Abstract
Two results are proved for real meromorphic functions in the plane. First, a lower bound is given for the distance between distinct non-real poles when the function and its second derivative have finitely many non-real zeros and the logarithmic derivative has finite lower order. Second, if the function has finitely many non-real zeros, and one of its higher derivatives has finitely many zeros in the plane, and if the multiplicities of non-real poles grow sufficiently slowly, then the function is a rational function multiplied by the exponential of a polynomial.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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.
W. Bergweiler and A. Eremenko, Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions, Acta Math. 197 (2006), 145–166.
W. Bergweiler, A. Eremenko and J. K. Langley, Real entire functions of infinite order and a conjecture of Wiman, Geom. Funct. Anal. 13 (2003), 975–991.
W. Bergweiler and J. K. Langley, Nonvanishing derivatives and normal families, J. Analyse Math. 91 (2003), 353–367.
T. Craven, G. Csordas and W. Smith, Zeros of derivatives of entire functions, Proc. Amer. Math. Soc. 101 (1987), 323–326.
T. Craven, G. Csordas and W. Smith, The zeros of derivatives of entire functions and the Pólya-Wiman conjecture, Ann. of Math. (2) 125 (1987), 405–431.
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.
A. N. Fletcher, J. K. Langley and J. Meyer, Nonvanishing derivatives and the MacLane class A, Illinois J. Math. 53 (2009), 379–390.
G. Frank, Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen, Math. Z. 149 (1976), 29–36.
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.
G. Frank, W. Hennekemper and G. Polloczek, Uber die Nullstellen meromorpher Funktionen und deren Ableitungen, Math. Ann. 225 (1977), 145–154.
G. Frank and J. K. Langley, Pairs of linear differential polynomials, Analysis 19 (1999), 173–194.
W. H. J. Fuchs, Proof of a conjecture of G. Pólya concerning gap series, Illinois Math. J. 7 (1963), 661–667.
A. A. Gol’dberg and I. V. Ostrovskii, Distribution of Values of Meromorphic Functions, Nauka, Moscow, 1970 (in Russian); English translation, Translations of Mathematical Monographs 236, Amer. Math. Soc., Providence, 2008.
G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), 88–104.
W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
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.
W. K. Hayman, The local growth of power series: a survey of the Wiman-Valiron method, Canad. Math. Bull. 17 (1974), 317–358.
W. K. Hayman, Subharmonic Functions Vol. 2, Academic Press, London, 1989.
S. Hellerstein and J. Williamson, Derivatives of entire functions and a question of Pólya, Trans. Amer. Math. Soc. 227 (1977), 227–249.
S. Hellerstein and J. Williamson, Derivatives of entire functions and a question of Pólya, II, Trans. Amer. Math. Soc. 234 (1977), 497–503.
S. Hellerstein and J. Williamson, The zeros of the second derivative of the reciprocal of an entire function, Trans. Amer. Math. Soc. 263 (1981), 501–513.
S. Hellerstein, L.-C. Shen and J. Williamson, Real zeros of derivatives of meromorphic functions and solutions of second order differential equations, Trans. Amer. Math. Soc. 285 (1984), 759–776.
A. Hinkkanen, Reality of zeros of derivatives of meromorphic functions, Ann. Acad. Sci. Fenn. 22 (1997), 1–38.
S. Hellerstein and J. Williamson, Zeros of derivatives of strictly non-real meromorphic functions, Ann. Acad. Sci. Fenn. 22 (1997), 39–74.
S. Hellerstein and J. Williamson, Iteration, level sets, and zeros of derivatives of meromorphic functions, Ann. Acad. Sci. Fenn. 23 (1998), 317–388.
J. D. Hinchliffe, The Bergweiler-Eremenko theorem for finite lower order, Result. Math. 43 (2003), 121–128.
H. Ki and Y.-O. Kim, On the number of nonreal zeros of real entire functions and the Fourier-Pólya conjecture, Duke Math. J. 104 (2000), 45–73.
Y.-O. Kim, A proof of the Pólya-Wiman conjecture, Proc. Amer. Math. Soc. 109 (1990), 1045–1052.
J. K. Langley, On differential polynomials, fixpoints and critical values of meromorphic functions, Result. Math. 35 (1999), 284–309.
J. K. Langley, The second derivative of a meromorphic function of finite order, Bull. London Math. Soc. 35 (2003), 97–108.
J. K. Langley, Non-real zeros of higher derivatives of real entire functions of infinite order, J. Analyse Math. 97 (2005), 357–396.
J. K. Langley, Non-real zeros of derivatives of real meromorphic functions, Proc. Amer. Math. Soc. 137 (2009), 3355–3367.
J. K. Langley, Zeros of derivatives of meromorphic functions, Comput. Methods Funct. Theory 10 (2010), 421–439.
J. K. Langley, Real meromorphic functions and linear differential polynomials, Sci. China Ser. A 53 (2010), 739–748.
J. K. Langley, Non-real zeros of real differential polynomials, Proc. Roy. Soc. Edinburgh Sect. A. 141 (2011), 631–639.
J. K. Langley, Non-real zeros of derivatives of real meromorphic functions of infinite order, Math. Proc. Camb. Phil. Soc. 150 (2011), 343–351.
J. K. Langley, Value distribution of differences of meromorphic functions, Rocky Mountain J. Math. 41 (2011), 275–291.
B. Ja. Levin, Distribution of Zeros of Entire Functions, GITTL, Moscow, 1956. 2-nd English transl., AMS, Providence RI, 1980.
B. Ja. Levin and I. V. Ostrovskii, The dependence of the growth of an entire function on the distribution of zeros of its derivatives, Sibirsk. Mat. Zh. 1 (1960), 427–455 (in Russian); English translation in: Amer. Math. Soc. Transl. (2) 32 (1963), 323–357.
D. A. Nicks, Value distribution of meromorphic functions and their derivatives, Ph.D. thesis, University of Nottingham, 2010.
—, Non-real zeroes of real entire functions and their derivatives, to appear in J. Analyse Math.
J. Rossi, The reciprocal of an entire function of infinite order and the distribution of the zeros of its second derivative, Trans. Amer. Math. Soc. 270 (1982), 667–683.
W. Schwick, Normality criteria for families of meromorphic functions, J. Analyse Math. 52 (1989), 241–289.
T. Sheil-Small, On the zeros of the derivatives of real entire functions and Wiman’s conjecture, Ann. of Math. 129 (1989), 179–193.
M. Tsuji, On Borel’s directions of meromorphic functions of finite order, I, Tohoku Math. J. 2 (1950), 97–112.
L. Zalcman, Normal families: New perspectives, Bull. Amer. Math. Soc. 35 (1998), 215–230.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Langley, J. Zeros of Derivatives of Real Meromorphic Functions. Comput. Methods Funct. Theory 12, 241–256 (2012). https://doi.org/10.1007/BF03321825
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03321825