1. Introduction
Let A denote the class of function
analytic in the open unit disk
and let S be the subclass of A consisting of functions univalent in U and have the form
(1.1)
The class of convex functions of order
in U, denoted as
is given by
Definition 1.1. The Hadamard product or convolution
of the func- tion
and
, where
is as defined in (1.1) and the function
is given by
is defined as:
(1.2)
Definition 1.2. Let
and
be analytic in the unit disk
. Then
is said to be subordination to
in
and written as:
if there exist a Schwarz function
, analytic in U with
,
such that
(1.3)
In particular, if the function
is univalent in U, then
is said to be subordinate to
if
(1.4)
Definition 1.3. The sequence
of complex numbers is said to be a subordinating factor sequence of the function
if whenever
in the form (1.1) is analytic, univalent and convex in the unit disk
, the subordination is given by
We have the following theorem:
Theorem 1.1. (Wilf [1] ) The sequence
is a subordinating factor sequence if and only if
(1.5)
Definition 1.4. A function
which is normalized by
is said to be in
if
The class
was studied by Janwoski [2] . The family
contains many interesting classes of functions. For example, for
, if
Then
is starlike of order
in U and if
Then
is convex of order
in U.
Let
be the subclass of
consisting of functions
such that
(1.6)
we have the following theorem
Theorem 1.2. [3] Let
be given by Equation (1.6) with
. If
then
,
.
It is natural to consider the class
Remark 1.1. [4] If
, then
consists of starlike functions of order
,
since
Our main focus in this work is to provide a subordination results for functions belonging to the class
2. Main Results
2.1. Theorem
Let
, then
(2.1)
where
and
is convex function.
Proof:
Let
and suppose that
that is
is a convex function of order
.
By definition (1.1) we have
(2.2)
Hence, by Definition 1.3…to show subordination (2.1) is by establishing that
(2.3)
is a subordinating factor sequence with
. By Theorem 1.1, it is sufficient to show that
(2.4)
Now,
Since (
), therefore we obtain
which by Theorem 1.1 shows that
is a subordinating factor, hence, we have established Equation (2.5).
2.2. Theorem
Given
, then
(2.6)
The constant factor
cannot be replaced by a larger one.
Proof:
Let
which is a convex function, Equation (2.1) becomes
Since
(2.7)
This implies
(2.8)
Therefore, we have
which is Equation (2.6).
Now to show that sharpness of the constant factor
We consider the function
(2.9)
Applying Equation (2.1) with
and
, we have
(2.10)
Using the fact that
(2.11)
We now show that the
(2.12)
we have
This implies that
and therefore
Hence, we have that
That is
which shows the Equation (2.12).
2.3. Theorem
Let
,
then
.
Proof:
Let
then by definition of the class
,
we have that
which gives that
hence