: Since the function f (z) belongs to the class S*
(A, B, b), we have
1
1 ( ) 1 ( )
1 1 (1 )
1 ( )
D f z Aw z
b z Bw z
λ
µ µ
+
   +
+ − = − +   
+  
, (1.9)
where
1
( ) n
n
n
w z b z
∞
=
= ∑ , is analytic in U and satisfies the conditions
w (0) = 0, ( ) 1w z < .
From (1.9), we have
{ }1
1
( ) / 1
( )
( )
( ) 1
D f z z
w z
D f z
A B b B
z
λ
λ
µ
+
+
−
=
 
− − − 
 
=
1
2
1
2
( , )
( ) ( , )
n
n
n
n
n
n
n a z
A B b B n a z
α λ
µ α λ
∞
−
=
∞
−
=
− −
∑
∑
=
2 2 2
2 2
2 3
{ ( ,2)}1
( ,2) ( ,3) ...
( ) ( )
B a z
a z a z
A B b A B b
α λ
α λ α λ
µ µ
 
+ + + 
− − 
and then comparing the coefficients of z and z2
on both sides, we have
and
2
1
2 2
3 2
2 2 2 2
( ,2)
( )
( ,3) { ( ,2)}
( ) ( )
a
b
A B b
a B a
b
A B b A B b
α λ
µ
α λ α λ
µ µ
=
−
= +
− −
Thus
1
2
( )
( ,2)
A B bb
a
µ
α λ
−
=
and

101
2
2 1
3
( ) ( )
( ,3)
A B b b Bb
a
µ
α λ
− −
=
Hence
2 2
3 2 2 1
( )
( ),
( ,3)
A B b
a a b db
µ
δ
α λ
−
− = −
where
2
2
{ ( ,2)} ( ) { ( ,3)}
{ ( ,2)}
B A B b
d
α λ µ δ α λ
α λ
+ −
=
therefore
2 2
3 2 2 1
( )
( ,3)
A B b
a a b db
µ
δ
α λ
−
− = −
Using Lemma 1 in the above equation, we get
2
3 2
( )
max{1, },
( ,3)
A B b
a a d
µ
δ
α λ
−
− ≤ .
Hence the result.
Theorem 2: If the function f (z) defined by
2
( ) n
n
n
f z z a z
∞
=
= + ∑ ,
belongs to the class S*
(A, B, b), then
( )
n
A B b
a
n
−
≤ (2.0)
The estimates are sharp.
Proof: Since the function f(z) belongs to the class S*
(A, B, b), we have
{ }
1 1 ( )
1 '( ) 1
1 ( )
Aw z
f z
b Bw z
+
+ − =
+
(2.1)
where the function w(z) defined by (1.1) is regular in U and satisfies the conditions w(0)=0,│w(z)│< 1 , for z
belongs to U. Now, from (2.1) we have
1 1
2 2
1
( ) ( / ) ( )n n
n n
n n
B A B b na z w z na z
b
∞ ∞
− −
= =
 
− + = − 
 
∑ ∑
i.e.
1 1
2 1 2
1
( ) ( / ) n n n
n n n
n n n
A B B b na z b z na z
b
∞ ∞ ∞
− −
= = =
   
− − =   
   
∑ ∑ ∑ (2.2)
now equating the corresponding coefficients in (2.2), we observe that the coefficient an on the RHS of (2.2)
depends only on a2, a3,…..,an-1 on the LHS of (2.2). Hence for n ≥ 2, it follows from (2.2) that
1
1 1 1
2 2 1
1
( ) ( / ) ( )
k k
n n n
n n n
n n n k
A B B b na z w z na z d z
b
− ∞
− − −
= = = +
 
− − = + 
 
∑ ∑ ∑
This yields
1
1 1 1
2 2 1
1
( ) ( / )
k k
n n n
n n n
n n n k
A B B b na z na z d z
b
− ∞
− − −
= = = +
 
− − = + 
 
∑ ∑ ∑ (2.3)
Now squaring both sides of (2.3) and integrating round │z│= r , 0 < r <1, we obtain
2 1
22 2 2 2
2
2
( )
k
n
n
n
B
A B n a r
b
−
−
=
− + ∑
1
2 22 2 2 2 2
2
2 1
1 k
n n
n n
n n k
n a r d r
b
− ∞
− −
= = +
≥ +∑ ∑
by assuming r → 1, we get

102
2 1
2 22 2 2
22
2 2
1
( )
k k
n n
n n
B
A B n a n a
b b
−
= =
− + ≥∑ ∑
or
1
2 2 22 2 2 2
2
(1 ) ( )
k
n n
n
B n a n a A B b
−
=
− + ≤ −∑ (2.4)
Since 1 1,B− ≤ < , we obtain from (2.4)
2 22 2
( )nn a A B b≤ −
This gives
( )
n
A B b
a
n
−
≤
This shows the result is sharp.
References:
1. F. R. Keogh and E. P. Merkes: A coefficient inequality for certain classes of analytic functions Proc.
Amer. Math. Soc. 20, 8-14 (1969).
2. I. S.Jack: Functions Starlike and convex of order α, J. London Math. Soc. 3, 469-474 (1971).
3. H.Silverman: Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51, 109-116
(1975).
4. P. Singh and M. Tygel: On some univalent functions in the unit disc, Indian Jour. pure app. Math. 12(4)
(1981), 513-520 (1981).
5. V. Singh: On some criteria for univalence and starlikeness Ind.Jour. of Pure and App. Math. 34(4), 569-
577 (2003).
6. Y. Polatoglu and M. Bolcal: A coefficient inequality for the class of analytic functions in the unit disc,
I.J.M.M.S. Vol. 59, 3753–3759 (2003).
7. R.K. Sharma and D. Singh: "An Inequality of Subclasses of Univalent Functions Related to Complex
Order by Convolution Method", General Mathematics Notes Vol. 3, No. 2, (2011).

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