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International Journal of Differential Equations
Volume 2009 (2009), Article ID 597292, 18 pages
http://dx.doi.org/10.1155/2009/597292
Research Article

An Extension to the Owa-Srivastava Fractional Operator with Applications to Parabolic Starlike and Uniformly Convex Functions

School of Mathematical Sciences, Faculty of Science and Technology, University Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia

Received 8 October 2008; Accepted 6 January 2009

Academic Editor: Shaher Momani

Copyright © 2009 Oqlah Al-Refai 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.

Abstract

Let be the class of analytic functions in the open unit disk 𝕌. We define by where is the fractional derivative of of order . If , then a function in is said to be in the class if is a parabolic starlike function. In this paper, several properties and characteristics of the class are investigated. These include subordination, characterization and inclusions, growth theorems, distortion theorems, and class-preserving operators. Furthermore, sandwich theorem related to the fractional derivative is proved.

1. Introduction and Definitions

Let be the class of functions analytic in the open unit disk and let be the subclass of consisting of functions of the formand be the class of functions in of the formLet be the subclass of consisting of functions of the formA function in is said to be uniformly convex in if is a univalent convex function along with the property that, for every circular arc contained in , with center also in , the image curve is a convex arc. The class of uniformly convex functions is denoted by (for details, see [1]). It is well known from [2, 3] thatCondition (1.4) implies thatlies in the interior of the parabolic regionfor every value of . A function in is said to be in the class of parabolic starlike functions, denoted by (cf. [3]), if

Let the function be given bywhere is the Pochhammer symbol defined byFurther, let (cf. [4, 5])In terms of Hadamard product or convolution, note that is the identity operator andIt is well known that if , then maps into itself. We also need the following definitions of a fractional derivative.

Definition 1.1 (cf. [5, 6], see also [7, 8]). Let the function be analytic in a simply connected domain of the -plane containing the origin. The fractional derivative of of order is defined by where the multiplicity of is removed by requiring to be real when .

Using Definition 1.1 and its known extensions involving fractional derivatives and fractional integrals, Owa and Srivastava [5] introduced the operator defined byNote that .

Corresponding to the operator defined in (1.13), Srivastava and Mishra [9] studied the class of functions satisfying the inequality

In Definition 1.2, we generalize the Owa-Srivastava operator defined in (1.13) as follows.

Definition 1.2. Let be in . One defines an operator bywhere is the fractional derivative of of order .

From Definition 1.2, we note that

In the present paper, we study a class of analytic functions, related to , , and , using the operator defined in Definition 1.2.

Definition 1.3. Let , where be the class of functions satisfying the inequality

It follows that

Remark 1.4. if and only if is uniformly convex function.

Using the definition of , we start with proving sandwich theorem related to the fractional derivative. Then, we investigate several properties and characteristics of the general class using similar techniques to [9]. These include subordination, inclusions and characterization, growth theorems, and class-preserving operators (like the Hadamard product and various integral transforms).

2. Sandwich Theorem

In order to prove our sandwich result, we need first to recall the principle of subordination between analytic functions, let the functions and be in . We say that is subordinate to or is superordinate to in , written as , if is univalent in ,Let and let . If and are univalent and satisfies the first-order differential superordination,then is a solution of the differential superordination (2.2). An analytic function is called a subordination if for all satisfying (2.2). A univalent subordinant that satisfies for all subordinations of (2.2) is said to be the best subordinant. An analytic function is said to be dominant if for all satisfyingA univalent dominant that satisfies for all dominants of (2.3) is said to be the best dominant.

We also need the following definition and lemma.

Definition 2.1 (see [10, page 817, Definition 2]). Denoted by , the set of all functions that are analytic and injective on , whereand are such that for .

Lemma 2.2 (see [11]). Let be two nonzero univalent functions in , and let . Further assume that and for , is starlike univalent in . If ,   , is the th derivative of and is univalent in , thenimpliesand are, respectively, the best subordinant and the best dominant.

As an application of Lemma 2.2, we prove the following theorem.

Theorem 2.3. Let be two nonzero univalent functions in , and let , and . Further, assume that and are starlike univalent in . If ,thenimpliesand are, respectively, the best subordinant and the best dominant.

Proof. Let be defined as in Definition 1.2, where and . Then from (1.17), we have for ,This yieldsBy applying Lemma 2.2 for and , we get the result.

Putting in Theorem 2.3, we get the following corollary.

Corollary 2.4. Let be two nonzero univalent functions in , and let ,  . Further, assume that and are starlike univalent in . If ,thenimpliesand are, respectively, the best subordinant and the best dominant.

In particular, for , Corollary 2.4 reduces to the following remark.

Remark 2.5. Let be two nonzero univalent functions in , and assume that and are starlike univalent in . For , if , is univalent in and , thenimpliesand are, respectively, the best subordinant and the best dominant.

Remark 2.6. Taking in Theorem 2.3 yields that Corollary 2.4 and Remark 2.5 are also hold true for , where .

3. Some Properties of the Class

We need the following results in our investigation of the class .

Lemma 3.1 (see [12]). Let and be univalent convex functions in . Then the Hadamard product is also univalent convex in .

Lemma 3.2 (see [13]). Let and be univalent convex functions in . Also let and . Then .

Lemma 3.3 (see [12]). Let each of the functions and be univalent starlike of order . Then, for every function ,where denotes the closed convex hull.

Theorem 3.4. If and , then

Proof. Let . ThenAlso it is known that (cf. [14])Since is a convex region, using Lemma 3.3, we getThus, . This completes the proof of Theorem 3.4.

Corollary 3.5. Let and . ThenIn particular, the functions in are parabolic starlike and they are uniformly convex when .

Corollary 3.6. Let and . Then

It can be verified that the Riemann map of onto the region , satisfying and , is given by

We define the function by

Theorem 3.7. Let and let be defined by (3.8). Then is a convex univalent function. Furthermore, if , then

Proof. We first note thatwhere each member of the Hadamard product in (3.10) is known to be a convex univalent function (cf. [2, 14]). Therefore, by Lemma 3.1, is a univalent convex function. Next, if , thenThus, there exists a function satisfying the Schwarz Lemma such thatSince is a univalent convex function, a result of [15] (see also [16, page 50]) yieldsIt now follows from a known result of [14, page 508, Theorem 2] thatThe proof of Theorem 3.7 is evidently completed.

Remark 3.8. (i) Letting or equal to zero in Theorem 3.7, we immediately obtain a subordination result due to Srivastava and Mishra (see [9]).
(ii) Taking in Theorem 3.7, we get a result of [2, page 169, Theorem 3].

Theorem 3.9. Let . If , thenwhere is defined by (3.8). Equality holds true in (3.15) and (3.16) for some if and only if is a rotation of .

Proof. Let . Then, by Theorem 3.7 and the Lindelöf principle of subordination, we getSince is a univalent convex function and has real coefficients, is a convex region symmetric with respect to real axis. Hence,Thus, (3.17) gives the assertion (3.15) of Theorem 3.9. Also, we readily have the assertion (3.16) of Theorem 3.9. The sharpness in (3.15) and (3.16) is also a consequence of the principle of subordination. This completes the proof of Theorem 3.9.

Corollary 3.10. Let , where . Then,The result is sharp.

Remark 3.11. (i) Letting or equal to zero in Theorem 3.9, we obtain a result due to Srivastava and Mishra (see [9]).
(ii) Taking in Theorem 3.9, we get a result of [2, page 170, Corollary 3].

Next, we investigate characterization for to be in the class . We need first the following lemma.

Lemma 3.12. If , where , then

Proof. Suppose . We can writeThen, there exists an integer such thatFor , we haveSince , there exists a real number , , such that . Hence, is not univalent.

Theorem 3.13. Let . Then, a function if and only if

Proof. First, considerwhere . Hence, if (3.24) holds, then the above expression is less than , and consequentlyConversely, if and is real, we getLet along the real axis, then we getUsing Lemma 3.12, we haveTherefore, the denominator in (3.28) is positive, and hence (3.24) holds. This completes the proof of Theorem 3.13.

Remark 3.14. Theorem 3.13 is sharp for functions of the form

Corollary 3.15. If , thenIn particular, if and only if and if and only if

Corollary 3.16. If , where , then

Corollary 3.17. If where , then

Proof. Let , where . Clearly,Therefore,

Remark 3.18. Under the hypothesis of Corollary 3.17, lies in a disc centered at the origin with radius given byIn particular, we have
(i) if , then lies in a disc centered at the origin with radius (ii) if , then lies in a disc centered at the origin with radius ;(iii)if , then lies in a disc centered at the origin with radius ;(iv)if , then lies in a disc centered at the origin with radius .

Consequently, let be defined byThen, belonging to the corresponding class implies lies in a disc centered at the origin with radius given by

4. Class-Preserving Operators and Transforms

Theorem 4.1. Let be univalent starlike function of order    and let . Then,In particular, if is univalent starlike function of order and , then

Proof. Let be univalent starlike function of order and . By definition,The commutative and associative properties of the Hadamard product yieldTherefore, using Lemma 3.3, we getThis completes the proof of Theorem 4.1.

Taking and in Theorem 4.1, then we have the following corollary.

Corollary 4.2. If and , then . In particular, if and , then . Moreover, if and , then .

Taking or , and in Theorem 4.1, then we have the following corollary.

Corollary 4.3. If and , then . In particular, if and , then . Moreover, if and , then .

Corollary 4.4 (see [9]). If and , thenIn particular, if and , then

Corollary 4.5 (see [3]). If and , then . In particular, if and , then .

Theorem 4.6. Let and , where . Then, .

Proof. The proof of Theorem 4.6 is similar to that of Theorem 4.1. Let and . We first note thatTherefore, using Lemma 3.3, we getThus, . This completes the proof of Theorem 4.6.

Corollary 4.7. The class is closed under convolution, and in particular the classes and are so.

Theorem 4.8. Let and . Then, . In particular, the class is closed under convolution.

Proof. The proof of Theorem 4.8 is similar to that of Theorem 4.6. Let and . We first note thatTherefore, using Lemma 3.3, we getThus, . By taking , we see that the class is closed under convolution. This completes the proof of Theorem 4.8.

Theorem 4.9. Let . Also letDefine a function byThen, .

Proof. Let and let be defined by (4.13). Direct calculation givesThus, by Definition 1.3, . This completes the proof of Theorem 4.9.

Theorem 4.10. Let , where . Then, the function defined by the integral transformis also in the class .

Proof. We begin by noting thatUsing a result of Bernardi [17], it can be verified thatAlso, by hypothesis, . Thus, using Lemma 3.3, we getwhich completes the proof of Theorem 4.10.

Acknowledgment

The work presented here was supported by EScienceFund 04-01-02-SF0425, Academy of Sciences, Malaysia.

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