In this paper, we introduce a new generalized class of analytic functions involving the Mittag-Leffler operator and Bazilevic̆ functions. We examine inclusion properties, radius problems, and an application of the generalized Bernardi–Libera–Livingston integral operator for this function class.

1. Introduction

Let be the family of all functions of the formwhich are analytic in the open unit disc . Denote by the subfamily of consisting of functions that are univalent in . Let , , and be the well-known subclasses of consisting of functions that are, respectively, convex, starlike (with respect to the origin), and close-to-convex in . The class consists of analytic functions that are analytic in and satisfies the conditions and in and is also well-known in the theory of univalent functions. For definitions, properties, and history of these classes, one may refer to a survey article by the first author [1, 2]. Recently, Ali et al. [3] and Anand et al. [4] studied these classes to find various radius problems.

The Mittag-Leffler function , defined bywas introduced in 1903 by Mittag-Leffler [5, 6] in connection with his method of summation of some divergent series. A general form of this special function (2) given bywhich was studied by Wiman [7] in 1905. During these last twenty-five years, interest in Mittag-Leffler type functions (2) and (3) has significantly increased among engineers and scientists due to their applications in numerous applied problems, such as fluid flow, diffusive transport skin to diffusion, electric networks, probability, and statistical distribution theory. For detailed account of various properties and references related to applications, one may refer to [8, 9]. Motivated by Sivastava and Tomovski [10], Attiya [11] studied certain applications of the generalized Mittag-Leffler operator involving differential subordination.

Corresponding to the function , Elhaddad et al. [12] introduced the Mittag-Leffler linear operator given bywhere , , , , and when and . From (4), the following recurrence formula can be easily obtained:where and . For suitable values of the parameters , and , we may get several linear operators; for example,(1)For and , we get Al-Oboudi operator [13].(2)For , , and , we get Sălăgean operator [14].(3)For and , we get the operator

Let us denote by or briefly denote by , a class of functions for which , , and real numbers with such thatwhere powers are taken as principal values. Bazilevich [15] proved that . In fact, it is known that . For and , Ponnusamy and Karunakaran [16] showed that

For , these authors observed that

In view of (7), Singh [17] observed that , , , and are subclasses of . For further details, one may refer to [15, 18].

In 1976, Padmanabhan and Parvatham [19] introduced the class of analytic functions defined in satisfying the properties andwhere , , and . . In fact, he proved the following important result.

Lemma 1 (see [19]). If , thenwhere is a function with bounded variation on such that

From (10), it is observed that if and only if there exists such that (see [18])

We note that, for , we obtain the class defined by Pinchuk [20]. For , we get the class of analytic functions with positive real part greater than , and for and , we have the class of functions with positive real part.

Motivated by many researchers in [57, 1012, 15, 17, 18, 21], we introduce the following generalized class of Mittag-Leffler–Bazilevic̆ operator involving the class .

Definition 1. Let , , , , , and be such that . Then, a function given by (1) is in the class if it satisfies the conditionwhere .
This new class involves several subclasses of the family of Bazilevic̆ functions defined by (7); see [15]. For example, , , and are subclasses of ; see [15, 17]. In fact, for different values of and , the class reduces to many important subclasses studied by various researchers. For instance,(i)Setting , , , and , we get (see [18])(ii)Setting , , , , , and , we get (see [15])(iii)Setting , , , , , , and , we get (see [21])In view of the above examples, we conclude that the notion of generalized class unifies several known subclasses of .
In this paper, we study various properties of the class . In particular, we investigate inclusion properties, radius problem, and an application of the generalized Bernardi–Libera–Livingston integral operator for this function class.
We list some preliminary lemmas required for proving our main results.

Lemma 2 (see [22]). Let and , and suppose is a complex function satisfying the following conditions:(i) is continuous in a domain (ii) and (iii), whenever and

If is an analytic function in such that and for , then in .

Lemma 3 (see [23]). If is in , and if is a complex number satisfying , then implies thatwhere is an increasing function of and . This estimate cannot be improved in general.

2. Inclusion Properties

In this section, we examine some inclusion properties for the class .

Theorem 1. Let and . Then,where is given by

Proof. In view of (20), letwhere andThus, by using (14) and (22), we obtainTaking logarithmic differentiation of (22), we getBy using the identity (5) in the last expression, we obtainSubstituting (26) into (24) and using (13), we arrive atSince , it follows thatThat is,To prove the theorem, we will show that . We form the functional by taking and such thatUsing (29), it is easy to show that the first two conditions of Lemma 2 are satisfied. To verify condition (iii), we obtainwhere . Now, ifIt follows from , given by (21) and that . In view of Lemma 2, for , , we get . This proves the result.
For , , , and , Theorem 1 reduces to the following new result.

Corollary 1. Let and . Then, , where is given by

By using the inclusion relation given in Theorem 1, we prove the following result.

Theorem 2. Let and . Then, .

Proof. Let . Then, we haveIn view of Theorem 1, we conclude thatThus, for , we haveBecause the class is a convex set (see [18]), it follows that the right side of (36) belongs to , and therefore .
For , , , and , Theorem 2 reduces to the following inclusion result.

Remark 1 (see [18]). Let and . Then, .

3. Radius Problem

In this section, we examine certain radius problems.

Theorem 3. If a function satisfiesthen for , where

Proof. In view of (37), we havewhere . Hence, by using (5) and (39), we easily getwhere and .
Now, by using well-known estimates (see [2]) for the class given bywe haveThe right hand side of the last inequality is positive if , where is given by (38).
Letting , , , and , Theorem 3 reduces to the following new result.

Corollary 2. If a function satisfies , then for , where

Remark 2. If , , and in Theorem 3, then is in for . This result was proved in Theorem 3.4 in [24].

4. Application of an Integral Operator

In this section, we consider an application of the generalized Mittag-Leffler operator given by (4) involving the generalized Bernardi–Libera–Livingston integral operator given by

From this operator, we easily get

For several special cases of this operator and related operators, one may refer to a survey article by the first two authors [25] and related references therein.

Theorem 4. Let and be given by (44). Ifthenwhere is given by

Proof. Consider the functionDifferentiating both sides and using (45), we getIf we use the identity given by (46), we obtainThis implies thatBy using Lemma 3, we see that , where is given by (48). Thus, we arrive at . This completes the proof.
Setting , , , and in operator , Theorem 4 gives the following result.

Corollary 3. Let and be given by (44). Ifthen , where is given by

5. Conclusion

We conclude our investigation by remarking that the defined new generalized class of analytic functions involving the Mittag-Leffler operator and Bazilevic̆ functions gives various well-known subclasses of Bazilevic̆ functions as particular cases which in turn yields many known results as corollaries.

Data Availability

No data were used to support this study.


A preprint of an earlier version is available at https://arxiv.org/abs/2109.135 09.

Conflicts of Interest

The authors declare that they have no conflicts of interest.