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International Journal of Analysis
Volume 2013 (2013), Article ID 153128, 10 pages
http://dx.doi.org/10.1155/2013/153128
Research Article

A New Class of Analytic Functions Defined by Using Salagean Operator

1Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt
2Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 33516, Egypt
3Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt

Received 14 August 2012; Revised 6 November 2012; Accepted 12 November 2012

Academic Editor: Yaozhong Hu

Copyright © 2013 R. M. El-Ashwah 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

We derive some results for a new class of analytic functions defined by using Salagean operator. We give some properties of functions in this class and obtain numerous sharp results including for example, coefficient estimates, distortion theorem, radii of star-likeness, convexity, close-to-convexity, extreme points, integral means inequalities, and partial sums of functions belonging to this class. Finally, we give an application involving certain fractional calculus operators that are also considered.

1. Introduction

Let denote the class of functions of the form that are analytic and univalent in the open unit disc .

For , Salagean [1] introduced the following differential operator: , , .

We note that

Definition 1 (subordination principle). For two functions and , analytic in , we say that the function is subordinate to in and write , if there exists a Schwarz function , which (by definition) is analytic in with and , such that . Indeed it is known that Furthermore, if the function is univalent in , then we have the following equivalence [2, page 4]:

Definition 2 (see [3]). Let denote the subclass of consisting of functions of the form (1) and satisfy the following subordination: Specializing the parameters , and , we obtain the following subclasses studied by various authors:
(i) (see Eker and Owa [4]);
(ii) (see Shams et al. [5, 6]);
(iii)  (see Janowski [7] and Padmanabhan and Ganesan [8]).

Also we note that

Let denote the subclass of functions of of the form

Further, we define the class by

For suitable choices of the parameters , and , we can get various known or new subclasses of . For example, we have the following:(i) (see Rosy and Murugusundaramoorthy [9] and Aouf [10]);(ii) and (see Bharati et al. [11]);(iii) and (see Silverman [12]).

2. Coefficient Estimates

Unless otherwise mentioned, we assume in the reminder of this paper that and .

Now, we will need the following lemma which gives a sufficient condition for functions belonging to the class .

Lemma 3 (see [13]). A function of the form (1) is in the class if

In Theorem 4, it is shown that the condition (12) is also necessary for functions of the form (10) to be in the class .

Theorem 4. Let . Then if and only if

Proof. In view of Lemma 3, we only need to prove the only if part of Theorem 4. Since , for functions , we can write then Since , then we obtain Now choosing to be real and letting , we obtain Or, equivalently This completes the proof of Theorem 4.

Remark 5. (i) The result obtained by Theorem 4 corrects the result obtained by Li and Tang [3, Theorem 1].
(ii) Putting and in Theorem 4, we correct the result obtained by Eker and Owa [4, Theorem  2.1].
(iii) Putting , and in Theorem 4, we obtain the result obtained by Rosy and Murugusudaramoorthy [9, Theorem 2].

Corollary 6. Let the function be defined by (10) and let it be in the class . Then The result is sharp for the function

3. Distortion Theorems

Theorem 7. Let the function defined by (10) be in the class . Then The result is sharp.

Proof. In view of Theorem 4, since is an increasing function of , we have that is Thus we have Similarly, we get Finally the result is sharp for the function at and . This completes the proof of Theorem 7.

Theorem 8. Let the function defined by (10) be in the class . Then The result is sharp.

Proof. Similarly is an increasing function of , in view of Theorem 4, we have that is Thus we have Similarly Finally, we can see that the assertions of Theorem 8 are sharp for the function defined by (27). This completes the proof of Theorem 8.

4. Radii of Starlikeness, Convexity, and Close-to-Convexity

In this section radii of close-to-convexity, starlikeness, and convexity for functions belonging to the class are obtained.

Theorem 9. Let the function defined by (10) be in the class ; then(i) is starlike of order in , where (ii) is convex of order in , where (iii)is close-to-convex of order in , where Each of these results is sharp for the function given by (20).

Proof. It is sufficient to show that where is given by (33). Indeed we find from (10) that Thus we have if and only if But, by Theorem 4, (39) will be true if that is, if Or This completes the proof of (33).

To prove (34) and (35) it is sufficient to show that respectively.

5. Extreme Points

Theorem 10. Let Then if and only if it can be expressed in the following form: where

Proof. Suppose that Then, from Theorem 4, we have Thus, in view of Theorem 4, we find that .
Conversely, let us suppose that , then, since Set Thus clearly, we have This completes the proof of Theorem 10.

Corollary 11. The extreme points of the class are given by

6. Integral Means Inequalities

In 1925, Littlewood [14] proved the following subordination lemma.

Lemma 12. If the functions and are analytic in with then for and , We now make use of Lemma 12 to prove Theorem 13.

Theorem 13. Suppose that ,, and is defined by Then for , we have

Proof. For , (55) is equivalent to prove that By applying Littlewood’s subordination lemma (Lemma 12), it would suffice to show that By setting and using (13), we obtain This completes the proof of Theorem 13.

7. Partial Sums

In this section partial sums of functions in the class are obtained, also we will obtain sharp lower bounds for the ratios of real part of to .

Theorem 14. Define the partial sums and by Let the function be given by (1) and let it satisfy the condition (12) and where, for convenience, Then

Proof. For the coefficients given by (64) it is not difficult to verify that Therefore we have By setting and applying (68), we find that Now if From the condition (12), it is sufficient to show that which is equivalent to which readily yields the assertion (65) of Theorem 14. In order to see that gives sharp result, we observe that for that as . Similarly, if we take and making use of (68), we can deduce that which leads us immediately to the assertion (66) of Theorem 14.
The bound in (66) is sharp for each with the extremal function given by (75). Then the proof of Theorem 14 is completed.

8. Distortion Theorems Involving Fractional Calculus

In this section, we will prove several distortion theorems for functions belonging to the class . Each of these theorems would involve certain operators of fractional calculus (i.e., fractional integrals and fractional derivatives), which are defined as follows (see, for details, [1518]). For our present investigation, we recall the following definitions.

Definition 15. The fractional integral of order is defined, for a function , by where the function is analytic in a simply connected domain of the complex -plane containing the origin, and the multiplicity of is removed by requiring to be real when .

Definition 16. The fractional derivative of order is defined, for a function , by where the function is constrained, and the multiplicity of is removed as in Definition 15.

Definition 17. Under the hypotheses of Definition 16, the fractional derivative of order is defined, for a function , by Using Definitions 15, 16, and 17, we obtain in terms of Gamma functions.

Theorem 18. Let the function defined by (10) be in the class . Then The results are sharp.

Proof. Let where Since is a decreasing function of , we can write Furthermore, in view of Theorem 4, we have Then Therefore, by using (85) and (87), we can see that and similarly which prove Theorem 18.
Finally, the equalities are attained for the function defined by or, equivalently, by given by (27).
Then the results are sharp, and the proof of Theorem 18 is completed.

Corollary 19. Under the hypothesis of Theorem 20, is included in a disk with its center at the origin and radius given by

Theorem 20. Let the function defined by (10) be in the class . Then Each of these results is sharp.

Proof. Let where Since is a decreasing function of , we can write Furthermore, in view of Theorem 4, we have Then Therefore, by using (95) and (97), we can see that and similarly which together prove the two assertions of Theorem 20.
Finally, the equalities are attained for the function defined by or, equivalently, by given by (27).
Then the result is sharp, and the proof of Theorem 20 is completed.

Corollary 21. Under the hypothesis of Theorem 20, is included in a disk with its center at the origin and radius given by

Acknowledgment

The authors thank the referee for his valuable suggestions which led to improvement of this study.

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