Abstract
An estimate is obtained for the growth of a meromorphic function near to a logarithmic singularity of the derivative. This estimate is applied to show that if f is meromorphic of finite lower order in the plane, such that the second derivative f″ has finitely many zeros and the multiplicities of the poles z of f grow at most polynomially in ¦z¦, then f has finitely many poles. Subsequent results then consider the zeros of linear differential polynomials F = f (k)+a k−1 f (k−1)+…+ a 0 f, where f is transcendental and meromorphic of finite order in the plane, and the coefficients a j are constants.
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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.
F. Brüggemann, Proof of a conjecture of Frank and Langley concerning zeros of meromorphic functions and linear differential polynomials, Analysis 12 (1992) no. 1/2, 5–30.
G. Frank, Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen, Math. Zeit. 149 (1976), 29–36.
G. Frank, Über die Nullstellen von linearen Differentialpolynomen mit meromorphen Koeffizienten, Complex Methods on Partial Differential Equations, 39–48, Math. Res. 53, Akademie-Verlag, Berlin 1989.
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, Über die Nullstellen meromorpher Funktionen and ihrer Ableitungen, Math. Ann. 225 (1977), 145–154.
G. Frank and G. Weissenborn, Rational deficient functions of meromorphic functions, Bull. London Math. Soc. 18 (1986), 29–33.
W. K. Hayman, Meromorphic Functions, Oxford at the Clarendon Press, 1964.
J. D. Hinchliffe, The Bergweiler-Eremenko theorem for finite lower order, Result. Math. 43 (2003), 121–128.
F. R. Keogh, A property of bounded schlicht functions, J. London Math. Soc. 29 (1954), 379–382.
J. K. Langley, Proof of a conjecture of Hayman concerning f and f″, J. London Math. Soc. (2)48 (1993), 500–514.
J. K. Langley, On second order linear differential polynomials, Result. Math. 26 (1994), 51–82.
J. K. Langley, On the zeros of the second derivative, Proc. Roy. Soc. Edinburgh 127A (1997), 359–368.
J. K. Langley, The second derivative of a meromorphic function of finite order, Bulletin London Math. Soc. 35 (2003), 97–108.
J. K. Langley and John Rossi, Critical points of certain discrete potentials, Complex Variables 49 (2004), 621–637.
J. K. Langley and D. F. Shea, On multiple points of meromorphic functions, J. London Math. Soc. (2) 57 (1998), 371–384.
R. Nevanlinna, Eindeutige analytische Funktionen, 2. Auflage, Springer, Berlin, 1953.
G. Pólya, Über die Nullstellen sukzessiver Derivierten,:Math. Z. 12 (1922), 36–60.
C. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren der Mathematischen Wissenschaften 299, Springer, Berlin, 1992.
N. Steinmetz, On the zeros of a certain Wronskian, Bull. London Math. Soc. 20 (1988), 525–531.
M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.
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Research supported by Engineering and Physical Sciences Research Council grant EP/D065321/1.
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Langley, J. Logarithmic Singularities and the Zeros of the Second Derivative. Comput. Methods Funct. Theory 9, 565–578 (2009). https://doi.org/10.1007/BF03321745
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DOI: https://doi.org/10.1007/BF03321745