Research Article | Open Access
Applications of Fractional -Calculus to Certain Subclass of Analytic -Valent Functions with Negative Coefficients
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.
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 , 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  introduced the generalized class of starlike functions of order and Raghavendar and Swaminathan  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 . 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 , Al-Oboudi defined the generalized Salagean operator by using the technique of convolution structure. In , 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  and fractional differential operator. Moreover, Bulut  modified the Al-Oboudi and Al-Amoudi operator  by introducing nonnegative parameter in that operator. Recently, Selvakumaran et al.  introduced the fractional -differintegral operator by using the fractional -calculus operators involving the generalized Al-Oboudi and Al-Amoudi operator . 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  which generalized Selvakumaran et al. operator . 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 ). 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 . 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 , 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 .
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
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 .
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 . Also, when the operator reduces to the operator introduced by Bulut . Moreover Bulut  noticed that, for suitable parameters , and , the operator generalizes many operators introduced by several authors, for instance, Salagean , Al-Oboudi , Al-Oboudi and Al-Amoudi , Acu and Owa , Acu et al. , Cătaş , Cho and Srivastava , Cho and Kim , Kumar et al. , Owa and Srivastava , and Uralegaddi and Somanatha .
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 . 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 , Al-Oboudi in , and Tăut et al.  and the special cases of the class , as , have been proved by Altintas .
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.
Next, we derive some sharp coefficient inequalities contained in the following theorem that are useful in the main results.
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.
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.
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 .
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. . 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
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
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
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
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