About this Journal Submit a Manuscript Table of Contents
Journal of Mathematics
Volume 2013 (2013), Article ID 194053, 4 pages
http://dx.doi.org/10.1155/2013/194053
Research Article

Sufficient Conditions for -Spirallike and -Robertson Functions of Complex Order

Department of Mathematics, Faculty of Science, Al al-Bayt University, P.O. Box 130095, Mafraq, Jordan

Received 7 January 2013; Revised 12 February 2013; Accepted 12 February 2013

Academic Editor: Abdellatif Agouzal

Copyright © 2013 B. A. Frasin. 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 obtain several sufficient conditions for -spirallike and -Robertson functions of complex order by making use of the well-known Jack Lemma.

1. Introduction and Definitions

Let denote the class of functions of the form which are analytic in the open unit disk and . A function is said to be the -spirallike function of complex order and type in , denoted by if and only if for some real numbers with and . Furthermore, a function is also said to be the -Robertson function of complex order and type in if and only if for some real numbers with and . We denote this class by .

Noting that the above function classes include several subclasses which have important role in the analytic and geometric function theory. From this reason, we want to state some of them. (i) is the class of the -spirallike function of complex order introduced and studied by Al-Oboudi and Haidan [1].(ii) is the class of the -Robertson function of complex order (see, [2]).(iii) is the class of starlike functions of complex order and type introduced and studied by the author [3], and is said to be the starlike functions class of order and was studied by Robertson [4].(iv) is the class of convex functions of complex order and type introduced and studied by the author [3], and is said to be the convex functions class of order and was studied by Robertson [4].(v) is known the -spirallike univalent functions class and was defined by Spacek [5], is said to be the starlike functions class of complex order and was studied by Nasr and Aouf [6], and is known the -spirallike functions class of order and was studied by Libera [7].(vi) is known the -Robertson type functions class and was first studied by Robertson [8], is called the convex functions class of complex order and was studied by Wiatrowski [9], Nasr and Aouf [10] and Aouf [11], and is known the -Robertson type functions class of order and was studied by Chichra [12].

In this paper, we obtain several sufficient conditions for the analytic functions belonging to the classes ,  ,  ,  ,  ,  ,  , and   by making use of the well-known Jack Lemma [13].

2. Main Result

In order to derive our main result, we have to recall here the following Jack Lemma.

Lemma 1 (see [13]). Let be analytic in such that . Then, if attains its maximum value on circle at a point , one has where is a real number.

Now, with the help of Lemma 1, we can prove the following result.

Theorem 2. Let ,   and , and let be defined by If satisfies any of the following inequalities: then The powers are taken by their principal value.

Proof. Define a function by
Then, is analytic in and . It follows from (12) that
Thus, we have
We claim that in . For otherwise, by Lemma 1, there exists such that , where and . Therefore, (14)–(18) yield which contradicts our assumptions (6)–(10), respectively. Therefore, holds true for all . We finally have thus, we have

Remark 3. Taking different choices of ,,  , and in Theorem 2, we obtain new sufficient conditions for functions to be in the classes , , , , ,  , and .

Acknowledgment

The author would like to thank the referee for his helpful comments and suggestions.

References

  1. F. M. Al-Oboudi and M. M. Haidan, “Spirallike functions of complex order,” Journal of Natural Geometry, vol. 19, no. 1-2, pp. 53–72, 2001. View at Zentralblatt MATH · View at MathSciNet
  2. M. K. Aouf, F. M. Al-Oboudi, and M. M. Haidan, “On some results for λ-Roberston functions ofcomplex order,” Publications de l'Institut Mathématique, vol. 75, no. 91, pp. 93–98, 2005.
  3. B. A. Frasin, “Family of analytic functions of complex order,” Acta Mathematica, vol. 22, no. 2, pp. 179–191, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. 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
  5. L. Spacek, “Contribution a la theori des function univalentes,” Casopsis pro Pestovani Matematiky a Fysiki, vol. 62, pp. 12–19, 1932.
  6. M. A. Nasr and M. K. Aouf, “Starlike function of complex order,” The Journal of Natural Sciences and Mathematics, vol. 25, no. 1, pp. 1–12, 1985. View at Zentralblatt MATH · View at MathSciNet
  7. R. J. Libera, “Univalent α-spiral functions,” Canadian Journal of Mathematics, vol. 19, pp. 449–456, 1967. View at MathSciNet
  8. M. S. Robertson, “Univalent functions f(z) for which zf'(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
  9. P. Wiatrowski, “The coefficients of a certain family of holomorphic functions,” Zeszyty Naukowe Uniwersytetu Lodzkiego, Nauki Matematyczno Przyrodnicze, Seria 2, Zeszyt, no. 39, pp. 75–85, 1971. View at MathSciNet
  10. M. A. Nasr and M. K. Aouf, “On convex functions of complex order,” Mansoura Science Bulletin, vol. 9, pp. 565–582, 1982.
  11. M. K. Aouf, “p-valent classes related to convex functions of complex order,” The Rocky Mountain Journal of Mathematics, vol. 15, no. 4, pp. 853–864, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. P. N. Chichra, “Regular functions f(z) for which zf'(z) is α-spiral-like,” Proceedings of the American Mathematical Society, vol. 49, pp. 151–160, 1975. View at MathSciNet
  13. I. S. Jack, “Functions starlike and convex of order α,” Journal of the London Mathematical Society, vol. 3, pp. 469–474, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet