International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 783084 | https://doi.org/10.1155/2011/783084

Lokesh Vijayvargy, Pranay Goswami, Bushra Malik, "On Some Integral Operators for Certain Classes of -Valent Functions", International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 783084, 10 pages, 2011. https://doi.org/10.1155/2011/783084

On Some Integral Operators for Certain Classes of -Valent Functions

Academic Editor: H. S. V. De Snoo
Received03 Dec 2010
Revised09 Feb 2011
Accepted03 Mar 2011
Published02 May 2011

Abstract

We study some generalized integral operators for the classes of p-valent functions with bounded radius and boundary rotation. Our work generalizes many previously known results. Many of our results are best possible.

1. Introduction

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

Let and be analytic functions in we say that is subordinate to , written as if there exists a Schwarz function in , with and , such that In particular, when is univalent, then the above subordination is equivalent to For functions , given by we define the Hadamard product (or convolution) of and by Janowski [1] defined the class as follows.

Let be a function, analytic in , with . Then is said to belong to the class , , if and only if, for , Or equivalently, we can say that , , if and only if, Geometrically, is in the class , if and only if, and the image of lies inside the open disc centered on the real axis with diameter end points, Clearly .

In the recent paper, Noor [2] introduced the class . We define it as follows. Let , be the class of functions with and satisfying the property where is a real-valued function of bounded variation on and and .

The classes and are related to the class and can be defined as We define a class as follows.

Let , , , denote the class of -valent analytic functions that are represented by where . For and , it reduces to the class and is the class of -valent analytic functions with , . Taking , , and , we have (see [3]), and is the class of functions with positive real part.

Definition 1.1. A function , analytic in , and given by (1.1) is said to be in the class ; , , if and only if, For , is introduced and studied by Noor [4]. We note that where is the class of functions with bounded radius rotation (see [5]). For , we have where is the class of -valent starlike functions. Similarly, we can define the class as follows.

Definition 1.2. A function , analytic in , and given by (1.1) is said to be in the class ; , , if and only if, It is clear that For , is the class introduced and studied by Noor [4]. It is easy to see that, where is the class of functions with bounded boundary rotation see [5]. Also where is the class of -valent convex functions.

Very recently, Frasin [6], introduced the following general integral operators for -valent functions, Clearly, we may see that for , these operators become the general integral operators introduced and studied by Breaz and Breaz [7] and Breaz et al. [8], (see also [9, 10]).

For , in (1.20), we obtain the integral operator studied in [11] and for , , in (1.21), we obtain the integral operator , studied in [12].

2. Main Results

Lemma 2.1. Let, , ,with . If then where , denotes the Gauss hypergeometric function. From (2.2), we can deduce the sharp result , where is defined in (2.4). This result is a special case of one given in [11].

Proof. To prove this Lemma we use Theorem 3.2j of [11, page 97]. Take , and where and .
Since is convex to apply Theorem 3.2j of [11, page 97] we only need to determine condition .
The range of under is a half plane. In order to satisfy the required condition this half plane needs to lie in the right half plane. This requirement will be satisfied if and .Or we can write it as When , , these conditions imply that , and if , then . Hence all the conditions of Theorem 3.2j of [11, page 97] are satisfied for , with , thus we have the required result.

To show that the solution can be represented in terms of hypergeometric functions we take , , in Theorem 3.3d of [11, page 109].

Lemma 2.2. Let , , . Then in , where This result is sharp.

Proof. Let for , , we have where , are analytic in with , , .
We define By using (2.8), with convolution technique, see [13], we have This implies that, Logarithmic differentiation of (2.8) yields, Since , , thus By using Lemma 2.1 (for and ), we deduce that , where is given in (2.7). This estimate is best possible because of the best dominant property of function , where

For , we have the sharp result proved in [14].

We begin with the following theorem.

Theorem 2.3. (i) Let , for all , and, . Then the integral operator in , where , .
(ii) Let , for all with , . If , then the integral operator defined by (1.20) also belongs to the class in , where is defined by (2.7). This result is sharp.

Proof (i). From (1.20), we can see that in , and Differentiating logarithmically and multiplying by , we obtain, Thus, we have or where , for all .
Since is a convex set, see [15], it follows that, where and therefore, This proves the result.

Substituting , in Theorem 2.3(i), we have the following corollary.

Corollary 2.4. Let , for all , , . Then the integral operator in .

Remark 2.5. Letting , , and in Corollary 2.4, we obtain a result due to Noor [4].

For , , and in Theorem 2.3(i), we have the following.

Corollary 2.6. Let in , , . Then the integral operator , .

Proof (ii). Taking , , with , we have for all , using part (i) of Theorem 2.3, we have Now using Lemma 2.2 for , implies that The sharpness of the result is clear from the function defined by (2.14).

For , we have the following corollary.

Corollary 2.7. Let , for all , with and , . Then the integral operator defined by (1.20) also belongs to the class in , where

Remark 2.8. Letting , , and in Corollary 2.7, we have the sharp result proved in [14].

For , , and , we have

Theorem 2.9. (i) Let , for all . If , then the integral operator defined by (1.21), also belongs to the class in , where , .
(ii) Let for, and , for all with , , . Then the integral operator in , where is defined by (2.7). This result is sharp.

Proof (i). From definition (1.20), we have or where , for all .
Since is a convex set, see [15], it follows that, where and therefore, This implies that .

Letting in Theorem 2.9(i), we have the following corollary.

Corollary 2.10. Let , for all and , . If , then in .

Proof (ii). Taking , , we have for all Now using part (i) of Theorem 2.9, we have Now using Lemma 2.2, for , we have The sharpness of the result is clear from the function defined by (2.14).

For , we have the following corollary.

Corollary 2.11. (i) Let , , , with and, . Then in , where and defined in (2.24).
Also for , , we have.
(ii) If for all , then in .

Acknowledgments

Authors are thankful to anonymous referees for their very constructive comments to improve this paper. Second author (P. Goswami) is also grateful to Professor S. P. Goyal for his guidance and constant encouragement.

References

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Copyright © 2011 Lokesh Vijayvargy 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.


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