- Open Access
Majorization properties for certain new classes of analytic functions using the Salagean operator
© Li et al.; licensee Springer 2013
- Received: 9 November 2012
- Accepted: 14 February 2013
- Published: 4 March 2013
In the present paper, we investigate the majorization properties for certain classes of multivalent analytic functions defined by the Salagean operator. Moreover, we point out some new and interesting consequences of our main result.
- analytic functions
- multivalent functions
- α-uniformly starlike functions of order β
- α-uniformly convex functions of order β
- majorization property
It may be noted here that (1.2) is closely related to the concept of quasi-subordination between analytic functions.
that are analytic and p-valent in the open unit disk Δ. Also, let .
In view of (1.6), it is clear that , and is a known operator introduced by Salagean .
() (α-uniformly starlike functions of order β);
() (α-uniformly convex functions of order β);
The classes and were introduced by Goswami and Aouf  and Li and Tang , respectively. The classes and were studied recently in  (see also [7–12]). The class was introduced by Akbulut et al. . Also, the classes and are said to be classes of starlike and convex of complex order in Δ which were considered by Nasr and Aouf  and Wiatrowski  (see also [16, 17]), and denotes the class of starlike functions of order β in Δ.
A majorization problem for the class has recently been investigated by Altintas et al. . Also, majorization problems for the classes and have been investigated by MacGregor  and Goswami and Aouf , respectively. Very recently, Goyal and Goswami  (see also ) generalized these results for the fractional derivative operator. In the present paper, we investigate a majorization problem for the class .
We begin by proving the following result.
which holds true for all .
Hence, upon setting in (2.11), we conclude that (2.1) of Theorem 2.1 holds true for , which completes the proof of Theorem 2.1. □
Setting in Theorem 2.1, we get the following result.
Remark 2.1 Corollary 2.1 improves the result of Goswami and Aouf [, Theorem 1].
Putting , , , , and in Theorem 2.1, we obtain the following result.
For , , putting , and , in Corollary 2.2, respectively, we obtain the following Corollaries 2.3 and 2.4.
Also, putting , , , and in Theorem 2.1, we obtain the following result.
Dedicated to Professor Hari M. Srivastava.
The present investigation is partly supported by the Natural Science Foundation of Inner Mongolia of People’s Republic of China under Grant 2009MS0113, 2010MS0117. The authors would like to thank the referees for their helpful comments and suggestions to improve our manuscript.
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