Convexity of Certain 𝑞-Integral Operators of 𝑝-Valent Functions

Abstract

By applying the concept (and theory) of fractional -calculus, we first define and introduce two new -integral operators for certain analytic functions defined in the unit disc . Convexity properties of these -integral operators on some classes of analytic functions defined by a linear multiplier fractional -differintegral operator are studied. Special cases of the main results are also mentioned.

1. Introduction and Preliminaries

The subject of fractional calculus has gained noticeable importance and popularity due to its established applications in many fields of science and engineering during the past three decades or so. Much of the theory of fractional calculus is based upon the familiar Riemann-Liouville fractional derivative (or integral). The fractional -calculus is the extension of the ordinary fractional calculus in the -theory. Recently, there was a significant increase of activity in the area of the -calculus due to applications of the -calculus in mathematics, statistics, and physics. For more details, one may refer to the books [1–4] on the subject. Recently, Purohit and Raina [5–7] have added one more dimension to this study by introducing certain subclasses of functions which are analytic in the open disk , by using fractional -calculus. Purohit [8] also studied similar work and considered new classes of multivalently analytic functions in the open unit disk.

The aim of this paper is to consider a linear multiplier fractional -differintegral operator and to define certain new subclasses of functions which are -valent and analytic in the open unit disk. The results derived include convexity properties of these -integral operators on some classes of analytic functions. Special cases of the main results are also mentioned.

Let denote the class of functions of the form which are analytic and -valent in the open unit disk . A function is said to be -valently starlike of order if and only if We denote by the class of all such functions. On the other hand, a function is said to be in the class of -valently convex of order if and only if Note that and are, respectively, the classes of -valently starlike and -valently convex functions in . Also, we note that and are, respectively, the usual classes of starlike and convex functions in . A function is said to be in the class of -uniformly -valent starlike of order if it satisfies Furthermore, a function is said to be in the class of -uniformly -valent convex of order if it satisfies For uniformly starlike and uniformly convex functions we refer to the papers [9–11]. Note that and , where the classes and are, respectively, the classes of -uniformly starlike of order and -uniformly convex of order studied in [12].

For the convenience of the reader, we now give some basic definitions and related details of -calculus which are used in the sequel.

For any complex number the -shifted factorials are defined as and in terms of the basic analogue of the gamma function where the -gamma function is defined by If , the definition (6) remains meaningful for as a convergent infinite product: In view of the relation we observe that the -shifted factorial (6) reduces to the familiar Pochhammer symbol , where . Also, the -derivative and -integral of a function on a subset of are, respectively, given by (see [2] pp. 19–22) Therefore, the -derivative of , where is a positive integer, is given by where and is called the -analogue of . As , we have .

The -analogues to the function classes , , , and are given as follows.

A function is said to be in the class of -valently starlike with respect to -differentiation of order if it satisfies Also, a function is said to be in the class of -valently convex with respect to -differentiation of order if it satisfies On the other hand, a function is said to be in the class of -uniformly -valent starlike with respect to -differentiation of order if it satisfies Furthermore, a function is said to be in the class of -uniformly -valent convex with respect to -differentiation of order if it satisfies

In the following, we define the fractional -calculus operators of a complex-valued function , which were recently studied by Purohit and Raina [5].

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 of the -plane containing the origin and the -binomial function is given by The series is single valued when and (see for details [2], pp. 104–106); therefore, the function in (18) 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.

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

Remark 4. It follows from Definition 2 that

2. The Operator

Using , we define a -differintegral operator as follows: where in (23) represents, respectively, a fractional -integral of of order when and a fractional -derivative of of order when . Here we note that .

We now define a linear multiplier fractional -differintegral operator as follows: If is given by (1), then by (24) we have It can be seen that, by specializing the parameters, the operator reduces to many known and new integral and differential operators. In particular, when , , and the operator reduces to the operator introduced by AL-Oboudi [13] and if , , , and it reduces to the operator introduced by Sǎlǎgean [14].

By using the operator defined by (24) and -differentiation, we introduce two new subclasses of analytic functions and as follows.

A function is said to be in the class if and only if

Furthermore, a function is said to be in the class if and only if

It is interesting to note that the classes and generalize several well-known subclasses of analytic functions. For instance, if , then(1),(2),(3),(4).

3. The -Valent -Integral Operators and

We now introduce two new -valent -integral operators as follows.

Definition 5. Let , and for all , . Then is defined as and is defined as where is given by (24).

It is interesting to observe that several well-known and new integral operators are special cases of the operators and . We list a few of them in the following remarks.

Remark 6. Letting for all and , the -integral operator reduces to the operator studied by Frasin in [15]. Upon setting , , , and , we obtain the integral operator studied by Breaz et al. in [16]. For , , and , the operator reduces to the operator which was studied by D. Breaz and N. Breaz in [17]. Observe that when , , , and , we obtain the integral operator studied by Pescar and Owa in [18]. Also, for , , , and , the -integral operator reduces to the Alexander integral operator studied in [19].

Remark 7. Letting for all and , the -integral operator reduces to the operator studied by Frasin in [15]. For , and , the operator reduces to the operator which was studied by Breaz et al. (see [20]). Also, for , , , and , the -integral operator reduces to the integral operator introduced and studied by Pfaltzgra (see [21]).

In this paper, we obtain the order of convexity with respect to -differentiation of the -integral operators and on the classes and . As special cases, the order of convexity of the operators and is also given.

4. Convexity of the Operator

First, we prove the following convexity result with respect to -differentiation of the operator .

Theorem 8. Let , , , , and for all , . If then the -integral operator defined by (28) is -valently convex with respect to -differentiation of order .

Proof. From (28), we observe that . On the other hand, it is easy to verify that Now by logarithmic -differentiation we have Therefore, Taking the real parts on both sides of the above equation, we have Since for all , from (26) we get As , for all , we obtain from the above This completes the proof.

Corollary 9. Let , , , and for all , . If for all , then the -integral operator defined by (28) is -valently convex with respect to -differentiation in .

Proof. From (35) and (37) we easily find that .

Letting , , , , , , and in Theorem 8, we have the following.

Corollary 10. Let , , , and . If , then the integral operator is convex of order in .

5. Convexity of the Operator

Now, we prove the following convexity result with respect to -differentiation of the operator .

Theorem 11. Let , , , , and for all , . If then the -integral operator defined by (29) is -valently convex with respect to -differentiation of order .

Proof. From (29), we observe that . On the other hand, it is easy to verify that Now by logarithmic -differentiation we have Therefore, Taking the real parts on both sides of the above equation, we have Since for all , from (27) we get As , for all , we obtain from the above This completes the proof.

Corollary 12. Let , , , , and for all , . If for all , then the -integral operator defined by (29) is -valently convex with respect to -differentiation in .

Proof. From (43) and (45) we easily find that .

Letting , , , , , , and in Theorem 11, we have the following.

Corollary 13. Let , , , and . If , then the integral operator is convex of order in .

We remark in conclusion that, by suitably specializing the parameters in Theorems 8 and 11, we can deduce the results obtained in [15, 22, 23].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.