Advertisement

Results in Mathematics

, Volume 35, Issue 3–4, pp 284–309 | Cite as

On differential polynomials, fixpoints and critical values of meromorphic functions

  • J.K. Langley
Article

Abstract

We prove some results concerning functions ƒ meromorphic of finite lower order in the plane, such that ƒƒ″−α(ƒ′)2 has few zeros, where α is a constant, using in part new estimates for the multipliers at fixpoints of certain functions. We go on to consider zeros of derivatives and the minimal lower growth of meromorphic functions with finitely many critical values.

A.M.S. Classification

30D35 

Key Words

Nevanlinna theory differential polynomials 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Bank and R. Kaufman, On meromorphic solutions of first-order differential equations, Comment. Math. Helv. 51 (1976), 289–299.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    P.D. Barry, The minimum modulus of small integral and subharmonic functions, Proc. London Math. Soc. (3) 12 (1962), 445–495.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    W. Bergweiler, An inequality for real functions with applications to function theory, Bull. London Math. Soc. 21 (1989), 171–175.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    W. Bergweiler, On the zeros of certain homogeneous differential polynomials, Archiv der Math. 64 (1995), 199–202.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995), 355–373.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    J. Clunie, A. Eremenko and J. Rossi, On equilibrium points of logarithmic and Newtonian potentials, J. London Math. Soc. (2) 47 (1993), 309–320.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    D. Drasin, Proof of a conjecture of F. Nevanlinna concerning functions which have deficiency sum two, Acta. Math. 158 (1987), 1–94.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    D. Drasin and J.K. Langley, On deficiencies and fixpoints of composite meromorphic functions, Complex Variables 34 (1997), 63–82.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    A. Edrei, The deficiencies of functions of finite lower order, Duke Math. J. 31 (1964), 1–22.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    A. Edrei and W.H. J. Fuchs, Bounds for the number of deficient values of certain classes of meromorphic functions, Proc. London Math. Soc. (3) 12 (1962), 315–344.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    A. Eremenko, Meromorphic functions with small ramification, Indiana Univ. Math. J. 42 (1994), 1193–1218.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Eremenko, A, Langley, JK, Rossi, J 1994On the zeros of meromorphic functions of the form \(\Sigma_{k=1}^\infty\ {a_{k} \over z-z_{k}}\) J. d’Analyse Math.62271286MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    G. Frank and S. Hellerstein, On the meromorphic solutions of nonhomogeneous linear differential equations with polynomial coefficients, Proc. London Math. Soc. (3) 53 (1986), 407–428.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    G. Frank, W. Hennekemper and G. Polloczek, Über die Nullstellen meromorpher Funktionen and ihrer Ableitungen, Math. Ann. 225 (1977), 145–154.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    W.H.J. Fuchs, Meromorphic functions of lower order less than 1, Math. Sci. Report 64, Inst. of Math. Sci., Madras 1969.Google Scholar
  16. [16]
    W.H.J. Fuchs, Proof of a conjecture of G. Pólya concerning gap series, Ill. Math. J. 7 (1963), 661–667.MATHGoogle Scholar
  17. [17]
    A.A. Gol’dberg and O.P. Sokolovskaya, Some relations for meromorphic functions of order or lower order less than one, Izv. Vyssh. Uchebn. Zaved. Mat. 31 no.6 (1987), 26–31. Translation: Soviet Math. (Izv. VUZ) 31 no.6 (1987), 29-35.MathSciNetGoogle Scholar
  18. [18]
    W.K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. 70 (1959), 9–42.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    W.K. Hayman, Meromorphic functions, Oxford at the Clarendon Press, 1964.Google Scholar
  20. [20]
    W.K. Hayman, On the characteristic of functions meromorphic in the plane and of their integrals, Proc. London Math. Soc. (3) 14A (1965), 93–128.MathSciNetCrossRefGoogle Scholar
  21. [21]
    W.K. Hayman, Subharmonic functions Vol. 2, Academic Press, London, 1989.Google Scholar
  22. [22]
    S. Hellerstein, J. Miles and J. Rossi, On the growth of solutions of ƒ″ + gƒ′ + hƒ = 0, Trans. Amer. Math. Soc. 324 (1991), 693–706.MathSciNetMATHGoogle Scholar
  23. [23]
    S. Hellerstein, J. Miles and J. Rossi, On the growth of solutions of certain linear differential equations, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 17 (1992), 343–365.MathSciNetMATHGoogle Scholar
  24. [24]
    F.R. Keogh, A property of bounded schlicht functions, J. London Math. Soc. 29 (1954), 379–382.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    I. Laine, Nevanlinna theory and complex differential equations, de Gruyter Studies in Math. 15, Walter de Gruyter, Berlin/New York 1993.Google Scholar
  26. [26]
    J.K. Langley, Proof of a conjecture of Hayman concerning ƒ and ƒ″, J. London Math. Soc. (2) 48 (1993), 500–514.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    J.K. Langley, On the multiple points of certain meromorphic functions, Proc. Amer. Math. Soc. 123, no. 6 (1995), 1787–1795.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    J.K. Langley, A lower bound for the number of zeros of a meromorphic function and its second derivative, Proc. Edinburgh Math. Soc. 39 (1996), 171–185.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    J.K. Langley, The zeros of the first two derivatives of a meromorphic function, Proc. Amer. Math. Soc. 124, no. 8 (1996), 2439–2441.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    J.K. Langley, The zeros of the second derivative of a meromorphic function, XVIth Rolf Nevanlinna Colloquium, Eds.: Laine/Martio, Walter de Gruyter, Berlin 1996.Google Scholar
  31. [31]
    J.K. Langley, On the zeros of the second derivative, Proc. Roy. Soc. Edinburgh 127A (1997), 359–368.MathSciNetCrossRefGoogle Scholar
  32. [32]
    J.K. Langley, On the zeros of ƒ(k)/ ƒ, to appear, Complex Variables.Google Scholar
  33. [33]
    J.K. Langley and D.F. Shea, On multiple points of meromorphic functions, to appear, J. London Math. Soc.Google Scholar
  34. [34]
    J.K. Langley and J.H. Zheng, On the fixpoints, multipliers and value distribution of certain classes of meromorphic functions, to appear, Ann. Acad. Sci. Fenn.Google Scholar
  35. [35]
    B.Ja. Levin, Distribution of zeros of entire functions, Amer. Math. Soc, Providence RI, 1980.Google Scholar
  36. [36]
    J. Miles and J. Rossi, Linear combinations of logarithmic derivatives of entire functions with applications to differential equations, Pacific J. Math. 174 (1996), 195–214.MathSciNetMATHGoogle Scholar
  37. [37]
    E. Mues, Über die Nullstellen homogener Differentialpolynome, Manuscripta Math. 23 (1978), 325–341.MathSciNetMATHCrossRefGoogle Scholar
  38. [38]
    R. Nevanlinna, Eindeutige analytische Funktionen, 2. Auflage, Springer, Berlin, 1953.MATHCrossRefGoogle Scholar
  39. [39]
    C. Pommerenke, Boundary behaviour of conformai maps, Grundlehren der Mathematischen Wissenschaften 299, Springer, Berlin 1992.Google Scholar
  40. [40]
    M. Reinders, On the zeros of \( W ({\cal G},{\cal G} {\prime},\dots,{\cal G} {(n-1)}) \), Analysis 15 (1995), 423–431.MathSciNetMATHGoogle Scholar
  41. [41]
    M. Reinders, On the zeros of the Wronskian of an entire or meromorphic function and its derivatives, Ann. Acad. Sci. Fenn. 22 (1997), 89–112.MathSciNetGoogle Scholar
  42. [42]
    D. Shea, On the frequency of multiple values of a meromorphic function of small order, Michigan Math. J. 32 (1985), 109–116.MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
    K. Tohge, On the zeros of a homogeneous differential polynomial, Kodai Math. J. 16 (1993), 398–415.MathSciNetMATHCrossRefGoogle Scholar
  44. [44]
    K. Tohge, Meromorphic functions which share the value zero with their first two derivatives, Complex Variables 28 (1996), 249–260.MathSciNetMATHCrossRefGoogle Scholar
  45. [45]
    K. Tohge, Meromorphic functions which share the value zero with their first two derivatives II, XVIth Rolf Nevanlinna Colloquium, Eds.: Laine/Martio, Walter de Gruyter, Berlin 1996.Google Scholar
  46. [46]
    M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo, 1959.MATHGoogle Scholar
  47. [47]
    A. Weitsman, A theorem on Nevanlinna deficiencies, Acta Math. 128 (1972), 41–52.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 1999

Authors and Affiliations

  • J.K. Langley
    • 1
  1. 1.Department of MathematicsUniversity of Nottingham

Personalised recommendations