Abstract and Applied Analysis

Normal Families of Meromorphic Functions Concerning Higher Derivative and Shared Values

Wei Chen, Wenjun Yuan, and Honggen Tian

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Abstract

We study the normal families related to a Hayman conjecture of higher derivative and concerning shared values and get two normal criteria. Our results improve the related theorems which were obtained independently, respectively by Fang and Yuan (2001), Yuan et al. ((2011) and (2012)), Wang et al. (2011), and Qiu et al. (2012). Meanwhile, some examples are given to show the sharpness of our results.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 368321, 8 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393512101

Digital Object Identifier
doi:10.1155/2013/368321

Mathematical Reviews number (MathSciNet)
MR3121530

Zentralblatt MATH identifier
1292.30018

Citation

Chen, Wei; Yuan, Wenjun; Tian, Honggen. Normal Families of Meromorphic Functions Concerning Higher Derivative and Shared Values. Abstr. Appl. Anal. 2013 (2013), Article ID 368321, 8 pages. doi:10.1155/2013/368321. https://projecteuclid.org/euclid.aaa/1393512101


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References

  • C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, vol. 557 of Mathematics and Its Applications, Science Press, Beijing, China; Kluwer Academic Publishers, New York, NY, USA, 2003.
  • Y. X. Gu, X. C. Pang, and M. L. Fang, Theory of Normal Family and Its Applications, Science Press, Beijing, China, 2007 (Chinese).
  • W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964.
  • L. Yang, Value Distribution Theory, Springer, Berlin, Germany, 1993.
  • W. Schwick, “Normality criteria for families of meromorphic functions,” Journal d'Analyse Mathématique, vol. 52, pp. 241–289, 1989.
  • X. C. Pang and L. Zalcman, “Normal families and shared values,” The Bulletin of the London Mathematical Society, vol. 32, no. 3, pp. 325–331, 2000.
  • X. C. Pang and L. Zalcman, “Normality and shared values,” Arkiv för Matematik, vol. 38, no. 1, pp. 171–182, 2000.
  • Q. C. Zhang, “Normal families of meromorphic functions concerning shared values,” Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 545–551, 2008.
  • W. K. Hayman, Research Problems in Function Theory, The Athlone Press University of London, London, UK, 1967.
  • D. Drasin, “Normal families and the Nevanlinna theory,” Acta Mathematica, vol. 122, pp. 231–263, 1969.
  • X. C. Pang, “On normal criterion of meromorphic functions,” Science in China A, vol. 33, no. 5, pp. 521–527, 1990.
  • H. H. Chen and M. L. Fang, “On a theorem of Drasin,” Advances in Mathematics, vol. 20, pp. 504–504, 1991 (Chinese).
  • Y. S. Ye, “A new normality criterion and its application,” Chinese Annals of Mathematics A, vol. 12, supplement, pp. 44–49, 1991.
  • H. H. Chen and Y. X. Gu, “An improvement of Marty's criterion and its application,” Science in China A, vol. 36, no. 6, pp. 674–681, 1993.
  • S. Y. Li, “On normality criterion of a class of the functions,” Journal of Fujian Normal University, vol. 2, pp. 156–158, 1984.
  • X. J. Li, “Proof of Hayman's conjecture on normal families,” Science in China A, vol. 28, no. 6, pp. 596–603, 1985.
  • J. K. Langley, “On normal families and a result of Drasin,” Proceedings of the Royal Society of Edinburgh A, vol. 98, no. 3-4, pp. 385–393, 1984.
  • L. Zalcman, On Some Questions of Hayman, 1994.
  • M. L. Fang and W. J. Yuan, “On the normality for families of meromorphic functions,” Indian Journal of Mathematics, vol. 43, no. 3, pp. 341–351, 2001.
  • W. J. Yuan, J. J. Wei, and J. M. Lin, “A note on normal families of meromorphic functions concerning shared values,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 463287, 10 pages, 2011.
  • J. H. Wang, Q. Lu, and Q. L. Liao, “Normal criteria and shared values by differential polynomials,” Advances in Pure Mathematics, vol. 1, no. 4, pp. 210–217, 2011.
  • W. J. Yuan, Z. R. Li, and J. M. Lin, “Some normality criteria of function families concerning shared values,” Mathematical Methods in the Applied Sciences, vol. 35, no. 17, pp. 2095–2100, 2012.
  • H. L. Qiu, D. Liu, and M. L. Fang, “Hayman's question concerning normal families and shared values,” Scientia Sinica Mathematica, vol. 42, no. 6, pp. 603–610, 2012 (Chinese).
  • L. Zalcman, “A heuristic principle in complex function theory,” The American Mathematical Monthly, vol. 82, no. 8, pp. 813–817, 1975.
  • L. Zalcman, “Normal families: new perspectives,” Bulletin of the American Mathematical Society, vol. 35, no. 3, pp. 215–230, 1998.