Two New Criteria for Normal Families

Abstract

Let a 0, …, a k−1 be analytic functions on a domain Ω. Let F be a family of meromorphic functions f defined on Ω such that f ≠ 0 and f (k) + a k−1 f (k−1) + … + a 0 f ≠ 0 on Ω, for all fF. Then f′ / f: fF is a normal family. Furthermore, let a 0,…, a k−1 be meromorphic functions on a domain Ω. Let F be a family of meromorphic functions f on Ω such that f≠ 0, f′ ≠ 0 and f (k) + a k− f (k−1) + … + a 0 f ≠ 0 on Ω, for all fF. Then f′ / f: fF is a normal family. These two new criteria for normal families extend a recent result of Bergweiler and Langley, [1, Corollary 1.1].

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References

  1. 1.

    W. Bergweiler and J. K. Langley, Nonvanishing derivatives and normal families, J. Anal. Math. 91 (2003), 353–367. MR 2 037 414

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    F. Brüggemann, Proof of a conjecture of Frank and Langley concerning zeros of meromorphic functions and linear differential polynomials, Analysis 12 (1992), 5–30.

    MathSciNet  MATH  Google Scholar 

  3. 3.

    J. Clunie, On integral and meromorphic functions, J. London Math. Soc. 37 (1962), 17–27. MR 26 #1456

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    G. Frank, Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen, Math. Z. 149 no.1 (1976), 29–36. MR 54 # 10601

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    G. Frank and S. Hellerstein, On the meromorphic solutions of nonhomogeneous linear differential equations with polynomial coefficients, Proc. London Math. Soc. (3) 53 no.3 (1986), 407–428. MR 88k:30032

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    G. Frank, W. Hennekemper and G. Polloczek, Über die Nullstellen meromorpher Funktionen und deren Ableitungen, Math. Ann. 225 no.2 (1977), 145–154. MR 55 #3257

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. (2) 70 (1959), 9–42. MR 22 #1675

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    J. K. Langley, The Tsuji characteristic and zeros of linear differential polynomials, Analysis 9 no.3 (1989), 269–282. MR 90k:30056

    MathSciNet  MATH  Google Scholar 

  9. 9.

    J. K. Langley, An application of the Tsuji characteristic, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38 no.2 (1991), 299–318. MR 92m:30060

    MathSciNet  MATH  Google Scholar 

  10. 10.

    J. K. Langley, Proof of a conjecture of Hayman concerning f and f″, J. London Math. Soc. (2) 48 no.3 (1993), 500–514. MR 94k:30075

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    X. Pang, Shared values and normal families, Analysis (Munich) 22 no.2 (2002), 175–182. MR 2003h:30043

    MathSciNet  MATH  Google Scholar 

  12. 12.

    X. and L. Zalcman, Normal families and shared values, Bull. London Math. Soc. 32 no.3 (2000), 325–331. MR 2001e:30059

    MathSciNet  Article  Google Scholar 

  13. 13.

    G. Pólya, Über die Nullstellen sukzessiver Derivierten, Math. Z. 12 (1922), 36–60.

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    L. A. Rubel, Four counterexamples to Bloch’s principle, Proc. Amer. Math. Soc. 98 no.2 (1986), 257–260. MR 87i:30064

    MathSciNet  MATH  Google Scholar 

  15. 15.

    W. Schwick, Normality criteria for families of meromorphic functions, J. Analyse Math. 52 (1989), 241–289. MR 90k:30061

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    N. Steinmetz, On the zeros of (f (p) + a p−1 f (p−1) + … + a 0 f) f, Analysis 7 no.3–4 (1987), 375–389. MR 89e

    MathSciNet  MATH  Google Scholar 

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Correspondence to Eleanor F. Clifford.

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Clifford, E.F. Two New Criteria for Normal Families. Comput. Methods Funct. Theory 5, 65–76 (2005). https://doi.org/10.1007/BF03321086

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Keywords

  • Normal families
  • meromorphic functions
  • Nevanlinna theory

2000 MSC

  • 32A19
  • 32A20
  • 32A22