ISRN Mathematical Analysis

Volume 2013 (2013), Article ID 384170, 8 pages

http://dx.doi.org/10.1155/2013/384170

Research Article

Some Inclusion Relationships of Certain Subclasses of -Valent Functions Associated with a Family of Integral Operators

M. K. Aouf,¹ R. M. El-Ashwah,² and Ahmed M. Abd-Eltawab³

¹Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

²Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt

³Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt

Received 16 June 2013; Accepted 24 August 2013

Academic Editors: G. Gripenberg and B. Wang

Copyright © 2013 M. K. Aouf et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By making use of the new integral operator , we introduce and investigate several new subclasses of -valent starlike, -valent convex, -valent close-to-convex, and -valent quasi-convex functions. In particular, we establish some inclusion relationships associated with the aforementioned integral operators. Some of the results established in this paper would provide extensions of those given in earlier works.

1. Introduction

Let denote the class of functions of the form

which are analytic and -valent in the unit disc , and and let .

A function is said to be in the class of -valent starlike functions of order in if and only if

The class was introduced by Patil and Thakare [1].

Owa [2] introduced the class of -valent convex of order in if and only if

It is easy to observe from (2) and (3) that

We denote by and where and are the classes of -valently starlike functions and -valently convex functions, respectively, (see Goodman [3]).

For a function , we say that if there exists a function such that

Functions in the class are called -valent close-to-convex functions of order and type . The class was studied by Aouf [4] and the class was studied by Libera [5].

Noor [6, 7] introduced and studied the classes and as follows.

A function is said to be in the class of quasi-convex functions of order and type if there exists a function such that

It follows from (5) and (6) that

For functions given by (1) and given by

the Hadamard product (or convolution) of and is given by

For the function , we introduced the operator as follows:

From (10), it is easy to verify that

Remark 1. Consider (i) For ,

where the operator was introduced and studied by Liu and Owa [8], and , where the operator was introduced and studied by Jung et al. [9];

(ii) For and ,

where is the familiar integral operator, which was defined by Cho and Kim [10]. The operator was introduced by Bernardi [11] and we note that was introduced and studied by Libera [12] and Livingston [13].

The main object of this paper is to investigate the various inclusion relationships for each of the following subclasses of the normalized analytic function class which are defined here by means of the operator given by (10).

Definition 2. In conjunction with (2) and (10),

Definition 3. In conjunction with (3) and (10),

Definition 4. In conjunction with (5) and (10),

Definition 5. In conjunction with (6) and (10),

Remark 6. Consider (I) For , in the above definitions, we have

(II) For and , in the above definitions, we have

(III) For , in the above definitions, we have

where the classes , , , and were introduced and studied by Gao et al. [14].

In order to establish our main results, we need the following lemma due to Miller and Mocanu [15].

Lemma 7 (see [15]). Let be a complex-valued function such that

and let , . Suppose that satisfies the following conditions:

(i) is continuous in ;(ii) and ;(iii) for all such that .

Let

be analytic in such that for all . If

then

2. The Main Results

In this section, we give several inclusion relationships for analytic function classes, which are associated with the integral operator . Unless otherwise mentioned, we assume throughout this paper that , ,  , , and .

Theorem 8. Let . Then

Proof. Let and set

where is given by (22). By using identity (11), we obtain

Differentiating (27) logarithmically with respect to , we obtain

We now choose and , and define the function by

Then, clearly satisfies the following conditions:

(i) is continuous in ;(ii) and ;(iii)for all such that we have

which shows that the function satisfies the hypotheses of Lemma 7. Consequently, we easily obtain the inclusion relationship (25).

Theorem 9. Let , . Then

Proof. Let . Then, from Definition 3, we have

Furthermore, in view of the relationship (4), we find that

that is, that

Thus, by using Definition 2 and Theorem 8, we have

which implies that

The proof of Theorem 9 is thus completed.

Theorem 10. Let , . Then

Proof. Let . Then there exists a function such that

We put

so that we have

We next put

where is given by (22). Thus, by using identity (11), we obtain

Since , then from Theorem 8 we have , so that we can put

where

Then

We thus find from (41) that

Differentiating both sides of (46) with respect to , we obtain

By substituting (47) into (45), we have

The remainder of our proof of Theorem 10 is much akin to that of Theorem 8. We, therefore, choose to omit the details involved.

Theorem 11. Let , . Then

Proof. Just as we derived Theorem 9 as a consequence of Theorem 8 by using the equivalence (4), we can also prove Theorem 11 by using Theorem 10 in conjunction with the equivalence (7).

Our main results in Theorems 8–11, can thus be applied with a view to deducing the following corollaries.

Taking in Theorems 8–11 above, we obtain the following corollary.

Corollary 12. Let , . Then

Remark 13. Taking in Corollary 12, we obtain the results obtained by Gao et al. [14, Theorems 1-4].

Taking and in Theorems 8–11, we obtain the following corollary.

Corollary 14. Let , . Then

Remark 15. Taking in Corollary 14, we obtain the results obtained by Gao et al. [14, Corollary  1–4].

3. A Set of Integral-Preserving Properties

In this section, we present several integral-preserving properties of the analytic function classes introduced here. In order to obtain the integral-preserving properties involving the integral operator defined by (13).

Theorem 16. Let be any real number and . If , then , where is defined by (13).

Proof. From (13), we have

Let and set

where is given by (22). By using identity (52), we obtain

Differentiating (54) logarithmically with respect to , we obtain

We now choose , and , and define the function by

It is easy to see that the function satisfies the conditions of Lemma 7, and the remaining part of the proof of Theorem 16 is similar to that of Theorem 8.

Taking in Theorem 16, we obtain the following corollary.

Corollary 17. Let be any real number and . If , then , where is defined by (13).

Theorem 18. Let be any real number and . If , then , where is defined by (13).

Proof. By applying Theorem 16 in conjunction with the equivalence (4), it follows that

which proves Theorem 18.

Taking in Theorem 18, we obtain the following corollary.

Corollary 19. Let be any real number and . If , then , where is defined by (13).

Theorem 20. Let be any real number and . If , then , where is defined by (13).

Proof. Let . Then there exists a function such that

We put

so that we have

We next put

where is given by (22). Thus, by using identity (52), we obtain

Since , then from Theorem 16 we have , so that we can put

where

Then

We thus find from (61) that

Differentiating both sides of (66) with respect to , we obtain

By substituting (67) into (65), we have

The remainder of our proof of Theorem 20 is much akin to that of Theorem 8. We, therefore, choose to omit the details involved.

Taking in Theorem 20, we obtain the following corollary.

Corollary 21. Let be any real number and . If , then , where is defined by (13).

Theorem 22. Let be any real number and . If , then , where is defined by (13).

Proof. Just as we derived Theorem 18 as a consequence of Theorem 16 by using the equivalence (4), we can also prove Theorem 22 by using Theorem 20 in conjunction with the equivalence (7).

Taking in Theorem 22, we obtain the following corollary.

Corollary 23. Let be any real number and . If , then , where is defined by (13).

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