In this paper, we investigate a subordination property and the coefficient inequality for the class M(1, b), The lower bound is also provided for the real part of functions belonging to the class M(1, b).
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
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:
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
In particular, if the function is univalent in U, then is said to be subordinate to if
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  ) The sequence is a subordinating factor sequence if and only if
Definition 1.4. A function which is normalized by is said to be in if
The class was studied by Janwoski  . 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
we have the following theorem
Theorem 1.2.  Let be given by Equation (1.6) with . If
then , .
It is natural to consider the class
Remark 1.1.  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
Let , then
where and is convex function.
and suppose that
that is is a convex function of order .
By definition (1.1) we have
Hence, by Definition 1.3…to show subordination (2.1) is by establishing that
is a subordinating factor sequence with . By Theorem 1.1, it is sufficient to show that
Since ( ), therefore we obtain
which by Theorem 1.1 shows that is a subordinating factor, hence, we have established Equation (2.5).
Given , then
The constant factor cannot be replaced by a larger one.
which is a convex function, Equation (2.1) becomes
Therefore, we have
which is Equation (2.6).
Now to show that sharpness of the constant factor
We consider the function
Applying Equation (2.1) with and , we have
Using the fact that
We now show that the
This implies that
Hence, we have that
which shows the Equation (2.12).
then by definition of the class ,
we have that
which gives that