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.
Keywordsanalytic functions multivalent functions α-uniformly starlike functions of order β α-uniformly convex functions of order β subordination majorization property
1 Introduction and definitions
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 .
2 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.
- MacGregor TH: Majorization by univalent functions. Duke Math. J. 1967, 34: 95–102. 10.1215/S0012-7094-67-03411-4MathSciNetView ArticleGoogle Scholar
- Frasin BA: Neighborhoods of certain multivalent functions with negative coefficients. Appl. Math. Comput. 2007, 193: 1–6. 10.1016/j.amc.2007.03.026MathSciNetView ArticleGoogle Scholar
- Salagean GS: Subclasses of univalent functions. Lecture Notes in Math. 1013. In Complex Analysis - Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981). Springer, Berlin; 1983:362–372.View ArticleGoogle Scholar
- Goswami P, Aouf MK: Majorization properties for certain classes of analytic functions using the Salagean operator. Appl. Math. Lett. 2010, 23: 1351–1354. 10.1016/j.aml.2010.06.030MathSciNetView ArticleGoogle Scholar
- Li S-H, Tang H: Certain new classes of analytic functions defined by using the Salagean operator. Bull. Math. Anal. Appl. 2010, 4(2):62–75.Google Scholar
- Kanas S, Srivastava HM: Linear operators associated with k -uniformly convex functions. Integral Transforms Spec. Funct. 2000, 9: 121–132. 10.1080/10652460008819249MathSciNetView ArticleGoogle Scholar
- Kanas S, Yaguchi T: Subclasses of k -uniformly convex and starlike functions defined by generalized derivative, I. Indian J. Pure Appl. Math. 2001, 32(9):1275–1282.MathSciNetGoogle Scholar
- Kanas S: Integral operators in classes k -uniformly convex and k -starlike functions. Mathematica (Cluj-Napoca, 1992) 2001, 43(66)(1):77–87.MathSciNetGoogle Scholar
- Kanas S, Wiśniowska A: Conic regions and k -uniform convexity, II. Zeszyty Nauk. Politech. Rzeszowskiej Mat. 1998, 170: 65–78.Google Scholar
- Kanas S: Differential subordination related to conic sections. J. Math. Anal. Appl. 2006, 317(2):650–658. 10.1016/j.jmaa.2005.09.034MathSciNetView ArticleGoogle Scholar
- Ramachandran C, Srivastava HM, Swaminathan A: A unified class of k -uniformly convex functions defined by the Salagean derivative operator. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 2007, 55: 47–59.MathSciNetGoogle Scholar
- Shams S, Kulkarni SR, Jahangiri JM: On a class of univalent functions defined by Ruscheweyh derivatives. Kyungpook Math. J. 2003, 43: 579–585.MathSciNetGoogle Scholar
- Akbulut S, Kadioglu E, Ozdemir M: On the subclass of p -valently functions. Appl. Math. Comput. 2004, 147(1):89–96. 10.1016/S0096-3003(02)00653-7MathSciNetView ArticleGoogle Scholar
- Nasr MA, Aouf MK: Starlike function of complex order. J. Nat. Sci. Math. 1985, 25(1):1–12.MathSciNetGoogle Scholar
- Wiatrowski P: On the coefficients of some family of holomorphic functions. Zeszyry Nauk. Univ. Lodz. Nauk. Mat.-Przyrod. Ser. II 1970, 39: 75–85.Google Scholar
- Kanas S, Darwish HE: Fekete-Szego problem for starlike and convex functions of complex order. Appl. Math. Lett. 2010, 23: 777–782. 10.1016/j.aml.2010.03.008MathSciNetView ArticleGoogle Scholar
- Kanas S, Sugawa T: On conformal representation of the interior of an ellipse. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 2006, 31: 329–348.MathSciNetGoogle Scholar
- Altintas O, Ozkan O, Srivastava HM: Majorization by starlike functions of complex order. Complex Var. Theory Appl. 2001, 46: 207–218. 10.1080/17476930108815409MathSciNetView ArticleGoogle Scholar
- Goyal SP, Goswami P: Majorization for certain classes of analytic functions defined by fractional derivatives. Appl. Math. Lett. 2009, 22(12):1855–1858. 10.1016/j.aml.2009.07.009MathSciNetView ArticleGoogle Scholar
- Prajapat JK, Aouf MK: Majorization problem for certain class of p -valently analytic function defined by generalized fractional differential operator. Comput. Math. Appl. 2012, 63: 42–47.MathSciNetGoogle Scholar
- Nehari Z: Conformal Mapping. McGraw-Hill, New York; 1955.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.