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