On the maximum points and deviations of meromorphic minimal
surfaces

A. Kowalski; I. I. Marchenko

doi:10.15330/ms.46.2.137-151

We provide an upper estimate of the magnitude of deviation for a meromorphic minimal surface of finite lower order. The estimate is given in terms of the number of separated maximum points of the norm of the surface. The examples showing that this estimate is sharp are also presented.

1. Baernstein A., Integral means, univalent functions and circular simmetrization, Acta Math., 133 (1974), 139–169.

2. Beckenbach E.F., Hutchison G.A., Meromorphic minimal surfaces, Pac. Journ. of Math., 28 (1969), 14–47.

3. Beckenbach E.F., Cootz T., The second fundamental theorem for meromorphic minimal surfaces, Bull. Amer. Math. Soc., 76 (1970), 711–716.

4. Blashke W., Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitatstheorie. – Russian transl., ONTI, Moscow, 1935, 703 p.

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

6. Ciechanowicz E., Marchenko I.I., On the maximum modulus points of entire and meromorphic functions, Mat. Stud., 21 (2004), №1, 25–34.

7. Ciechanowicz E., Marchenko I.I., On the separated maximum modulus points of meromorphic functions, Mat. fizika, analiz, geometriya, 12 (2005), №2, 218–229.

8. Ciechanowicz E., Marchenko I.I., A note on the separated maximum modulus points of meromorphic fuctions, Ann. Pol. Math., 110 (2014), №3, 295–310.

9. Essen M., Shea D.F., Applications of Denjoy integral inequalities and differential inequalities to growth problems for subharmonic and meromorphic functions, Proc. Roy. Irish Acad. Sect., A82 (1982), 201– 216.

10. Gariepy R., Lewis J.L., Space analogues of some theorems for subharmonic and meromorphic functions, Ark. Mat., 13 (1975), 91–105.

11. Goldberg A.A., Ostrovskii I.V., Distribution of values of meromophic functions, Nauka, Moscow, 1970, (in Russian); Engl. transl. : AMS Transl., 236, Providence, 2008.

12. Hayman W.K., Multivalent functions, Cambridge Univ. Press, Cambridge, 1958.

13. Marchenko I.I., Growth of meromorphic minimal surfaces, Teor. Funkts., Funkts. Anal., Prilozh., 34 (1979), 95–98. (in Russian)

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

15. Marchenko I.I., On the maximum modulus points of entire and meromorphic functions and the problem of Erdos, Mat. Stud., 38 (2012), №2, 212–215.

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

17. Petrenko V.P., Growth and distribution of values of minimal surfaces, Dokl. Akad. Nauk SSSR, 256 (1981), №1, 40–43. (in Russian)

18. Vekua I.N., The basics of tensor analysis and theory of covariants. – Nauka, Moscow, 1978. (in Russian)

	On the maximum points and deviations of meromorphic minimal surfaces
Author	A. Kowalski, I. I. Marchenko akowalski@zut.edu.pl, iwan.marczenko@usz.edu.pl School of Mathematics, West Pomeranian University of Technology, Szczecin, Poland; Faculty of Mathematics and Physics, University of Szczecin, Poland
Abstract	We provide an upper estimate of the magnitude of deviation for a meromorphic minimal surface of finite lower order. The estimate is given in terms of the number of separated maximum points of the norm of the surface. The examples showing that this estimate is sharp are also presented.
Keywords	meromorphic minimal surfaces; maximum modulus point; subharmonic function; Nevanlinna theory
DOI	doi:10.15330/ms.46.2.137-151
Reference	1. Baernstein A., Integral means, univalent functions and circular simmetrization, Acta Math., 133 (1974), 139–169. 2. Beckenbach E.F., Hutchison G.A., Meromorphic minimal surfaces, Pac. Journ. of Math., 28 (1969), 14–47. 3. Beckenbach E.F., Cootz T., The second fundamental theorem for meromorphic minimal surfaces, Bull. Amer. Math. Soc., 76 (1970), 711–716. 4. Blashke W., Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitatstheorie. – Russian transl., ONTI, Moscow, 1935, 703 p. 5. Ciechanowicz E., Marchenko I.I., Maximum modulus points, deviations and spreads of meromorphic functions, Value Distribution Theory and Related Topics, Kluwer, 2004, 117–129. 6. Ciechanowicz E., Marchenko I.I., On the maximum modulus points of entire and meromorphic functions, Mat. Stud., 21 (2004), №1, 25–34. 7. Ciechanowicz E., Marchenko I.I., On the separated maximum modulus points of meromorphic functions, Mat. fizika, analiz, geometriya, 12 (2005), №2, 218–229. 8. Ciechanowicz E., Marchenko I.I., A note on the separated maximum modulus points of meromorphic fuctions, Ann. Pol. Math., 110 (2014), №3, 295–310. 9. Essen M., Shea D.F., Applications of Denjoy integral inequalities and differential inequalities to growth problems for subharmonic and meromorphic functions, Proc. Roy. Irish Acad. Sect., A82 (1982), 201– 216. 10. Gariepy R., Lewis J.L., Space analogues of some theorems for subharmonic and meromorphic functions, Ark. Mat., 13 (1975), 91–105. 11. Goldberg A.A., Ostrovskii I.V., Distribution of values of meromophic functions, Nauka, Moscow, 1970, (in Russian); Engl. transl. : AMS Transl., 236, Providence, 2008. 12. Hayman W.K., Multivalent functions, Cambridge Univ. Press, Cambridge, 1958. 13. Marchenko I.I., Growth of meromorphic minimal surfaces, Teor. Funkts., Funkts. Anal., Prilozh., 34 (1979), 95–98. (in Russian) 14. Marchenko I.I., On magnitudes of deviations and spreads of meromorphic functions of fnite lower order, Mat. Sb., 186 (1995), 85–102, (in Russian); Engl. transl.: Sb. Mat., 186 (1995), 391–408. 15. Marchenko I.I., On the maximum modulus points of entire and meromorphic functions and the problem of Erdos, Mat. Stud., 38 (2012), №2, 212–215. 16. Petrenko V.P., Growth of meromorphic functions. – Kharkov, Vyshcha Shkola, 1978, 136 p. (in Russian) 17. Petrenko V.P., Growth and distribution of values of minimal surfaces, Dokl. Akad. Nauk SSSR, 256 (1981), №1, 40–43. (in Russian) 18. Vekua I.N., The basics of tensor analysis and theory of covariants. – Nauka, Moscow, 1978. (in Russian)
Pages	137-151
Volume	46
Issue	2
Year	2016
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On the maximum points and deviations of meromorphic minimal surfaces