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Non-real zeros of derivatives of real meromorphic functions of infinite order

Published online by Cambridge University Press:  20 September 2010

J. K. LANGLEY
J. K. LANGLEY*
Affiliation:
School of Mathematical Sciences, University of Nottingham, NG 7 2RD. e-mail: jkl@maths.nott.ac.uk
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Abstract

Let f be a real meromorphic function of infinite order in the plane, with finitely many zeros and non-real poles. Then f″ has infinitely many non-real zeros.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 2 , March 2011 , pp. 343 - 351
Copyright
Copyright © Cambridge Philosophical Society 2010

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References

REFERENCES

[1]Ålander, M.Sur les zéros extraordinaires des dérivées des fonctions entières réelles. Ark. för Mat., Astron. och Fys. 11, No. 15 (1916), 1–18.Google Scholar
[2]Ålander, M.Sur les zéros complexes des dérivées des fonctions entières réelles. Ark. för Mat., Astron. och Fys. 16, No. 10 (1922), 1–19.Google Scholar
[3]Bergweiler, W.On the zeros of certain homogeneous differential polynomials. Arch. Math. (Basel) 64 (1995), 199–202.CrossRefGoogle Scholar
[4]Bergweiler, W. and Eremenko, A.On the singularities of the inverse to a meromorphic function of finite order. Rev. Mat. Iberoamericana 11 (1995), 355–373.CrossRefGoogle Scholar
[5]Bergweiler, W. and Eremenko, A.Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions. Acta Math. 197 (2006), 145–166.CrossRefGoogle Scholar
[6]Bergweiler, W., Eremenko, A. and Langley, J.K.Real entire functions of infinite order and a conjecture of Wiman. Geom. Funct. Anal. 13 (2003), 975–991.CrossRefGoogle Scholar
[7]Craven, T., Csordas, G. and Smith, W.Zeros of derivatives of entire functions. Proc. Amer. Math. Soc. 101 (1987), 323–326.CrossRefGoogle Scholar
[8]Craven, T., Csordas, G. and Smith, W.The zeros of derivatives of entire functions and the Pólya–Wiman conjecture. Ann. of Math. (2) 125 (1987), 405–431.CrossRefGoogle Scholar
[9]Hayman, W. K.Meromorphic Functions (Clarendon Press, Oxford, 1964).Google Scholar
[10]Xing, Gu Yong A criterion for normality of families of meromorphic functions (Chinese). Sci. Sinica Special Issue 1 on Math. (1979), 267–274.Google Scholar
[11]Hellerstein, S. and Williamson, J.The zeros of the second derivative of the reciprocal of an entire function. Trans. Amer. Math. Soc. 263 (1981), 501–513.CrossRefGoogle Scholar
[12]Hellerstein, S., Shen, L.-C. and Williamson, J.Real zeros of derivatives of meromorphic functions and solutions of second order differential equations. Trans. Amer. Math. Soc. 285 (1984), 759–776.CrossRefGoogle Scholar
[13]Hinkkanen, A.Reality of zeros of derivatives of meromorphic functions. Ann. Acad. Sci. Fenn. Math. 22 (1997), 1–38.Google Scholar
[14]Hinkkanen, A.Zeros of derivatives of strictly non-real meromorphic functions. Ann. Acad. Sci. Fenn. Math. 22 (1997), 39–74.Google Scholar
[15]Hinkkanen, A.Iteration, level sets, and zeros of derivatives of meromorphic functions. Ann. Acad. Sci. Fenn. Math. 23 (1998), 317–388.Google Scholar
[16]Ki, H. and Kim, Y.-O.On the number of nonreal zeros of real entire functions and the Fourier-Pólya conjecture. Duke Math. J. 104 (2000), 45–73.CrossRefGoogle Scholar
[17]Kim, Y.-O.A proof of the Pólya–Wiman conjecture. Proc. Amer. Math. Soc. 109 (1990), 1045–1052.Google Scholar
[18]Langley, J. K.Non-real zeros of higher derivatives of real entire functions of infinite order. J. Anal. Math. 97 (2005), 357–396.CrossRefGoogle Scholar
[19]Langley, J. K.Non-real zeros of linear differential polynomials. J. Anal. Math. 107 (2009), 107–140.CrossRefGoogle Scholar
[20]Langley, J. K.Non-real zeros of derivatives of real meromorphic functions. Proc. Amer. Math. Soc. 137 (2009), 3355–3367.CrossRefGoogle Scholar
[21]Langley, J. K.Zeros of derivatives of meromorphic functions. Comput. Methods Funct. Theory 10 (2010), 421–439.CrossRefGoogle Scholar
[22]Langley, J. K. Non-real zeros of real differential polynomials. Submitted manuscript, 2010.Google Scholar
[23]Levin, B.Ja. and Ostrovskii, I. V.The dependence of the growth of an entire function on the distribution of zeros of its derivatives. Sibirsk. Mat. Zh. 1 (1960), 427–455. English transl. AMS Transl. (2) 32 (1963), 323–357.Google Scholar
[24]Mues, E.Über die Nullstellen homogener Differentialpolynome. Manuscripta Math. 23 (1978), 325–341.CrossRefGoogle Scholar
[25]Nicks, D. A. Non-real zeroes of real entire functions and their derivatives. Submitted manuscript 2010.Google Scholar
[26]Pólya, G.On the zeros of the derivatives of a function and its analytic character. Bull. Amer. Math. Soc. 49 (1943), 178–191.CrossRefGoogle Scholar
[27]Rossi, J.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.Google Scholar
[28]Schwick, W.Normality criteria for families of meromorphic functions. J. Anal. Math. 52 (1989), 241–289.CrossRefGoogle Scholar
[29]Sheil–Small, T.On the zeros of the derivatives of real entire functions and Wiman's conjecture. Ann. of Math. 129 (1989), 179–193.CrossRefGoogle Scholar
[30]Tsuji, M.On Borel's directions of meromorphic functions of finite order I. Tôhoku Math. J. 2 (1950), 97–112.CrossRefGoogle Scholar