Research Article | Open Access

Ma'moun Harayzeh Al-Abbadi, Maslina Darus, "Differential Subordination Defined by New Generalised Derivative Operator for Analytic Functions", *International Journal of Mathematics and Mathematical Sciences*, vol. 2010, Article ID 369078, 15 pages, 2010. https://doi.org/10.1155/2010/369078

# Differential Subordination Defined by New Generalised Derivative Operator for Analytic Functions

**Academic Editor:**Dorothy Wallace

#### Abstract

A new generalised derivative operator is introduced. This operator generalised many well-known operators studied earlier by many authors. Using the technique of differential subordination, we will study some of the properties of differential subordination. In addition we investigate several interesting properties of the new generalised derivative operator.

#### 1. Introduction and Preliminaries

Let denote the class of functions of the form

which are analytic in the open unit disc on the complex plane . Let denote the subclasses of consisting of functions that are univalent, starlike of order and convex of order in , respectively. In particular, the classes are the familiar classes of starlike and convex functions in , respectively. A function if Furthermore a function analytic in is said to be convex if it is univalent and is convex.

Let be the class of holomorphic function in unit disc .

Let

with .

For and we let

Let be analytic in the open unit disc . Then the Hadamard product (or convolution) of the two functions , is defined by

Next, we state basic ideas on subordination. If and are analytic in , then the function is said to be subordinate to , and can be written as

if and only if there exists the Schwarz function , analytic in , with andsuch that,

Furthermore if is univalent in , then if and only if and (see [1, page 36]).

Let and let be univalent in . If is analytic in and satisfies the (second-order) differential subordination

then is called a solution of the differential subordination.

The univalent function is called a dominant of the solutions of the differential subordination, or more simply a dominant, if for all satisfying (1.6). A dominant that satisfies for all dominants of (1.6) is said to be the best dominant of (1.6). (Note that the best dominant is unique up to a rotation of .)

Now, denotes the Pochhammer symbol (or the shifted factorial) defined by

To prove our results, we need the following equation throughout the paper:

where , , and is analytic function given by

Here is the generalized derivative operator which we shall introduce later in the paper. Moreover, we need the following lemmas in proving our main results.

Lemma 1.1 (see [2, page 71]). * Let be analytic, univalent, and convex in , with , and, . If and
**
then
**
where *

The function is convex and is the best -dominant.

Lemma 1.2 (see [3]). *Let be a convex function in and let
**
where and is a positive integer. **If **
is holomorphic in and
**
then
**
and this result is sharp.*

Lemma 1.3 (see [4]). *Let , if
**
then
**
belongs to the class of convex functions. *

#### 2. Main Results

In the present paper, we will use the method of differential subordination to derive certain properties of generalised derivative operator Note that differential subordination has been studied by various authors, and here we follow similar works done by Oros [5] and G. Oros and G. I. Oros [6].

In order to derive our new generalised derivative operator, we define the analytic function

where and Now, we introduce the new generalised derivative operator as follows.

*Definition 2.1. *For the operator is defined by
where denotes the Ruscheweyh derivative operator [7], given by
where

If is given by (1.1), then we easily find from equality (2.2) that

where .

*Remark 2.2. *Special cases of this operator include the Ruscheweyh derivative operator in two cases and [7], the Salagean derivative operator [8], the generalised Ruscheweyh derivative operator in two cases and [9], the generalised Salagean derivative operator introduced by Al-Oboudi [10], and the generalised Al-Shaqsi and Darus derivative operator that can be found in [11].

Now, we remind the well-known Carlson-Shaffer operator [12] associated with the incomplete beta function , defined by

where

is any real number, and .

It is easily seen that

and also

where

Next, we give the following.

*Definition 2.3. *For and let denote the class of functions which satisfy the condition
Also let denote the class of functions which satisfy the condition

*Remark 2.4. *It is clear that , and the class of functions satisfying
is studied by Ponnusamy [13] and others.

Now we begin with the first result as follows.

Theorem 2.5. *Let
**
be convex in , with h(0)=1 and . If and the differential subordination
**
holds, then
**
where is given by
**
The function is convex and is the best dominant.*

*Proof. *By differentiating (1.8), with respect to , we obtain
Using (2.16) in (2.13), differential subordination (2.13) becomes
Let
Using (2.18) in (2.17), the differential subordination becomes
By using Lemma 1.1, we have
where is given by (2.15), so we get
The function is convex and is the best dominant. The proof is complete.

Theorem 2.6. *If and then one has
**
where
**
and is given by (2.15).*

*Proof. *Let , then from (2.9) we have
which is equivalent to
Using Theorem 2.5, we have
Since is convex and is symmetric with respect to the real axis, we deduce that
from which we deduce This completes the proof of Theorem 2.6.

*Remark 2.7. *Special case of Theorem 2.6 with was given earlier in [11].

Theorem 2.8. *Let be a convex function in , with and let
**
If and and satisfies the differential subordination
**
then
**
and this result is sharp.*

*Proof. *Using (2.18) in (2.16), differential subordination (2.29) becomes
Using Lemma 1.2, we obtain
Hence
And the result is sharp. This completes the proof of the theorem.

We give a simple application for Theorem 2.8.

*Example 2.9. *For and and applying Theorem 2.8, we have
By using (1.8) we find that
Now,
A straightforward calculation gives the following:
Similarly, using (1.8), we find that
then
By using (2.37) we obtain
We get
From Theorem 2.8 we deduce that
implies that

Theorem 2.10. *Let be a convex function in , with and let
**
If and and satisfies the differential subordination
**
then
**
And the result is sharp.*

*Proof. *Let
Differentiating (2.47), with respect to , we obtain
Using (2.48), (2.45) becomes
Using Lemma 1.2, we deduce that
and using (2.47), we have
This proves Theorem 2.10.

We give a simple application for Theorem 2.10.

*Example 2.11. *For and and applying Theorem 2.10, we have
From Example 2.9, we have
so
Now, from Theorem 2.10 we deduce that
implies that

Theorem 2.12. *Let
**
be convex in , with and . If and the differential subordination holds as
**
then
**
The function is convex and is the best dominant.*

*Proof. *Let
Differentiating (2.60), with respect to , we obtain
Using (2.61), the differential subordination (2.58) becomes
From Lemma 1.1, we deduce that
Using (2.60), we have
The proof is complete.

From Theorem 2.12, we deduce the following corollary.

Corollary 2.13. * If , then
*

*Proof. * Since , from Definition 2.3 we have
which is equivalent to
Using Theorem 2.12, we have
Since is convex and is symmetric with respect to the real axis, we deduce that

Theorem 2.14. * Let , with , which satisfy the inequality
**
If and and satisfies the differential subordination
**
then
*

*Proof. *Let
Differentiating (2.73), with respect to , we have
Using (2.74), the differential subordination (2.71) becomes
From Lemma 1.1, we deduce that
and using (2.73), we obtain
From Lemma 1.3, we have that the function is convex, and from Lemma 1.1, is the best dominant for subordination (2.71). This completes the proof of Theorem 2.14.

#### 3. Conclusion

We remark that several subclasses of analytic univalent functions can be derived and studied using the operator .

#### Acknowledgment

This work is fully supported by UKM-ST-06-FRGS0107-2009, MOHE, Malaysia.

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#### Copyright

Copyright © 2010 Ma'moun Harayzeh Al-Abbadi and Maslina Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.