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COEFFICIENT BOUNDS FOR A CERTAIN
SUBCLASS OF CLOSE-TO-CONVEX FUNCTIONS
ASSOCIATED WITH THE KOEBE FUNCTION
Sidik Rathi, Shaharuddin Cik Sohand Ajab Akbarally
Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia
[email protected], [email protected], [email protected]
ABSTRACT
We consider here the functions
2
n n n
f z z a z
which are analytic and univalent in the open unit disc U
z : z 1
, normalized by f
0 0 and f
0 1. By C
,
,we denote a new subclass of close-to-convex function such that
Re ei zf z g z
for
which cos , 0 1, 2
and
21 z g z
z
. In this paper, we give the
representation theorem and obtain the coefficient bounds for functions in C
,
.Keywords: Close-to-convex functions, coefficient bounds, Koebe function, representation
theorem, univalent functions.
1. Introduction
Let A denote the class of functions f of the form
2
( ) n n
n
f z z a z
(1) analytic in the open unit disc U and normalized by f
0 0 and f
0 1. Also, we denote S to be the class of functions in A containing univalent function of the form (1). According to Duren (1983), the function f z
S is called a convex function and starlike function if itsatisfies Re 1 "( ) 0 '( ) zf z f z
and
'( )
Re 0
( ) zf z
f z
respectively. Alexander in 1915
definedf S to be close-to-convex if it satisfy Re '( ) 0 ( ) zf z h z
with h z( )S*(Goodman,
1983). Later, Kaplan (1952) stated that f S is said to be close-to-convex if and only if '( )
Re 0
'( ) f z g z
for z U and gK. Silverman and Silvia (1996) studied the class C of
close-to-convex
functions where functions in this class satisfy
Re ei f z 0 g z
2
, z U , g z
z. Then, with the same g z
, Mohamad (1998) defined G( , ) asthe generalized class of close-to-convexfunctions satisfying Re
e fi
z
, ,z U and cos . Soh and Mohamad (2012) and Akbarally et al. (2011) also studied the class of close-to-convexfunctions but with '( ) 1
1 z g z
z
and
1 '( )
1 g z
z
respectively. Later, Yahya et al. (2012) studied the class GSt( , ) such that
Re ei zf z g z
,
, cos and
1 2 z g zz
.
In this paper, we investigate a similar class of function with gS*. We define
,
C as the class of functions that satisfy
Re ei zf z g z
, 2
, cos ,
0 1 and
21 z g z
z
, where g z
is the well-known Koebe function.2. Representation Theorem
Let P be the class of functions analytic in U and having the form
1
( ) 1 n n
n
p z p z
. (2) Since p z( ) in P satisfiesRe ( )p z 0, for zU, we say thatp z( )is the Caratheodory function. We relate the functions in C
,
to functions in P as given in the following theorem.Theorem 2.1
Let fS be given by (1). Then for z U , f C
,
if and only if
'( ) ( )
sin
i
e zf z g z i A p z
where A cos .
Proof.
Let f C
,
, then
1'
1 n n
n
zf z
b z g z
. Multiplying both sides of the equation with eigives
1
1
'( )
, ( )
cos sin .
i i i n
n n
i n
n n
zf z
e e e b z
g z
i e b z
1
'( )
sin cos
( )
i i n
n n
zf z
e i e b z
g z
and
1
'( ) sin ( )
1 cos
i
n n n
zf z
e i
g z
p z
with
cos
i
n n
e
p b
. Replacing A cos gives
ei
zf z g z'( ) ( )
isin A p z
(3)as required. Conversely, from (3) taking
1
1 n
n n
p z p z
we can work backward to see that the conditions for functions to be in the class C
,
is satisfied.Next, we obtain the representation function for fC
,
as given in the following theorem by using the Herglotz Representation Theorem for functions in P.Theorem 2.2
Let fC
,
. Then for some probability measureon the unit circle X , f can be represented as
21 1 1
( ) 2 1 1 ( )
1 1
i i i
X
f z e e A e d x
xz xz x
for x 1. Conversely, if f is given by the above equation, then fC
,
.Proof.
According to Herglotz formula, for some probability measure on the unit circle X ,
1
( ) ( )
1
X
xz
p P p z d x
xz
.Replacing ( ) 2 (1 )
z g z
z
in (3) and with the aid of Herglotz formula, yields
21
( ) sin 1
1
i
X
e xz
f z A d x i
xz z
1 2 ( 2 ) 2 ( ).(1 ) 1
i
i i
X
e A
f z e e d x
z xz
(4)Integrating (4) with respect to z, gives
2
0
2 1
( ) ( 2 ) ( ) ,
(1 ) 1
z i
i i
X
e A
f z e e d x d
x x
21 1 1
2 1 1 ( ),
1 1
i i i
X
e e e A d x
xz xz x
(5)which is the desired representation function. By this result, we note that the extreme points of
,
C are the unit point masses
21 1 1
( ) 2 1 1
1 1
i i i
x
f z e e e A
xz xz x
with x1 and the derivatives of the extreme points of C
,
are the point masses2
2
1 ( 2 ) 1
' ( )
1 (1 )
i i
x
e e xz
f z
xz xz
, x 1.
3. Coefficient Bound
Using the representation formula established in Theorem 2.2, we now obtain the coefficient bound of functions in C
,
.Theorem 3.1
Suppose that fS is given by (1), then the sharp inequality an 1 A(n1) holds for 2,3, 4,...
n . Equality is attained for each n when f is an extreme point of C
,
.Proof.
From (5) we write
2 3
0
2
( 2 )
( ) ( )
(1 ) (1 )
z i i i
X
e A e e
f z d x d
x x
and since
2 0
1
( 1)( ) (1 )
n
n x
x
and 30
1 1
( 1)( 2)( )
(1 ) 2
n
n n x
x
,0 0 0
1
( ) ( 2 ) ( 1)( ) 2 ( 1)( 2)( ) ( ) 2
z
i i n i n
X
f z e e n x A e n n x d x d
, 1 1 01 ( 2 ) ( 1) ( 1)( 2) ( )
z
i i n n i n n
X
e e n x A e n n x d x d
,
1 0 1 01 ( 2 ) ( 1) ( )
( 1)( 2) ( ) .
z
i i n n
X z
i n n
X
e e n x d x d
A e n n x d x d
Integrating f z( ) with respect to gives
1 11 1
( ) ( 2 ) ( 1) ( ) ( 1)( 2) ( ) ,
1 1
n n
i i n i n
n n
X X
z z
f z z e e n x d x A e n n x d x
n n
1 12 2
( 2 ) ( ) ( 1) ( )
n n
i i n i n
n n
X X
z z
z e e nx d x A e n n x d x
n n
and upon factorization we obtain
12
( ) ( 2 ) ( 1) ( )
n
i i i n
n X
z
f z z e e A e n nx d x
n
. (6)
Now, comparing (6) and (1) gives
1
1
1
( 2 ) ( 1) ( ),
1 2 ( 1) ( ),
1 ( 1) ( ).
i i i n
n
X
i i n
X
i n
X
a e e A e n x d x
Ae A e n x d x
A e n x d x
Then, 1 11 i ( 1) n ( ) 1 i ( 1) n ( ) 1 ( 1)
n
X X
a A e n
x d x A e n
x d x A nReferences
Akbarally, A., Mohamad, D., Soh, S. C. & Kaharudin, N. (2011). On the properties of a new class of -close-to-convex functions. Int. Journal of Math. Analysis, vol 5, no. 8, 387-396.
Duren, P.L. (1983). Univalent Function. Springer Verlag New York Inc.
Goodman, A. W. (1983). Univalent Function. Mariner Publishing Co. Inc. Tampa, University of south Florida.
Kaplan, W. (1952). Close-to-convex schlicht function. Michigan Mathematical Journal, 1. 169-185.
Mohamad, D. (1998). Function whose derivative maps the unit disc into a half plane. PhD Thesis. University of Wales Swansea, UK.
Silverman, H., & Silvia, E. M. (1996). On α-close-to-convex Functions. Publicationes Mathematicae Debrecen, 49, 532-537.
Soh, S. C., & Mohamad. D. (2008). Coefficient bounds for certain classes of close-to-convex functions. International Journal of Mathematical Analysis, Vol. 2, No. 27, 1343-1351.
