Journal of Mathematics

Volume 2013 (2013), Article ID 630934, 4 pages

http://dx.doi.org/10.1155/2013/630934

## Coefficient Inequalities of Analytic Functions Related to Robertson Functions

Department of Mathematics, Abdul Wali Khan University, 23200 Mardan, Khyber Pakhtunkhwa, Pakistan

Received 5 January 2013; Revised 19 March 2013; Accepted 2 April 2013

Academic Editor: S. T. Ali

Copyright © 2013 Muhammad Arif and Mumtaz Ali. 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 introduce and study a subclass of analytic functions related to Robertson functions. Here we discuss the coefficient estimate for function in this class.

#### 1. Introduction

Let be the class of functions of the form which are analytic in the open unit disc . Also let and denote the well-known classes of starlike and convex functions, respectively.

For any two analytic functions given by (1) and with the convolution (Hadamard product) is given by Using the concept of convolution, Ruscheweyh [1] introduced a differential operator given by with where is a Pochhammer symbol given as It is obvious that , , and The following identity can easily be established: Now with the help of Ruscheweyh derivative, we define a class of analytic functions as follows.

*Definition 1. * Let . Then, , if and only if
where , , , is real with , and .

By giving specific values to , , , , and in , we obtain many important subclasses studied by various authors in earlier papers, see for details [2–5], and list some of them as follows: (i) and , studied by Spacek [6] and Robertson [7], respectively; for the advancement work, see [8, 9].(ii) and , studied by both Owa et al. and Shams et al. [10, 11].(iii), , introduced by Ravichandran et al. [12]. (iv), considered by Latha [13].(v), , the well-known classes of starlike and convex functions of order .

From the above special cases, we note that this class provides a continuous passage from the class of starlike functions to the class of convex functions.

We will assume throughout our discussion, unless otherwise stated, that , , , is real with , and .

#### 2. Some Properties of the Class

Theorem 2. * If with , then
*

*Proof. *Since for any complex number , implies that
which implies that
And hence, we obtain the required result.

Put , , and in Theorem 2; we obtain the following result.

Corollary 3 (see [10]). * If with , then
**
Set , , and in Theorem 2; one has the following result.*

Corollary 4 (see [10]). *If with , then
*

Theorem 5. * If , then
**
where
*

*Proof. * We note that for ,
Let us define the function by
Then, is analytic in with and . Let
Then, (19) can be written as
and using (8), we have
which implies that
where we have used (4) and (5). Now applying the coefficient estimates for Caratheodory function [14], we obtain

For ,
which proves (15).

For ,
Therefore, (16) holds for . Assume that (16) is true for all and consider
Thus, the result is true for , and hence by induction, (16) holds for all .

If we set , , and in Theorem 5, we get the result proved in [10].

Corollary 6. *If , then
*

*Remark 7. *If we take in Corollary 6, we have
which was proved by Robertson [15].

By setting , , and in Theorem 5, we obtain the result in [10].

Corollary 8. * If , then
*

*Remark 9. *Letting in Corollary 8, we have
given by Robertson [15].

#### Acknowledgment

The principle author would like to thank Prof. Dr. Ihsan Ali, Vice Chancellor Abdul Wali Khan University Mardan for providing excellent research facilities and financial support.

#### References

- S. Ruscheweyh, “New criteria for univalent functions,”
*Proceedings of the American Mathematical Society*, vol. 49, no. 1, pp. 109–115, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Aqlan, J. M. Jahangiri, and S. R. Kulkarni, “New classes of $k$-uniformly convex and starlike functions,”
*Tamkang Journal of Mathematics*, vol. 35, no. 3, pp. 261–266, 2004. View at Google Scholar · View at MathSciNet - S. Kanas and A. Wisniowska, “Conic regions and $k$-uniform convexity,”
*Journal of Computational and Applied Mathematics*, vol. 105, no. 1-2, pp. 327–336, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - S. Kanas and A. Wiśniowska, “Conic domains and starlike functions,”
*Revue Roumaine de Mathématiques Pures et Appliquées*, vol. 45, no. 4, pp. 647–657, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Nishiwaki and S. Owa, “Certain classes of analytic functions concerned with uniformly starlike and convex functions,”
*Applied Mathematics and Computation*, vol. 187, no. 1, pp. 350–355, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Spacek, “Prispevek k teorii funkei prostych,”
*Časopis pro Pěstování Matematiky a Fysik*, vol. 62, pp. 12–119, 1933. View at Google Scholar - M. S. Robertson, “Univalent functions $f(z)$ for which $z{f}^{\text{'}}(z)$ is spirallike,”
*The Michigan Mathematical Journal*, vol. 16, pp. 97–101, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Arif, W. Haq, and M. Ismail, “Mapping properties of generalized Robertson functions under certain integral operators,”
*Applied Mathematics*, vol. 3, no. 1, pp. 52–55, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - K. I. Noor, M. Arif, and A. Muhammad, “Mapping properties of some classes of analytic functions under an integral operator,”
*Journal of Mathematical Inequalities*, vol. 4, no. 4, pp. 593–600, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Owa, Y. Polatoğlu, and E. Yavuz, “Coefficient inequalities for classes of uniformly starlike and convex functions,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 7, no. 5, article 160, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Shams, S. R. Kulkarni, and J. M. Jahangiri, “Classes of uniformly starlike and convex functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2004, no. 55, pp. 2959–2961, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Ravichandran, C. Selvaraj, and R. Rajagopal, “On uniformly convex spiral functions and uniformly spirallike functions,”
*Soochow Journal of Mathematics*, vol. 29, no. 4, pp. 393–405, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Latha, “Coefficient inequalities for certain classes of Ruscheweyh type analytic functions,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 9, no. 2, p. Article 52, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Caratheodory, “Über den variabilitätsbereich der Fourier'schen konstanten von possitiven harmonischen funktionen,”
*Rendiconti del Circolo Matematico di Palermo*, vol. 32, pp. 193–217, 1911. View at Publisher · View at Google Scholar - M. I. S. Robertson, “On the theory of univalent functions,”
*Annals of Mathematics*, vol. 37, no. 2, pp. 374–408, 1936. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet