Research Article | Open Access
Differential Subordination Defined by New Generalised Derivative Operator for Analytic Functions
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 .
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 ). Let be a convex function in and let
where and is a positive integer.
is holomorphic in and then and this result is sharp.
Lemma 1.3 (see ). 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  and G. Oros and G. I. Oros .
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.
Remark 2.2. Special cases of this operator include the Ruscheweyh derivative operator in two cases and , the Salagean derivative operator , the generalised Ruscheweyh derivative operator in two cases and , the generalised Salagean derivative operator introduced by Al-Oboudi , and the generalised Al-Shaqsi and Darus derivative operator that can be found in .
Now, we remind the well-known Carlson-Shaffer operator  associated with the incomplete beta function , defined by
is any real number, and .
It is easily seen that
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  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.
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.
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.
We give a simple application for Theorem 2.10.
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.
From Theorem 2.12, we deduce the following corollary.
Corollary 2.13. If , then
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.
We remark that several subclasses of analytic univalent functions can be derived and studied using the operator .
This work is fully supported by UKM-ST-06-FRGS0107-2009, MOHE, Malaysia.
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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.