Applications of Fractional <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.38903pt" id="M1" height="12.3091pt" version="1.1" viewBox="-0.0657574 -7.92007 8.72646 12.3091" width="8.72646pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g50-114" d="M481 430L460 436C440 442 393 451 366 451C348 451 325 448 313 446C267 437 195 409 147 367C79 308 24 211 24 105C24 37 56 -12 93 -12C121 -12 156 1 199 32C246 66 313 122 349 168H351L286 -147C274 -205 261 -215 210 -224L199 -226L193 -255L438 -243L442 -214L417 -212C364 -208 359 -200 370 -140L433 204C455 324 469 389 481 430ZM391 385C382 336 366 260 358 238C321 178 209 59 143 59C129 59 115 83 115 131C115 207 152 317 220 373C243 392 282 403 314 403C348 403 374 395 391 385Z"/></g></svg>-Calculus to Certain Subclass of Analytic <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.38907pt" id="M2" height="13.4115pt" version="1.1" viewBox="-0.0657574 -9.02243 10.4144 13.4115" width="10.4144pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g50-113" d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z"/></g></svg>-Valent Functions with Negative Coefficients

Wongsaijai, Ben; Sukantamala, Nattakorn

doi:https://doi.org/10.1155/2015/273236

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2015 | Article ID 273236 | https://doi.org/10.1155/2015/273236

Applications of Fractional -Calculus to Certain Subclass of Analytic -Valent Functions with Negative Coefficients

Ben Wongsaijai¹and Nattakorn Sukantamala¹

Academic Editor: Mohamed K. Aouf

Received19 Jan 2015

Accepted06 Apr 2015

Published22 Apr 2015

Abstract

By making use of the concept of fractional -calculus, we firstly define -extension of the generalization of the generalized Al-Oboudi differential operator. Then, we introduce new class of -analogue of -valently closed-to-convex function, and, consequently, new class by means of this new general differential operator. Our main purpose is to determine the general properties on such class and geometric properties for functions belonging to this class with negative coefficient. Further, the -extension of interesting properties, such as distortion inequalities, inclusion relations, extreme points, radii of generalized starlikeness, convexity and close-to-convexity, quasi-Hadamard properties, and invariant properties, is obtained. Finally, we briefly indicate the relevant connections of our presented results to the former results.

1. Introduction

The formulation of fractional calculus began shortly after the classical calculus was established. Since its definition is based on the concept of a noninteger order either integral or derivative, the fractional calculus had been considered as a subject in pure mathematics with no real applications for a long time. However, the role of fractional calculus has been changed in recent decades. Its applications take place in many fields of mathematical sciences. Extended from the fractional calculus, the fractional -calculus is the -extension of the ordinary fractional calculus. Many results of the study on theory of -calculus operators in recent decades have been applied in various areas such as problems in the ordinary fractional calculus, optimal control, solutions of -difference equations, -differential equations, -integral equations, and -transform analysis and also in the geometric function theory of complex analysis.

In the field of geometric function theory, various subclasses of analytic functions have been studied from different viewpoints. The fractional -calculus is the important tools that are used to investigate subclasses of analytic functions. For example, the extension of the theory of univalent functions can be described by using the theory of -calculus. In [1], Ismail et al. introduced the generalized class of starlike functions by using the -difference operator and replaced the right-half plane by a suitable domain. In a similar way, Agrawal and Sahoo [2] introduced the generalized class of starlike functions of order and Raghavendar and Swaminathan [3] also introduced the class of -analogue to close-to-convex functions. Moreover, the -calculus operators, such as fractional -integral and fractional -derivative operators, are used to construct several subclasses of analytic functions (see, e.g., [4–8]).

In addition, the differential operators have been extensively investigated in the field of geometric function theory. The well-known differential operator defined on the class of analytic functions is introduced by Salagean [9]. This operator was successfully used by many authors and it led to the investigation of several properties of certain known and new classes of analytic functions (see, e.g., [10–14]). However, there are many generalized Salagean operators defined by several authors. In [15], Al-Oboudi defined the generalized Salagean operator by using the technique of convolution structure. In [16], Al-Oboudi and Al-Amoudi used the extension of fractional derivative and fractional integral to define linear multiplier fractional differential operator which yields the Al-Oboudi operator [15] and fractional differential operator. Moreover, Bulut [17] modified the Al-Oboudi and Al-Amoudi operator [16] by introducing nonnegative parameter in that operator. Recently, Selvakumaran et al. [8] introduced the fractional -differintegral operator by using the fractional -calculus operators involving the generalized Al-Oboudi and Al-Amoudi operator [16]. For some recent investigations of these operators on the classes of analytic functions and related topics, such as coefficient estimate, distortion theorem, extreme points, and subordination, we refer to [18–23] and the references cited therein.

This paper is organized as follows. In Section 2, we propose the -extension of the Bulut operator [17] which generalized Selvakumaran et al. operator [8]. We also define new class to by using this new general differential operator together with -analogue to -valent closed-to-convex function. In Section 3, we give linear combination property and coefficient estimate for function belonging to . By making use of the coefficient estimate, the -extension of geometric properties for function with negative coefficients is given in Section 4. Then, we finish our paper by observations and concluding remarks.

2. Preliminaries and Definitions

Let be a positive integer, and let be the class of analytic functions and -valent in the unit disk that are of the formLet be a subclass of consisting of functions of the form In particular, we set and . For , let be the subclass of consisting of all functions which satisfy in . The functions in are called functions of bounded turning. All of those are univalent and close-to-convex in (see [24]). Similarly, we denote by , where , the class of all functions in which satisfy (see more details in [25, 26]).

For the convenience, we provide some basic definitions and concept details of -calculus which are used in this paper. In the theory of -calculus, the -shifted factorial is defined for , as a product of factors by and in terms of the basic analogue of the gamma function where the -gamma function [27, 28] is defined byWe note that if , the -shifted factorial (3) remains meaningful for as a convergent infinite product: Here, we recall the following -analogue definitions given by Gasper and Rahman [27]. The recurrence relation for -gamma function is given by where , and is called -analogue of . It is well known that as , where is the ordinary Euler gamma function.

In view of the relation we observe that the -shifted factorial (3) reduces to the familiar Pochhammer symbol , where .

Let be fixed. A set is called a -geometric set if, for , . Let be a function defined on a -geometric set. Jackson’s -derivative and -integral of a function on a subset of are, respectively, given by (see Gasper and Rahman [27], pp. 19–22)In case , the -derivative of , where is a positive integer, is given by As and , we have .

We now recall the definition of the fractional -calculus operators of a complex-valued function , which were recently studied by Purohit and Raina [29].

Definition 1 (fractional -integral operator). The fractional -integral operator of a function of order is defined by where is analytic in a simply connected region in the -plane containing the origin. Here, the term is a -binomial function defined by

According to Gasper and Rahman [27], the series is single-valued when . Therefore, the function in (12) is single-valued when , , and .

Definition 2 (fractional -derivative operator). The fractional -derivative operator of a function of order is defined by where is suitably constrained and the multiplicity of is removed as in Definition 1 above.

Definition 3 (extended fractional -derivative operator). Under the hypotheses of Definition 2, the fractional -derivative for a function of order is defined by where , .

In addition, the extension of -differintegral operator , for , , and , is defined bywhere in (15) represents, respectively, a fractional -integral of of order when and a fractional -derivative of of order when . We note that when , the operator reduces to the operator introduced by Owa and Srivastava [30].

Now, we define the -extension of Al-Oboudi type differential operator , for , , and , which is defined byWe note that if is given by (1), then by (16) we havewhere We note that, by setting appropriated values for the parameters in the operator , this operator reduces to many known differential operators. For example, in case the operator is exactly the Selvakumaran et al. operator in [8]. Also, when the operator reduces to the operator introduced by Bulut [17]. Moreover Bulut [17] noticed that, for suitable parameters , and , the operator generalizes many operators introduced by several authors, for instance, Salagean [9], Al-Oboudi [15], Al-Oboudi and Al-Amoudi [16], Acu and Owa [31], Acu et al. [32], Cătaş [33], Cho and Srivastava [34], Cho and Kim [35], Kumar et al. [36], Owa and Srivastava [30], and Uralegaddi and Somanatha [37].

Next, we define the-analogous to the function class by . A function is said to be in the class of -valently closed-to-convex with respect to -differentiation if and only ifwhere . In particular, we set . Note that the class generalizes the class (with the function ) which was introduced by Raghavendar and Swaminathan [3]. Moreover, we see that , as . This implies that an inequality becomes . Hence, the class clearly reduces to and satisfies

Furthermore, by using the operator defined by (16) and -differentiation, we introduce a new class as follows.

Let , , , and . Denote by the class of all functions satisfying the condition

Denote by the class obtained by taking intersection of the class with the class . That is, In particular, we set . The special cases of the class , as , have been studied by Bulut [38], Al-Oboudi in [15], and Tăut et al. [39] and the special cases of the class , as , have been proved by Altintas [40].

3. Main Results

3.1. General Properties

We begin to derive the linear combination property on in the following result.

Theorem 4. The class is convex.

Proof. Let of the formIt is sufficient to show that the function , where , is in the class . By (23), we see that Hencewhere is defined by (18). Since , we have Applying (26) to (25), we obtain Now, the proof is completed.

Remark 5. In case , by letting , we obtain Theorem 4.1 in [38]. For with , we obtain Theorem in [15]. Moreover, for , , and , we obtain Theorem 2.1 in [39].

Next, we derive some sharp coefficient inequalities contained in the following theorem that are useful in the main results.

Theorem 6. Let be defined by (1) and satisfy the inequality where is defined in (18). Then, . Moreover, the converse also holds if . The result is sharp.

Proof. Let the function be defined by (1). To prove this, we considerBy assumption (28), (29) can be rewritten asTherefore, we infer that . To prove the converse, we let function be defined by (2) and belong to the class . Then, we haveOr, equivalently, In (32), by letting on the real axis, we obtain inequality (28) as desired. Finally, we note that assertion (28) is sharp, the extremal function beingNow, the proof is completed.

Corollary 7. If , then for , where is defined in (18).

4. Geometric Properties for the Class

By observation, Theorem 6 gives the necessary and sufficient conditions via coefficient bounded for functions to be in the multivalently analytic function class . Using this result, we will discuss standard properties for that class in sense of -theory, such as distortion inequalities, inclusion relations, extreme points, radii of close-to-convexity, starlikeness and convexity, quasi-Hadamard property, and invariant properties. However, some of the mentioned properties can be obtained only in case because the monotonicity of the sequence is required to prove those results. The following lemmas guarantee the monotone increasing property for the sequence in case and monotone decreasing in case .

Lemma 8. Let the sequence be defined by (i)If , then is a nondecreasing sequence and for .(ii)If , then is a decreasing sequence and for .

Proof. It is clear that the sequence is nonnegative for . We have that So, by using (7), we get Since , we see that Then, for , we conclude that is a nondecreasing sequence and satisfying for all . Also, for , the sequence is a decreasing sequence and satisfying for all .

Lemma 9. If , then the sequence defined in (18) is an increasing sequence and satisfying for all .

Proof. The result is directly obtained by Lemma 8 and the following inequality: where and .

4.1. Distortion Inequalities

Next, we derive the distortion inequalities for functions in the multivalently analytic functions class that will be given by the following results.

Theorem 10. For , suppose that is defined by (2). If , then Generally,where . The estimations in (40)–(42) are sharp.

Proof. Let the function be defined by (2) and belong to the class . In virtue of Theorem 6 and Lemma 9, we haveFrom (43), the consequence is thatSince , it is easy to see thatThe conjunction of (44) and (45) yields assertions (40) of Theorem 10. Hence, (41) and (42) follow fromFinally, we note that assertions (40)–(42) are sharp, since equalities are attained by the functionNow, the proof is completed.

Remark 11. By letting , Theorem 10 demonstrates that the disk is mapped onto a domain that contains the diskunder any multivalently analytic function , and onto a domain that contains the diskby any .

4.2. Inclusion Relation

In the following results, we obtain some inclusion relation for the parameters , , and of the class .

Theorem 12. If and , thenand if those parameters satisfy either or .

Proof. The inclusion relation is directly obtained by Theorem 6 and the inequalityIn case or , we see that belongs to the class but does not belong to the class , which implies that . Now, the proof is completed.

Applying Theorem 6 and Lemma 9, we obtain another inclusion relation as follows.

Theorem 13. If , then

4.3. Extreme Points

Now, let us determine extreme points of the class .

Theorem 14 (extreme points). Let andThen is in the class if and only if it can be expressed in the formwhere and .

Proof. Let the function be defined by (2). Since , we then have Now, we obtain Thus, by Theorem 6. Conversely, suppose that . We may set and . Then we have . This completes the proof of Theorem 14.

4.4. Radii of Generalized Close-to-Convexity, Starlikeness, and Convexity

Now, the discussions on radii of generalized close-to-convexity, starlikeness, and convexity for the class are given by the following results. In order to establish, we will also require the use of those classes of functions. First of all, a function is said to be in the class of -valently starlike with respect to -differentiation of order if it satisfies the inequalityFurthermore, a function is said to be in the class of -valently convex with respect to -differentiation of order if it satisfies the inequalityBoth and were introduced by Selvakumaran et al. [8]. However, we consider the case instead of . The definition of -analogous of -valently closed-to-convex was already recalled in (19).

Theorem 15. For , if , then is -valently closed-to-convex with respect to -differentiation of order .

Proof. By Theorem 13, we obtainThis completes the proof.

In general, for , the function does not necessarily belong to the class . We then derive the radii of generalized close-to-convexity order for the function .

Theorem 16. For , if , then is -valently closed-to-convex with respect to -differentiation of order in , where

Proof. It is sufficient to show that . That is,Since and by application of Theorem 6, we obtainHence, (63) is true if This completes the proof.

Next, we obtain the radii of generalized starlikeness of order in the following result.

Theorem 17. For , if , then is -valently starlike with respect to -differentiation of order in where

Proof. We have to show that . That is,Hence, (67) is true ifBy using (64), we can say (68) is true ifwhich completes the proof.

Corollary 18. If , then

Proof. By Theorem 17, we see that a function is -valently starlike with respect to -differentiation () in where It is easy to see that is a decreasing sequence and . This impliesMoreover, by using Lemma 9, we obtainThen, we have . That is, is -valently starlike with respect to -differentiation in . The proof is completed.

Next, we obtain the radii of generalized convexity of order , where , in the following result.

Theorem 19. For and , if , then is -valently convex with respect to -differentiation of order in , where

Proof. We have to show that . That is,Hence, (75) is true ifor equivalentlySince , by using (64), we can say (77) is true ifThe proof is completed.

4.5. Quasi-Hadamard Properties

In this section, we derive the quasi-Hadamard (convolution) properties. Before we derive the result, we recall the definition of the quasi-Hadamard properties. For any functions , of the formthe quasi-Hadamard product is defined by

Next, we derive the quasi-Hadamard properties for the class . Using the techniques of Schild and Silverman [41] with Theorem 6, we prove the following results.

Theorem 20. For , suppose that , ; then , where

Proof. To prove this theorem, we use the principle of mathematical induction on . Let the functions , for of the formfor and . Since for , by Theorem 6, we see thatAccording to Theorem 6, it is sufficient to prove that where is defined in (81). Applying Cauchy-Schwarz’s inequality to (83) for , we have the following inequality:From (84) and (85), if the following inequality for all , is satisfied, it can be concluded that . Now, applying Corollary 7, inequality (86) will be held if By Lemma 9, we see that This yields our desired inequality (86). Now, we have . Next, we let the functions , for and for . Suppose that where Then, by means of the above technique, it can be shown that where By Lemma 9, we have . This completes the proof of the theorem.

4.6. Invariant Properties

In the following results, we discuss invariant properties of the class via Theorem 6. We consider the formerly studied operators in terms of the standard convolution formula; we choose as a fixed function in such that exists for any . For various choices of we get different linear operators that have been studied in the recent past.

According to Theorem 6, we easily obtain the following properties.

Theorem 21. For , if the function is of the form where for , then for each .

Next, we recall the definition of Bernardi’s integral operator. For nonnegative real number and , Bernardi-Libera’s integral operator is defined as follows:which was studied by Bernardi in [42]. Also, their properties for are reported in [43, 44]. By using the concept of -calculus, we introduce -analogous to Bernardi’s integral operator defined by From (95), we have verified that That is, where . It is clear that for . Then, we obtain the invariant properties under integral operator as follows.

Theorem 22. For , and , the class is invariant under the integral operator defined in (95). That is,

Moreover, in the view of the definition of fractional -integral and Lemma 8, we obtain the invariant properties under fractional -integral.

Theorem 23. For , the class is invariant under the integral operator defined in (11). That is,

5. Observation and Concluding Remark

In this section we briefly point out some consequences of the results derived in the proceeding sections. If we let , we observe that the function classes and provide the -extensions of both known and newly obtained. By assigning appropriated values to the parameters , and , we can derive the corresponding results for several simpler subclasses of the class from each of our theorems, especially as indicated in Altintas [40]. Therefore, it leads to the -extension of the former results.

Furthermore, we let , where is a subclass of consisting of functions of the form Theorem 6 can indeed be generalized further by considering the class of multivalent function in place of . That leads to the following corollary.

Corollary 24. Let be defined by (100); then if and only if satisfies the inequality where is defined in (18). Moreover, the result is sharp.

By previous argument in this paper, Corollary 24 can fruitfully be used in investigating the geometric properties for several subclasses of . Then the -extension of the former results, such as Sarangi and Uralegaddi [45], Aouf et al. [46], and Aouf [47], is obtained.

We conclude this paper by remarking that the results presented in this paper give various -extension properties of different classes of analytic and multivalent function. Here, our results generalize several formerly known results. We introduce -extension of the general differential operator, which is generalized from Bulut operator [17], in sense of -theory. The new subclass of multivalently analytic function is proposed consequently by joining the class of -analogue to close-to-convexity together with our -extension of Bulut operator [17]. We discuss the linear combination property and coefficient estimate for the class . By making use of the coefficient estimate and the concept of -theory, we obtain the -extension of geometric properties for the class . We give the -analogue to distortion properties and the radii of -analogue starlikeness and convexity which were defined in [8]. Moreover, we consider the radii of -analogue close-to-convexity that replace usual derivative by -derivative operator. We also use the concept of -theory to extend the Bernardi integral operator. As a consequence, the invariant property for under such -integral operator is obtained. Finally, the presented results can be extended to investigate the function on which gives generalized formerly known and newly obtained results.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgment

This research was supported by the Department of Mathematics, Faculty of Science, Chiang Mai University.

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Copyright © 2015 Ben Wongsaijai and Nattakorn Sukantamala. 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.

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