1. Introduction

A continuous function is a complex-valued harmonic function in a domain if both u and v are real harmonic in D. In any simply connected domain we can write where h and g are analytic in D. We call h the analytic part and g the co-analytic part of f. The harmonic function is sense-preserving and locally one-to-one in D if in D. See Clunie and Sheil-Small [3].

For denote by the class of functions that are harmonic multivalent and sense-preserving in the unit disk where are defined by

(1)

which are analytic and multivalent functions in U .

Also, denote by subclass of consisting of harmonic functions where and are of the form

(2)

In 1984, Clunie and Sheil-Small [3] investigated the class and as well as its geometric subclasses and obtained some coefficient bounds. Since then, there have been several related papers on and its subclasses. For more basic results one may refer to the following standard

introductory text book by Duren [4], see also Ahuja [1], Jahangiri et al. [5] and Ponnusamy and Rasila ([6], [7]).

The convolution of two functions of form

(3)

Let denote the subclass of consisting of functions that satisfy the condition

(4)

Define

Lemma (1) [2]: Let Then if and only if where w be any complex number.

2. Coefficient Estimates

We begin with a sufficient condition for the function in to be the class

Theorem (2.1): Let defined in (1). If

(5)

Where then is harmonic Multivalent sense-preserving in U and

Proof: Let

By lemma (1), we must show that (4) holds true. It suffices to show that

Substituting for w and resorting to simple calculation, we find

Theorem (2.2): Let with h and g given by (2). Then if and only if

(6)

Proof: From theorem (2.1) to prove the necessary part, let us assume that using (4), we get

If we choose z to be real and let we obtain the condition (6) and the proof is complete.

3. Convex Set

Theorem (3.1): The class is convex set.

Proof: Let the function be in the class It is sufficient to show that the function defined by:

(7)

is in the class we have

Since for

In view of Theorem (2.1), we have

Hence, . This completes the proof.

4. Distortion and Covering

The next Theorem is on the distortion and covering bounds for functions in the class

Theorem (4.1): If then

(8)

and

(9)

Proof: Assume that Then by Theorem (2.2), we obtain

Relation (9) can be proved by using the similar statements.

The covering result given in corollary (4.2) follows from the inequality (9) of this Theorem.

Corollary(4.2): If then

5. Weighted Mean

Definition (5.1): Let where

Then the weighted mean is given by

In the theorem below, we will show the weighted mean for this class:

Theorem (5.2): If and be in the class then the weighted mean of is also in the class

Proof: By Definition (5.1), we have

Since are in the class so by Theorem (2.2), we get

and

Then