Abstract
We investigate several results concerning the differential subordination of analytic and multivalent functions which is defined by using a certain fractional derivative operator. Some special cases are also considered.
1. Introduction and Definitions
Let denote the class of functions of the form
which are analytic in the open unit disk Also let denote the class of all analytic functions with which are defined on . If and are analytic in with , then we say that is said to be subordinate to in , written or , if there exists the Schwarz function , analytic in such that , and In particular, if the function is univalent, then the above subordination is equivalent to and .
Let , and be complex numbers with Then the Gaussian hypergeometric function is defined by
where is the Pochhammer symbol defined, in terms of the Gamma function, by
The hypergeometric function is analytic in and if or is a negative integer, then it reduces to a polynomial.
There are a number of definitions for fractional calculus operators in the literature (cf., e.g., [1, 2]). We use here the Saigo-type fractional derivative operator defined as follows (see [3]; see also [4]).
Definition 1.1. Let and . Then the generalized fractional derivative operator of a function is defined by The function is an analytic function in a simply-connected region of the -plane containing the origin, with the order for and the multiplicity of is removed by requiring that be real when .
Definition 1.2. Under the hypotheses of Definition 1.1, the fractional derivative operator of a function is defined by
With the aid of the above definitions, we define a modification of the fractional derivative operator by
for and . Then it is observed that also maps onto itself as follows:
It is easily verified from (1.8) that
Note that , and , where is the fractional derivative operator defined by Srivastava and Aouf [5, 6].
In this manuscript, we will use the method of differential subordination to derive certain properties of multivalent functions defined by fractional derivative operator .
2. Main Results
In order to establish our results, we require the following lemma due to Miller and Mocanu [7].
Lemma 2.1. Let be univalent in and let and be analytic in a domain containing with when . Set , and suppose that(1) is starlike (univalent) in ,(2).If is analytic in , with , , and then and is the best dominant.
We begin by proving the following
Theorem 2.2. Let and , and let , , , and . Suppose that is univalent in and satisfies If and then and is the best dominant.
Proof. Define the function by Then is analytic in with . A simple computation using (2.5) gives By applying the identity (1.9) in (2.6), we obtain Making use of (2.5) and (2.7), we have In view of (2.8), the subordination (2.3) becomes and this can be written as (2.1), where Since , we find from (2.10) that and are analytic in with . Let the functions and be defined by Then, by virtue of (2.2), we see that is starlike and Hence, by using Lemma 2.1, we conclude that , which completes the proof of Theorem 2.2.
Remark 2.3. If we put in Theorem 2.2, then we get new subordination result for the fractional derivative operator due to Srivastava and Aouf [5, 6].
Theorem 2.4. Let and , and let , , , and . Suppose that is univalent in and satisfies If and then and is the best dominant.
Proof. Define the function by Then is analytic in with . By a simple computation, we find from (2.16) that By using the identity (1.9) in (2.17), we obtain Applying (2.16) and (2.18), we have In view of (2.19), the subordination (2.14) becomes and this can be written as (2.1), where Since , it follows from (2.21) that and are analytic in with . Let the functions and be defined by Then, by virtue of (2.13), we see that is starlike and Hence, by using Lemma 2.1, we conclude that , which proves Theorem 2.4.
If we put in Theorem 2.4, then we have the following.
Corollary 2.5. Let and , and let . Suppose that is univalent in and satisfies If and then and is the best dominant.
By putting in Corollary 2.5, we obtain the following.
Corollary 2.6. Let and , and let . Suppose that is univalent in and satisfies If and then and is the best dominant.
By using Lemma 2.1, we obtain the following.
Theorem 2.7. Let and , and let , , , and . Suppose that is univalent in and satisfies If and then and is the best dominant.
Proof. Define the function by Then is analytic in with . A simple computation using (1.9) and (2.31) gives By using (2.29), (2.31), and (2.32), we get And this can be written as (2.1) when and . Note that and and are analytic in . Let the functions and be defined by Then, by virtue of (2.28), we see that is starlike and Hence, by applying Lemma 2.1, we observe that , which evidently proves Theorem 2.7.
Finally, we prove
Theorem 2.8. Let and , and let , , , and . Suppose that be univalent in and satisfies If and then and is the best dominant.
Proof. If we define the function by then is analytic in with . Hence, by using the same techniques as detailed in the proof of Theorem 2.2, we obtain the desired result.
By taking in Theorem 2.8 and after a suitable change in the parameters, we have the following.
Corollary 2.9. Let and . Suppose that is univalent in and satisfies If and then and is the best dominant.
Acknowledgment
This work was supported by Daegu National University of Education Research Grant in 2008.