Non-real zeros of derivatives of real meromorphic functions
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Abstract:
The main result of this paper determines all real meromorphic functions $f$ of finite order in the plane such that $f’$ has finitely many zeros while $f$ and $f^{(k)}$, for some $k \geq 2$, have finitely many non-real zeros.References
- M. Ålander, Sur les zéros extraordinaires des dérivées des fonctions entières réelles. Ark. för Mat., Astron. och Fys. 11, No. 15 (1916), 1–18.
- M. Ålander, Sur les zéros complexes des dérivées des fonctions entières réelles. Ark. för Mat., Astron. och Fys. 16, No. 10 (1922), 1–19.
- Walter Bergweiler and Alexandre Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995), no. 2, 355–373. MR 1344897, DOI 10.4171/RMI/176
- Walter Bergweiler and Alexandre Eremenko, Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions, Acta Math. 197 (2006), no. 2, 145–166. MR 2296054, DOI 10.1007/s11511-006-0010-8
- W. Bergweiler, A. Eremenko, and J. K. Langley, Real entire functions of infinite order and a conjecture of Wiman, Geom. Funct. Anal. 13 (2003), no. 5, 975–991. MR 2024413, DOI 10.1007/s00039-003-0437-4
- Walter Bergweiler and J. K. Langley, Nonvanishing derivatives and normal families, J. Anal. Math. 91 (2003), 353–367. MR 2037414, DOI 10.1007/BF02788794
- Albert Edrei and Wolfgang 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. MR 138765, DOI 10.1112/plms/s3-12.1.315
- Stephanie Edwards and Simon Hellerstein, Non-real zeros of derivatives of real entire functions and the Pólya-Wiman conjectures, Complex Var. Theory Appl. 47 (2002), no. 1, 25–57. MR 1884528, DOI 10.1080/02781070290002912
- A. È. Erëmenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989–1020 (English, with English and French summaries). MR 1196102
- Günter Frank, Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen, Math. Z. 149 (1976), no. 1, 29–36 (German). MR 422615, DOI 10.1007/BF01301627
- Günter Frank and Simon Hellerstein, On the meromorphic solutions of nonhomogeneous linear differential equations with polynomial coefficients, Proc. London Math. Soc. (3) 53 (1986), no. 3, 407–428. MR 868452, DOI 10.1112/plms/s3-53.3.407
- Günter Frank, Wilhelm Hennekemper, and Gisela Polloczek, Über die Nullstellen meromorpher Funktionen und deren Ableitungen, Math. Ann. 225 (1977), no. 2, 145–154. MR 430250, DOI 10.1007/BF01351718
- G. Frank and J. K. Langley, Pairs of linear differential polynomials, Analysis (Munich) 19 (1999), no. 2, 173–194. MR 1705364, DOI 10.1524/anly.1999.19.2.173
- A. A. Gol′dberg and I. V. Ostrovskiĭ, Raspredelenie znacheniĭ meromorfnykh funktsiĭ, Izdat. “Nauka”, Moscow, 1970 (Russian). MR 0280720
- Gary G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), no. 1, 88–104. MR 921748, DOI 10.1112/jlms/s2-37.121.88
- W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
- 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. MR 180679, DOI 10.1112/plms/s3-14A.1.93
- W. K. Hayman, The local growth of power series: a survey of the Wiman-Valiron method, Canad. Math. Bull. 17 (1974), no. 3, 317–358. MR 385095, DOI 10.4153/CMB-1974-064-0
- Simon Hellerstein and Jack Williamson, Derivatives of entire functions and a question of Pólya, Trans. Amer. Math. Soc. 227 (1977), 227–249. MR 435393, DOI 10.1090/S0002-9947-1977-0435393-4
- Simon Hellerstein and Jack Williamson, Derivatives of entire functions and a question of Pólya. II, Trans. Amer. Math. Soc. 234 (1977), no. 2, 497–503. MR 481004, DOI 10.1090/S0002-9947-1977-0481004-1
- Simon Hellerstein and Jack Williamson, The zeros of the second derivative of the reciprocal of an entire function, Trans. Amer. Math. Soc. 263 (1981), no. 2, 501–513. MR 594422, DOI 10.1090/S0002-9947-1981-0594422-9
- Simon Hellerstein, Li-Chien Shen, and Jack Williamson, Real zeros of derivatives of meromorphic functions and solutions of second order differential equations, Trans. Amer. Math. Soc. 285 (1984), no. 2, 759–776. MR 752502, DOI 10.1090/S0002-9947-1984-0752502-1
- Einar Hille, Ordinary differential equations in the complex domain, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. MR 0499382
- A. Hinkkanen, Reality of zeros of derivatives of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 22 (1997), no. 1, 21–38. MR 1430392
- A. Hinkkanen, Reality of zeros of derivatives of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 22 (1997), no. 1, 21–38. MR 1430392
- A. Hinkkanen, Iteration, level sets, and zeros of derivatives of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 2, 317–388. MR 1642118
- E. Laguerre, Sur les fonctions du genre zéro et du genre un, C. R. Acad. Sci. Paris 95 (1882); Oevres, t. 1, 174–177.
- Ilpo Laine, Nevanlinna theory and complex differential equations, De Gruyter Studies in Mathematics, vol. 15, Walter de Gruyter & Co., Berlin, 1993. MR 1207139, DOI 10.1515/9783110863147
- J. K. Langley, On the zeros of the second derivative, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 2, 359–368. MR 1447956, DOI 10.1017/S0308210500023672
- J. K. Langley, The second derivative of a meromorphic function of finite order, Bull. London Math. Soc. 35 (2003), no. 1, 97–108. MR 1934438, DOI 10.1112/S0024609302001558
- J. K. Langley, Non-real zeros of higher derivatives of real entire functions of infinite order, J. Anal. Math. 97 (2005), 357–396. MR 2274982, DOI 10.1007/BF02807411
- B. Ja. Levin and I. V. Ostrovskiĭ, The dependence of the growth of an entire function on the distribution of zeros of its derivatives, Sibirsk. Mat. Ž. 1 (1960), 427–455 (Russian). MR 0130979
- John Lewis, John Rossi, and Allen Weitsman, On the growth of subharmonic functions along paths, Ark. Mat. 22 (1984), no. 1, 109–119. MR 735882, DOI 10.1007/BF02384375
- Rolf Nevanlinna, Eindeutige analytische Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band XLVI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). 2te Aufl. MR 0057330
- D.A. Nicks, Real meromorphic functions and a result of Hinkkanen and Rossi, to appear, Illinois J. Math.
- G. Polya, On the zeros of the derivatives of a function and its analytic character, Bull. Amer. Math. Soc. 49 (1943), 178–191. MR 7781, DOI 10.1090/S0002-9904-1943-07853-6
- P. J. Rippon and G. M. Stallard, Iteration of a class of hyperbolic meromorphic functions, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3251–3258. MR 1610785, DOI 10.1090/S0002-9939-99-04942-4
- John 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), no. 2, 667–683. MR 645337, DOI 10.1090/S0002-9947-1982-0645337-X
- Wilhelm Schwick, Normality criteria for families of meromorphic functions, J. Analyse Math. 52 (1989), 241–289. MR 981504, DOI 10.1007/BF02820480
- T. Sheil-Small, On the zeros of the derivatives of real entire functions and Wiman’s conjecture, Ann. of Math. (2) 129 (1989), no. 1, 179–193. MR 979605, DOI 10.2307/1971490
- Masatsugu Tsuji, On Borel’s directions of meromorphic functions of finite order. I, Tohoku Math. J. (2) 2 (1950), 97–112. MR 41224, DOI 10.2748/tmj/1178245639
- M. Tsuji, Potential theory in modern function theory, Chelsea Publishing Co., New York, 1975. Reprinting of the 1959 original. MR 0414898
- Lawrence Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc. (N.S.) 35 (1998), no. 3, 215–230. MR 1624862, DOI 10.1090/S0273-0979-98-00755-1
Additional Information
- J. K. Langley
- Affiliation: School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
- MR Author ID: 110110
- Email: jkl@maths.nottingham.ac.uk
- Received by editor(s): January 7, 2009
- Published electronically: May 21, 2009
- Additional Notes: The author’s research was supported by the Engineering and Physical Sciences Research Council grant EP/D065321/1
- Communicated by: Mario Bonk
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 3355-3367
- MSC (2000): Primary 30D20, 30D35
- DOI: https://doi.org/10.1090/S0002-9939-09-09979-1
- MathSciNet review: 2515405