1. Introduction

Let denote the class of all functions of the from

(1)

which are analytic and multivalent in the open unit disk

Let denote the subclass of consisting of functions of the from

(2)

which are analytic and multivalent in the open unit disk

For the function given by (2) and defined by

(3)

we define the convolution (or Hadamard product) of by

(4)

A function is said to be p-valentlystarlike of order if and only if

(5)

A function is said to be p-valently convex of order if and only if

(6)

A function is said to be p-valently close-to-convex of order if and only if

(7)

Definition 1 [8]: Let and

Then we define the linear operator

(8)

Definition 2: Let be a fixed function defined by (3). The function given by (2) is said to be in the class if and only if

(9)

where

Some of the following properties studied for other class in [1], [2], [3], [4], [6] and [7].

2. Coefficient Inequalities

Theorem 1: Let . Then if and only if

(10)

where

The result is sharp for the function

(11)

Proof: Suppose that the inequality (10) holds true and Then we have

by hypothesis.

Hence, by maximum modulus principle,

Conversely, suppose that Then from (9), we have

Since for all we get

(12)

We choose the value of on the real axis, so that is real.

Letting through real values, we obtain inequality (10).

Finally, sharpness follows if we take

(13)

Corollary 1: Let Then

(14)

3. Growth and Distortion Theorems

Theorem 2: Let the function Then

(15)

Proof:

By Theorem 1, we get

(16)

Thus

also

and the proof is complete.

Theorem 3: Let Then

Proof: Notice that

(17)

from Theorem 1, thus

(18)

On the other hand

(19)

Combining (18) and (19), we get the result.

Closure Theorems:

Theorem 4: Let the function defined by

(20)

be in the class for every Then the function defined by

also belongs to the class where

Proof: Since then by Theorem 1, we have

(21)

for every

Hence

By Theorem 1, it follows that

Theorem 5: Let the function defined by (20) be in the class for every Then the function defined by

is also in the class

Proof: By definition of we have

Since are in the class for every we obtain

4. Radii of Starlikeness, Convexity and Close-to-Convexity

In the following theorems, we obtain the radii of starlikeness, convexity and close-to-convexity for the class

Theorem 6: If then is p-valentlystarlike of order in the dick where

Proof: It is sufficient to show that

for we have

Thus

if

(22)

Hence, by Theorem 1, (22) will be true if

or if

Setting we get the desired result.

Theorem 7: If Then is p-valently convex of order in the disk where

Proof: It is sufficient to show that

for we have

Thus

if

(23)

Hence by Theorem 1, (23) will be true if

and hence

Setting we get the desired result.

Theorem 8: Let the function Then is p-valently close-to-convex of order in the disk where

Proof: It is sufficient to show that

for we have

Thus

if

(24)

hence, by Theorem 1, (24) will be true if

and hence

Setting we get the desired result.

5. Weighted Mean

Definition 3: Let be in the class Then the weighted mean of is given by

Theorem 9: Let be in the class Then the weighted mean of is also in the class

Poof: By Definition 3, we have

(25)

Since are in the class so by Theorem 1, we get

and

Hence

Therefore

The proof is complete.

6. Neighborhoods and Partial Sums

Now we define the neighborhoods of the function by

(26)

For identity function

(27)

The concept of neighborhoods was first introduced by Goodman [5] and then generalized by Ruscheweyh [9].

Definition 4: A function is said to be in the class if there exist a function such that

Theorem 10: If and

(28)

Then

Proof: Let Then we have from (26) that

which readily implies the following coefficient inequality

Next, since we have from Theorem 1

so that

Then by Definition 3, for every given by (28).

Now, we introduce the partial sums.

Theorem 11: Let be given by (2) and define the by

and

(30)

suppose also that

(31)

Thus, we have

(32)

and

(33)

Each of the bounds in (32) and (33) is the best possible for

Proof: For the coefficients given by (31), it is not difficult to verity that

Therefore, by using the hypothesis (30), we have

(34)

By setting

and applying (34), we find that

This prove (32). Therefore, and we obtain

Now, in the same manner, we prove the assertion(33), by setting

and this completes the proof.