^{1}

^{*}

^{2}

In this paper we investigate the new subclass of starlike functions in the unit disk
*U*
={
*z*
∈□:|
*z*
|＜1} via the generalized salagean differential operator. Basic proper-ties of this new subclass are also discussed.

Let

which are analytic in the unit disk

Let

Definition 1 ( [

Remark 1. If

Remark 2. For

From (2), the following relations holds:

and from which, we get

Definition 2 ( [

with

This operator is a particular case of the operator defined in [

Next, we define the new subclasses of

Definition 3. A function

Remark 3.

Remark 4.

Definition 4. Let

i)

ii)

iii)

Several examples of members of the set

Let P denote the class of functions

Lemma 1 ( [

i)

ii)

More general concepts were discussed in [

Lemma 2 ( [

If the differential subordination:

has univalent solution

The formal solution of (6) is given as

where

and

see [

Lemma 3 ( [

Theorem 1. Let

Proof. From (4), we have

If we suppose

Now, let

Then

By (2) and (3) we have

Applying Lemma 2 with

Theorem 2. Let

where

is the best dominant.

Proof. Let

By (9), we have

where

To show that

Now, considering the differential equation

whose solution is obtained from (8). If we proof that

sult follows trivially from Lemma 2. Setting

i)

ii)

where

Therefore,

iii)

so that

Hence,

Theorem 3.

Proof. Let

From (9), let

Corollary 1. All functions in

Proof. The proof follows directly from Theorem 3 and Remark 4.W

Corollary 2. The class

Proof. The proof is obvious from the above corollary and Definition 4.W

The functions

Theorem 4. The class

Proof. let

Applying

Let

Let

Theorem 5. Let

for some

Proof. Let

But

Applying the operator in Definition 2, we have the result.W

With

Theorem 6. Let

The function

Proof. Let

Theorem 7. Let

and

where

Proof. Let

and

for

Also, upon differentiating

and

for

The authors appreciates the immense role of Dr. K.O. Babalola (a senior lecturer at University of Ilorin, Ilorin, Nigeria) in their academic development.

Afis, S. and Sidiq, M. (2016) On Starlike Functions Using the Generalized Salagean Differential Operator. Open Access Library Journal, 3: e2895. http://dx.doi.org/10.4236/oalib.1102895