On the maximum modulus points of entire and meromorphic functions and a problem of Erdos

Author I. I. Marchenko
Institute of Mathematics, University of Szczecin, Poland

Abstract Two conjectures concerning the number of maximum modulus points of an entire and a meromorphic function on the circle {z:|z|=r} are presented and discussed.
Keywords maximum modulus point; meromorphic function; Nevanlinna’s characteristic; entire function; lower order
Reference 1. J.M. Anderson, K.F. Barth, D.A. Brannan, Research problems in complex analysis, Bull. London Math. Soc., 9 (1977), ¹2, P. 129–162.

2. E. Ciechanowicz, I.I. Marchenko, Maximum modulus points, deviations and spreads of meromorphic functions, in Value Distribution Theory and Related Topics, Kluwer Academic publishers, 2004, 117–129.

3. E. Ciechanowicz, I.I. Marchenko, On the maximum modulus points of entire and meromorphic functions, Mat. Stud., 21 (2004), ¹1, 25–34.

4. E. Ciechanowicz, I.I. Marchenko, On the separated maximum modulus points of meromorphic functions, Matematicheskaya fizika, analiz, geometriya, 12, (2005), ¹2, P. 218–229.

5. A.E. Eremenko, An analogue of the defect relation for the uniform metric I, Complex Variables Theory Appl. 34 (1997), 83–97.

6. A.A. Gol’dberg, I.V. Ostrowskii, Distribution of values of meromorphic functions, Nauka, Moscow, 1970 (in Russian). English translated in Translations of Mathematical Monographs, 236, AMS, Providence, 2008.

7. W.K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.

8. F. Herzog, G. Piranian, The Counting Function for Points of Maximum Modulus, Proceedings of Symposia in Pure Mathematics, Entire Functions and Related Parts Analysis, Amer. Math. Soc., 11 (1968), 240–243.

9. I.I. Marchenko, On the magnitudes of deviations and spreads of meromorphic functions of finite lower order, Mat. Sb., 186 (1995), 85–102 (in Russian). Engl. transl.: Sb. Mat., 186 (1995), 391–408.

10. I.I. Marchenko, On the growth of entire and meromorphic functions, Mat. Sb., 189 (1998), .6, 59–84 (in Russian). English translated in Sb. Mat. 189 (1998), 875–900.

11. V.P. Petrenko, Growth of meromorphic functions, Kharkov, Vyshcha Shkola, 1978, 136p. (in Russian)

Pages 212-215
Volume 38
Issue 2
Year 2012
Journal Matematychni Studii
Full text of paper PDF
Table of content of issue HTML