Certain Families of Multivalent Analytic Functions Associated with Iterations of the Owa-Srivastava Fractional Differintegral Operator

Abstract

By making use of a multivalent analogue of the Owa-Srivastava fractional differintegral operator and its iterations, certain new families of analytic functions are introduced. Several interesting properties of these function classes, such as convolution theorems, inclusion theorems, and class-preserving transforms, are studied.

1. Introduction

Let denote the class of analytic functions in the open unit disk and let be the subclass of consisting of functions represented by the following Taylor-Maclaurin’s series:

In a recent paper Patel and Mishra [1] studied several interesting mapping properties of the fractional differintegral operator: defined by where is given by (2). In the particular case and the fractional-differintegral operator was earlier introduced by Owa and Srivastava [2] (see also [3]) and this is popularly known as the Owa-Srivastava operator [4–6]. Moreover, for and , was investigated by Srivastava and Aouf [7] which was further extended to the range ,   by Srivastava and Mishra [8]. The following are some of the interesting particular cases of : Furthermore The -iterates of the operator are defined as follows: and, for , Similarly, for , represented by (2), let the operator be defined by the following:

and, for , Very recently Srivastava et al. [6] considered the composition of the operators and and introduced the following operator: That is, for , given by (2), we know that The transformation includes, among many, the following two previously studied interesting operators as particular cases. (i)For , , , the fractional derivative operator was recently introduced and investigated by Al-Oboudi and Al-Amoudi [9, 10], in the context of functions represented by conical domains.(ii)For , , , , is the Sălăgean operator [11], which is, in fact, the -iterates of the popular Alexander’s differential transform [12].

We next recall the definition of subordination. Suppose that and in is univalent in . We say that is subordinate to in if and . Considering the function , it is readily checked that satisfies the conditions of the Schwarz lemma and In a broader sense the function is said to be subordinate to the function ( need not be univalent in ), written as if condition (15) holds for some Schwarz function (see [12] for details). We also need the following definition of Hadamard product (or convolution). For the functions and in , given by the following Taylor-Maclaurin’s series their Hadamard product (or convolution) is defined by It is easy to see that .

The study of iterations of entire and meromorphic functions, as the number of iterations tends to infinity, is a popular topic in complex analysis. However, investigations have been initiated only recently regarding iterations of certain transforms defined on classes of analytic and meromorphic functions. For example, Al-Oboudi and Al-Amoudi [9, 10] investigated properties of certain classes of analytic functions associated with conical domains, by making use of the operator . Their work generalized several earlier results of Srivastava and Mishra [13]. This theme has been further pursued in our more recent papers [6, 14, 15]. In the sequel to these current investigations, in the present paper, we define the following subclass of associated with the iterated operator and investigate its several interesting properties. Our work is also motivated by earlier works in [16–20], connecting subordination and Hadamard product.

Definition 1. The function is said to be in the class if the following subordination condition is satisfied: where is a complex number and is an analytic convex univalent function in .

The function class includes several previously studied subclasses of as particular cases. For example,(i)for ,   the class was recently studied by Liu [17];(ii)for ;  ;  ;  ; and , the class was earlier investigated by R. Singh and S. Singh [21];(iii)for ;  ;  ;  ; and   ,   reduces to which was investigated by Yang [22];(iv)for ;  ;  ;  ; and ,   reduces to , the class introduced and studied by Zhongzhu and Owa [23] and Jinlin [24]. In the present paper we primarily focus on a variety of convolution theorems for the class . We also find inclusion theorems and study behavior of the Libera-Livingston integral operator.

2. Some More Definitions and Preliminary Lemmas

We need the following definitions and results for the presentation of our results. Let and denote, respectively, the classes of univalent convex functions of order and starlike functions of order (see [12] for details). The function is said to be in the class consisting of prestarlike functions of order [25] if It is readily seen that Furthermore, it is well known [25] that We will also need the following lemmas in order to derive our main results.

Lemma 2 (see [26]; also see [27]). Let be analytic in and let be analytic and convex univalent in with . If where and , then and is the best dominant of (24).

Lemma 3 (see [25]). Let ,  , and . Then, for any analytic function in , where denotes the closed convex hull of .

Lemma 4 (see [28]). Let and be univalent convex functions in , and let and be functions in . Suppose that and in . Then in .

The following well known result is a consequence of the principle of subordination and can be found, for example, in [12, 29].

Lemma 5. Let the function satisfy and   . Then

3. Convolution Results

We state and prove the following convolution results.

Theorem 6. Let , and suppose that in satisfies the following: Then

Proof. For every and in , we have where Now, if , then
Furthermore, condition (27) is equivalent to Therefore, an application of Lemma 4 in (29) yields
This shows that . The proof of Theorem 6 is completed.

Remark 7. Taking ,  , in Theorem 6 we get a recent result of Liu ([17], Theorem 3). The choices ,  ,  ,  , and    yield a result of Yang ([22], Theorem 4).

Corollary 8. Let the function given by (2) be a member of and Then the function is in the class .

Proof. Let . We note that where Also, for , it is well known [18] that In view of (36) and (38), an application of Theorem 6 gives The proof of Corollary 8 is completed.

Theorem 9. Let the function in be such that is a prestarlike function of order . If , then

Proof. Let and . Then (29) gives where is defined as in (30). We noted in the proof of Theorem 6 that . Since ,  , and is convex univalent in , an application of Lemma 3 in (41) yields the following: Therefore, . The proof Theorem 9 is completed.

Taking in Theorem 9 we get the following.

Corollary 10. Let and suppose that in is such that . Then In particular, if is univalent convex then .

In the following theorem we discuss convolution properties of the function class when is a right half plane mapping.

Theorem 11. Let and suppose that each of the functions is a member of the class , where If is defined by the following then , where and is given by The bound on is the best possible.

Proof. We consider the case . Suppose that ,  , where is given by (44). By setting we see, in the light of Definition 1, that A routine calculation yields the following: Now, if is defined by (45), then using (50) we get that where Since by using Lemma 4, we get A simple calculation gives that Therefore, by the Lindelöff principle of subordination we have Since by using (52) in conjunction with (56) and Lemma 4 we get the following: This proves that , where the function is given by (46).
In order to show that the value of is the least possible, we take the functions    defined by for which we have Hence, for , given by (45), we obtain Finally, for the case , the proof of Theorem 11 is simple, so we choose to omit the details involved.

4. Properties of the Libera-Livingston Transform

For the function , the function defined by is popularly known as the Libera-Livingston transform of . We state and prove the following.

Theorem 12. Let . Then the function defined by (63) is in the class , where Consequently, . The function is the best dominant in (64).

Proof. We define the function on by Differentiating both the sides of (65) with respect to , we get Also, the defining relation (63) yields Now, a routine calculation using (65), (66), and (67) gives Since , we get the following from the preceding equation (68): Therefore, by applying Lemma 2 we have This last subordination (70) is equivalent to The proof of Theorem 12 is completed.

Theorem 13. Let and suppose that the function is defined as in (63). If then , where The function is the best dominant in (73).

Proof. We define the function on by Differentiation of both the sides of (74) combined with the identity gives By making use of (67) we simplify the subordinate of (72) as follows:
Next, by using (74) and (76), the above identity further simplifies to the following: The subordination (72) is thus equivalent to
Therefore, an application of Lemma 2 yields the assertion of Theorem 13. The proof of Theorem 13 is completed.

5. Inclusion Theorems

Theorem 14. Let ,  , and . If , then , where The bound is sharp when .

Proof. Let . By setting we have Therefore, by using Definition 1, we get and an application of Lemma 2 yields where It follows from (85) that if , where is given by (80), then Equivalently, Since and are both convex univalent functions in , using Lemma 4, we obtain from (84) that Therefore in view of (81) we have Or equivalently, .
In order to prove that the bound on is the best possible, set and let the function be defined on by It can be readily verified that and Thus . Also, for , and , we have which shows that Hence the bound cannot be increased when . The proof of Theorem 14 is completed.

Taking , we use the notation in the following theorem.

Theorem 15. Let . Then

Proof. Let and suppose that . We define
The function is analytic in with . Differentiating the expressions on both the sides of (96) with respect to and using Definition 1, we have Hence an application of Lemma 2 yields Noting that and is convex univalent in , it follows from (96), (97), and (98) that Thus . Consequently, The proof of Theorem 15 is completed.

Remark 16. The particular case of Theorem 15 gives a result of Liu ([17], Theorem 1).

Conflict of Interests

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