Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-29T02:06:57.740Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  20 September 2010

J. K. LANGLEY*
Affiliation:
School of Mathematical Sciences, University of Nottingham, NG 7 2RD. e-mail: [email protected]

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
Copyright
Copyright © Cambridge Philosophical Society 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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), 118.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), 119.Google Scholar
[3]Bergweiler, W.On the zeros of certain homogeneous differential polynomials. Arch. Math. (Basel) 64 (1995), 199202.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), 355373.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), 145166.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), 975991.CrossRefGoogle Scholar
[7]Craven, T., Csordas, G. and Smith, W.Zeros of derivatives of entire functions. Proc. Amer. Math. Soc. 101 (1987), 323326.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), 405431.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), 501513.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), 759776.CrossRefGoogle Scholar
[13]Hinkkanen, A.Reality of zeros of derivatives of meromorphic functions. Ann. Acad. Sci. Fenn. Math. 22 (1997), 138.Google Scholar
[14]Hinkkanen, A.Zeros of derivatives of strictly non-real meromorphic functions. Ann. Acad. Sci. Fenn. Math. 22 (1997), 3974.Google Scholar
[15]Hinkkanen, A.Iteration, level sets, and zeros of derivatives of meromorphic functions. Ann. Acad. Sci. Fenn. Math. 23 (1998), 317388.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), 4573.CrossRefGoogle Scholar
[17]Kim, Y.-O.A proof of the Pólya–Wiman conjecture. Proc. Amer. Math. Soc. 109 (1990), 10451052.Google Scholar
[18]Langley, J. K.Non-real zeros of higher derivatives of real entire functions of infinite order. J. Anal. Math. 97 (2005), 357396.CrossRefGoogle Scholar
[19]Langley, J. K.Non-real zeros of linear differential polynomials. J. Anal. Math. 107 (2009), 107140.CrossRefGoogle Scholar
[20]Langley, J. K.Non-real zeros of derivatives of real meromorphic functions. Proc. Amer. Math. Soc. 137 (2009), 33553367.CrossRefGoogle Scholar
[21]Langley, J. K.Zeros of derivatives of meromorphic functions. Comput. Methods Funct. Theory 10 (2010), 421439.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), 427455. English transl. AMS Transl. (2) 32 (1963), 323–357.Google Scholar
[24]Mues, E.Über die Nullstellen homogener Differentialpolynome. Manuscripta Math. 23 (1978), 325341.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), 178191.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), 667683.Google Scholar
[28]Schwick, W.Normality criteria for families of meromorphic functions. J. Anal. Math. 52 (1989), 241289.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), 179193.CrossRefGoogle Scholar
[30]Tsuji, M.On Borel's directions of meromorphic functions of finite order I. Tôhoku Math. J. 2 (1950), 97112.CrossRefGoogle Scholar