Abstract and Applied Analysis

Normality Criteria of Meromorphic Functions That Share a Holomorphic Function

Jie Ding, Jianming Qi, and Taiying Zhu

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Abstract

Let F be a family of meromorphic functions defined in D , let ψ ( 0 ) , a 0 , a 1 , . . . , a k - 1 be holomorphic functions in D , and let k be a positive integer. Suppose that, for every function f F , f 0 , P ( f ) = f ( k ) + a k - 1 f ( k - 1 ) + + a 1 f ' + a 0 f 0 and, for every pair functions ( f , g ) F , P ( f ) , P ( g ) share ψ , then F is normal in D .

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 582854, 11 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495628

Digital Object Identifier
doi:10.1155/2012/582854

Mathematical Reviews number (MathSciNet)
MR2872318

Zentralblatt MATH identifier
1239.30020

Citation

Ding, Jie; Qi, Jianming; Zhu, Taiying. Normality Criteria of Meromorphic Functions That Share a Holomorphic Function. Abstr. Appl. Anal. 2012 (2012), Article ID 582854, 11 pages. doi:10.1155/2012/582854. https://projecteuclid.org/euclid.aaa/1355495628


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References

  • W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964.
  • J. L. Schiff, Normal Families, Springer, New York, NY, USA, 1993.
  • L. Yang, Value Distribution Theory, Springer, Berlin, Germany, 1993.
  • Y. X. Gu, “A normal criterion of mermorphic families,” Scientia Sinica, vol. 1, pp. 267–274, 1979.
  • L. Yang, “Normal families and fixed-points of meromorphic functions,” Indiana University Mathematics, vol. 192, pp. 225–294, 2004.
  • W. Schwick, “Exceptional functions and normality,” The Bulletin of the London Mathematical Society, vol. 29, no. 4, pp. 425–432, 1997.
  • Y. Xu, “On a result due to Yang and Schwick,” Science China Mathematics A, vol. 40, no. 5, pp. 421–428, 2010 (Chinese).
  • M. L. Fang and J. M. Chang, “Normal families and multiple values,” Archiv der Mathematik, vol. 88, no. 6, pp. 560–568, 2007.
  • J. Y. Xia and Y. Xu, “Normal families of meromorphic functions with multiple values,” Journal of Math-ematical Analysis and Applications, vol. 354, no. 1, pp. 387–393, 2009.
  • Y. T. Li and Y. X. Gu, “On normal families of meromorphic functions,” Journal of Mathematical Analysis and Applications, vol. 354, no. 2, pp. 421–425, 2009.
  • X. Pang and L. Zalcman, “Normal families and shared values,” The Bulletin of the London Mathematical Society, vol. 32, no. 3, pp. 325–331, 2000.
  • L. Zalcman, “Normal families: new perspectives,” Bulletin of the American Mathematical Society, vol. 35, no. 3, pp. 215–230, 1998.
  • W. Bergweiler and X. Pang, “On the derivative of meromorphic functions with multiple zeros,” Jour-nal of Mathematical Analysis and Applications, vol. 278, no. 2, pp. 285–292, 2003.
  • Y. Xu, “Picard values and derivatives of meromorphic functions,” Kodai Mathematical Journal, vol. 28, no. 1, pp. 99–105, 2005.