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Multiplicities in Hayman's alternative

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  09 April 2009

Walter Bergwelier  and
J. K. Langley
Walter Bergwelier
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn Str. 4, D-24098 Kiel, Germany e-mail: [email protected]
J. K. Langley
Affiliation:
School of Mathematical Sciences, University of Nottingham, NG7 2RD, UK e-mail: [email protected]
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Abstract

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In 1959 Hayman proved an inequality from which it follows that if f is transcendental and meromorphic in the plane then either f takes every finite complex value infinitely often or each derivative f(k), k ≥1, takes every finite non-zero value infinitely often. We investigate the extent to which these values may be ramified, and we establish a generalization of Hayman's inequality in which multiplicities are not taken into account.

MSC classification

Secondary: 30D35: Distribution of values, Nevanlinna theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 78 , Issue 1 , February 2005 , pp. 37 - 58
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Bergweiler, W., ‘Normality and exceptional values of derivatives’, Proc. Amer. Math. Soc. 129 (2001), 121–129.CrossRefGoogle Scholar
[2]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
[3]Hui, Chen Huai and Xing, Gu Yong, ‘Improvement of Marty's criterion and its application’, Sci. China Ser. A 36 (1993), 674–681.Google Scholar
[4]Hua, Chen Zhen, ‘Normality of families of meromorphic functions with multiple valued derivatives’, Acta Math. Sinica 30 (1987), 97–105.Google Scholar
[5]Chuang, Chi Tai and Yang, L., ‘Distributions of the values of meromorphic functions’, in: Analytic functions of one complex variable, Contemp. Math. 48 (Amer. Math. Soc., Providence, RI, 1985) pp. 21–63.CrossRefGoogle Scholar
[6]Eremenko, A., ‘Distribution of zeros of some real polynomials and iteration theory’, preprint, (Institute for Low Temperature Physics and Engineering, Kharkov, 1989).Google Scholar
[7]Xing, Gu Yong, ‘A criterion for normality of families of meromorphic functions’, Sci. Sinica Special Issue on Math. 1 (1979), 267–274.Google Scholar
[8]Hayman, W. K., ‘Picard values of meromorphic functions and their derivatives’, Ann. of Math. (2) 70 (1959), 9–42.CrossRefGoogle Scholar
[9]Hayman, W. K., Meromorphic functions (Clarendon Press, Oxford, 1964).Google Scholar
[10]Heckner, M., ‘A remark on Langley's generalisation of Hayman's alternative’, Result. Math. 25 (1994), 54–59.CrossRefGoogle Scholar
[11]Langley, J. K., ‘On Hayman's alternative’, Mathematika 32 (1985), 139–146.CrossRefGoogle Scholar
[12]Langley, J. K., ‘A lower bound for the number of zeros of a meromorphic function and its second derivative’, Proc. Edinburgh Math. Soc. 39 (1996), 171–185.CrossRefGoogle Scholar
[13]Xuecheng, Pang, ‘Bloch's principle and normal criterion’, Sci. China Ser. A 32 (1989), 782–791.Google Scholar
[14]Xuecheng, Pang, ‘On normal criterion of meromorphic functions’, Sci. China Ser. A 33 (1990), 521–527.Google Scholar
[15]Schwick, W., ‘Normality criteria for families of meromorphic functions’, J. Analyse Math. 52 (1989), 241–289.CrossRefGoogle Scholar
[16]Steinmetz, N., Rational iteration, de Gruyter Studies in Mathematics 16 (Walter de Gruyter, Berlin, 1993).CrossRefGoogle Scholar
[17]Wang, Y. F. and Fang, M., ‘Picard values and normal families of meromorphic functions with multiple zeros’, Acta Math. Sinica New Ser. 14 (1998), 17–26.Google Scholar
[18]Yang, L., ‘Precise fundamental inequalities and sums of deficiencies’, Sci. China Ser. A 34 (1991), 157–165.Google Scholar
[19]Yang, L. and Zhang, G. H., ‘Recherches sur la normalité des families de fonctions analytiques á des valeurs multiples. I. Un nouveau critère et quelques applications’, Sci. Sinica 14 (1965), 1258–1271.Google Scholar
[20]Yang, L. and Zhang, G. H., ‘Recherches sur la normalité des familles de fonctions analytiques à des valeurs multiples. II. Généralisations’, Sci. Sinica 15 (1966), 433–453.Google Scholar
[21]Zalcman, L., ‘A heuristic principle in complex function theory’, Amer. Math. Monthly 82 (1975), 813–817.CrossRefGoogle Scholar
[22]Zalcman, L., ‘Normal families: new perspectives’, Bull. Amer. Math. Soc. (N.S.) 35 (1998), 215–230.CrossRefGoogle Scholar