On Certain Subclasses of Analytic Multivalent Functions Using Generalized Salagean Operator

Abstract

We introduce and study two subclasses of multivalent functions denoted by and . Further, by using the method of differential subordination, certain inclusion relations between the two subclasses aforementioned are given. Moreover, several consequences of the main results are also discussed.

1. Introduction

Let denote the class of the functions of the form which are analytic in the open unit disc , and let denote .

A function is said to be multivalent starlike functions of order in , if it satisfies the following inequality: and we denote this class by .

A function is said to be multivalent convex functions of order in , if it satisfies the following inequality: and we denote this class by .

For a function , Goyal et al. [1] introduced the following generalized Salagean differential operator: If is given by (1), then from (5) and (6) we have

Remark 1. For , the differential operator reduces to Salagean differential operator [2].

Definition 2. Let be the class of functions that satisfy the condition where ), and let be the class of functions that satisfy the conditions where

Remark 3. By specifying different values, we have some well-known subclasses of the classes and appearing from the families of the classes and . (i) is the class of multivalent starlike functions of order . (ii) is the class of starlike functions of order . (iii) is the class of multivalent convex functions of order . (iv) is the class of convex functions of order . (v) is the subclass of Bazilević functions.

Let be denoted by the class In this investigation, we focus on certain inequalities consisting of the following differential operator : that generalizes the expression used in the definition of class and we receive several properties of the expression including relations between classes and .

In order to prove our main results, we will need the following lemmas due to Miller and Mocanu [3].

Lemma 4. Let and suppose that the function satisfies for all , and . If is analytic in and for all , then

Lemma 5. Let and suppose that the function satisfies for all , and . If is analytic in and for all , then

2. Main Results

Following the same techniques and procedure given by Goswami et al. [4], we have the following results.

Theorem 6. Let with for all , where is given by (10), and also let . If where , then where the powers are the principal ones.

Proof. Let the function be defined by From the assumptions with for all , we have that . By a simple manipulation, we have Now letting we have from (17) and (14) that Further, for any , , and , since , we also have which shows that for all , , and . Therefore, according to Lemma 4, we obtain . Hence, (15) is proven.

Theorem 7. Let with for all , where is given by (10), and also let . If where and then

Proof. Suppose that Then, is analytic in . It is easily seen from (23) that Further, since it leads to Also, for any , and , we have that is, . Finally, by Lemma 5, we obtain that . The proof of Theorem 7 is complete.

3. Corollaries and Consequences

We will discuss some interesting consequences of the main theorems that extend some previous results obtained in ([4, 5]).

Putting in Theorems 6 and 7, we get the following corollaries.

Corollary 8. Let with for all , where is given by (7), and also let . If where , then where the powers are the principal ones.

Corollary 9. Let with for all , where is given by (7), and also let . If where and then where the powers are the principal ones.

Taking and in Corollaries 8 and 9, respectively, we obtain the following special cases.

Corollary 10. Let with for all , where is given by (7), and also let . If where , then where the powers are the principal ones.

Corollary 11. Let with for all , where is given by (7), and also let . If where and then where the powers are the principal ones.

Next, upon taking , in Theorems 6 and 7, we obtain the following results.

Corollary 12. Let with for all , where is given by (7), and also let . If where , then where the powers are the principal ones.

Corollary 13. Let with for all , where is given by (7), and also let . If where and then where the powers are the principal ones.

Taking and in Corollaries 12 and 13, respectively, we obtain the following special cases.

Corollary 14. Let with for all , where is given by (7), and also let . If where , then where the powers are the principal ones.

Corollary 15. Let with for all , where is given by (7), and also let . If where and then where the powers are the principal ones.

In the next result, we will find the relation between and . For this purpose, taking and in Theorem 7, we obtain the following result.

Corollary 16. Let with for all , where is given by (10), and also let . If where and then .

Taking and in the above corollary, we get the next special result.

Corollary 17. Let with for all , where is given by (10), and also let . If where and then .

Again, for the special cases of and , Theorems 6 and 7 reduce at once to some results obtained by [4, 5].

Remark 18. Taking and in (7) and and in (10), we get a known result obtained by Irmak et al. [5].

Remark 19. Taking in (7) and in (10), we get a known result obtained by Goswami et al. [4].

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

Both authors agreed with the contents of the paper.

Acknowledgments

The authors would like to acknowledge and appreciate the financial support received from Universiti Kebangsaan Malaysia (UKM) under Grant ERGS/1/2013/STG06/UKM/01/2. The first author is also supported by ZAMALAH scheme grant under postgraduate center, UKM.