1. Introduction and Definitions

Let denote the class of functions of the form

(1.1)

Which are analytic in the unit disk

Shu-Hai and Tang in 2010[1], introduced the classes and gave the following definition:

Definition 1. Let denote the subclass of consisting of functions which satisfy the following inequality:

(1.2)

Definition 2. Let be the subclass of consisting of function which satisfy the following condition:

(1.3)

Where is the Salagean derivative operator defined as:

For it is easy to see that

We denote by k the class of convex functions i.e.

Definition 3. (Hadamard product or convolution) Given two functions and where is as defined in (1.1) and is given by

(1.4)

The Hadamard product (or convolution) of and is defined by

(1.5)

Definition 4. (Subordination principle)

Let and be analytic in the unit disk . Then is said to be subordinate to in and we write , if there exists a Schwarz function , analytic in with such that

(1.6)

In particular, if the function is univalent in , then is subordinate to if

(1.7)

Definition 5. (Subordinating factor sequence)

A sequence of complex numbers is said to be a subordinating factor sequence if whenever of the form (1.1) is analytic, univalent and convex in , the subordination is given by

We have the following theorem

Theorem 1.1 (Wilf [3])

The sequence is a subordinating factor sequence if and only if

(1.8)

Theorem 1.2. [1]

(1.9)

(1.10)

Theorem 1.3. [1]

(1.11)

(1.12)

In view of Theorem 1.2 and Theorem 1.3, we now introduce the subclasses and which consist of functions whose Taylor-Maclaurin coefficients satisfy the inequalities (1.9) and (1.11) respectively.

In our proposed investigation of functions in the classes and we obtain sharp subordination results for these classes and also investigate some applications of the main results which give important results of analytic functions.

2. Main Theorem

Subordination results for the class

Theorem 2.1. Let

(2.1)

(2.2)

(2.3)

(2.4)

PROOF OF THEOREM 2.1

Let defined by (1.1) be any member of the class and suppose that

Then

(2.5)

Thus, by Definition 5 the subordination (2.2) will hold if the sequence,

(2.6)

is a subordinating factor sequence with .

Therefore by Theorem 1.1, it is sufficient to show that

(2.7)

Now,

(2.8)

because is an increasing function of

Thus,

(2.9)

Thus, (2.4) holds true in U and consequently proves (2.2).

Next we show that

Now taking

and in (2.2) we have that

(2.10)

Therefore since

(2.11)

which implies that

(2.12)

Hence, we have

For

(2.13)

By max/min principle

(2.14)

(2.15)

Hence,

(2.16)

(2.17)

which shows that the constant

is the best possible and thus complete the proof of theorem 2.1.

Subordination result for the class

Our proof of Theorem 2.2 below is much akin to that of Theorem 2.1. Here we make use of Theorem 1.3 in place of Theorem 1.2 and let

Theorem 2.2. Let

(2.18)

(2.19)

(2.20)

(2.21)

3. Some Applications

Taking in Theorem 2.1; we obtain the following:

Corollary1.

(3.1)

(3.2)

(3.3)

Remark 1:

and Karthikenyan[5], and singh[6]

Corollary2.

(3.4)

(3.5)

(3.6)

Remark 2:

The case and was obtained by Frasin [4], and Selvaraj and Karthikeyan [5].and Karthikeyan [5]

(3.7)