Some Properties of the Class of Univalent Functions with Negative Coefficients ()
1. Introduction
Let 
be the class of analytic functions in the open unit disc
and 
be the subclass of 
consisting of functions of the form
Let 
denote the class of functions
normalized by
(1)
which are analytic in the open unit disc. In particular,
For two functions 
given by (1) and 
given by
the Hadamard product (or convolution) 
is defined, as usual, by
Let the function 
be given by:
where 
denotes the Pochhammer symbol (or the shifted factorial) defined by:
Carlson and Shaffer [1] introduced a convolution operator on 
involving an incomplete beta function as:
(2)
Our work here motivated by Catas [2], who introduced an operator on 
as follows:
where
Now, using the Hadamard product (or convolution), the authors (cf. [3,4]) introduced the following linear operator:
Definition 1.1 Let
where
and 
is the Pochhammer symbol. We defines a linear operator 
by the following Hadamard product:
(3)
where
and 
the Pochhammer symbol .
Special cases of this operator include:
• 
see [1].
• the Catas drivative operator [2]: 
• the Ruscheweyh derivative operator [5] in the cases:
• 
• the Salagean derivative operator [6]: 
• the generalized Salagean derivative operator introduced by Al-Oboudi [7]: 
• Note that:
Let 
denote the class of functions 
of the form
(4)
which are analytic in the open unit disc.
Following the earlier investigations by [8] and [9], we define 
-neighborhood of a function 
by
or,
where 
Let 
denote the subclass of 
consisting of functions which satisfy
A function 
in 
is said to be starlike of order 
in
.
A function 
is said to be convex of order 
it it satisfies
We denote by 
the subclass of 
consisting of all such functions [10].
The unification of the classes 
and 
is provided by the class 
of functions 
which also satisfy the following inequality
The class 
was investigated by Altintas [11].
Now, by using 
we will define a new class of starlike functions.
Definition 1.2 Let
A function 
belonging to 
is said to be in the class 
if and only if
(6)
Remark 1.3 The class 
is a generalization of the following subclasses:
i) 
and
defined and studied by [12];
ii) 
and 
studied by [13] and [14];
iii) 
studied by [15];
iv) 
studied by [16].
Now, we shall use the same method by [17] to establish certain coefficient estimates relating to the new introduced class.
2. Coefficient Estimates
Theorem 2.1 Let the function 
be defined by (1). Then 
belongs to the class 
if and only if
(7)
where
(8)
The result is sharp and the extremal functions are
(9)
Proof: Assume that the inequality (7) holds and let
. Then we have
Consequently, by the maximum modulus theorem one obtains
Conversely,suppose that
.
Then from (6) we find that
Choose values of 
on the real axis such that
is real. Letting 
through real values, we obtain
or, equivalently
which gives (7).
Remark 2.2 In the special case 
Theorem 2.1 yields a result given earlier by [17].
Remark 2.3 In the special case 
Theorem 2.2 yields a result given earlier by [6].
Theorem 2.4 Let the function 
defined by (3) be in the class
. Then
(10)
and
(11)
The equality in (10) and (11) is attained for the function 
given by (9).
Proof: By using Theorem 2.2, we find from (6) that
which immediately yields the first assertion (10) of Theorem 2.3.
On the other hand, taking into account the inequality (6), we also have
that is
which, in view of the coefficient inequality (10), can be put in the form
and this completes the proof of (11).
3. Closure Theorem
Theorem 3.1 Let the function 
be defined by
for 
be in the class 
then the function 
defined by
also belongs to the class
, where
Proof: Since 
it follows from Theorem 2.1, that
Therefore,
Hence by Theorem 2.1, 
also.
Morever, we shall use the same method by [17] to prove the distrotion Theorems.
4. Distortion Theorems
Theorem 4.1 Let the function 
defined by (1) be in the class
. Then we have
(12)
and
(13)
for
, where 
and 
is given by (8).
The equalities in (12) and (13) are attained for the function 
given by
(14)
Proof: Note that 
if and only if
, where
By Theorem 2.2, we know that
that is
The assertions of (12) and (13) of Theorem 4.1 follow immediately. Finally, we note that the equalities (12) and (13) are attained for the function 
defined by
This completes the proof of Theorem 4.1.
Remark 4.2 In the special case 
Theorem 4.1 yields a result given earlier by [17].
Corollary 4.3 Let the function 
defined by (1) be in the class
. Then we have
(15)
and
(16)
for
. The equalities in (15) and (16) are attained for the function 
given in (14).
Corollary 4.4 Let the function 
defined by (1) be in the class
. Then we have
(17)
and
(18)
for
. The equalities in (17) and (18) are attained for the function 
given in (14).
Corollary 4.5 Let the function 
defined by (3) be in the class
. Then the unit disc is mapped onto a domain that contains the disc
The result is sharp with the extremal function 
given in (14).
5. Integral Operators
Theorem 5.1 Let the function 
defined by (1) be in the class 
and let 
be a real number such that 
Then 
defined by
also belongs to the class 
Proof: From the representation of 
it is obtained that
where
Therefore
since 
belongs to 
so by virtue of Theorem 2.1, 
is also element of
6. Acknowledgements
The work presented here was partially supported by LRGS/TD/2011/UKM/ICT/03/02.
NOTES