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

R. M. El-Ashwah,¹ M. K. Aouf,² A. A. M. Hassan,³ and A. H. Hassan³

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, [15–18]). 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.

References

G. S. Salagean, “Subclasses of univalent functions,” in Complex Analysis: Fifth Romanian-Finnish Seminar, vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, Berlin, UK, 1983.

S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, vol. 225 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.

S.-H. Li and H. Tang, “Certain new classes of analytic functions defined by using the Salagean operator,” Bulletin of Mathematical Analysis and Applications, vol. 2, no. 4, pp. 62–75, 2010.

S. S. Eker and S. Owa, “Certain classes of analytic functions involving Salagean operator,” Journal of Inequalities in Pure and Applied Mathematics, vol. 10, no. 1, pp. 12–22, 2009.

S. Shams, S. R. Kulkarni, and J. M. Jahangiri, “On a class of univalent functions defined by Ruschweyh derivatives,” Kyungpook Mathematical Journal, vol. 43, no. 4, pp. 579–585, 2003.

S. Shams, S. R. Kulkarni, and J. M. Jahangiri, “Classes of uniformly starlike and convex functions,” International Journal of Mathematics and Mathematical Sciences, no. 53, pp. 2959–2961, 2004.

W. Janowski, “Some extremal problems for certain families of analytic functions,” Annales Polonici Mathematici, vol. 28, pp. 648–658, 1973.

K. S. Padmanabhan and M. S. Ganesan, “Convolutions of certain classes of univalent functions with negative coefficients,” Indian Journal of Pure and Applied Mathematics, vol. 19, no. 9, pp. 880–889, 1988.

T. Rosy and G. Murugusundaramoorthy, “Fractional calculus and their applications to certain subclass of uniformly convex functions,” Far East Journal of Mathematical Sciences, vol. 15, no. 2, pp. 231–242, 2004.

M. K. Aouf, “A subclass of uniformly convex functions with negative coefficients,” Mathematica, vol. 52, no. 2, pp. 99–111, 2010.

R. Bharati, R. Parvatham, and A. Swaminathan, “On subclasses of uniformly convex functions and corresponding class of starlike functions,” Tamkang Journal of Mathematics, vol. 28, no. 1, pp. 17–32, 1997.

H. Silverman, “Univalent functions with negative coefficients,” Proceedings of the American Mathematical Society, vol. 51, pp. 109–116, 1975.

M. K. Aouf, R. M. El-Ashwah, A. A. M. Hassan, and A. H. Hassan, “On subordination results for certain new classes of analytic functions defined by using Salagean operator,” Bulletin of Mathematical Analysis and Applications, vol. 4, no. 1, pp. 239–246, 2012.

J. E. Littlewood, “On inequalities in the theory of functions,” Proceedings of the London Mathematical Society, vol. 23, no. 1, pp. 481–519.

H. M. Srivastava and M. K. Aouf, “A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients I,” Journal of Mathematical Analysis and Applications, vol. 171, no. 1, pp. 1–13, 1992.

H. M. Srivastava and M. K. Aouf, “A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients, II,” Journal of Mathematical Analysis and Applications, vol. 192, no. 3, pp. 673–688, 1995.

H. M. Srivastava and S. Owa, “Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions,” Nagoya Mathematical Journal, vol. 106, pp. 1–28, 1987.

H. M. Srivastava and S. Owa, Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press, Horwood Limited, Chichester, UK; John Wiley and Sons, New York, NY, USA, 1989.