Subharmonic functions and electric fields in ball layers. I

O.P.Gnatiuk; A.A.Kondratyuk

A two-parameter approach for investigation of subharmonic functions in ball layers is presented. The explicit forms of Green's function and of Poisson-Jensen's formula as well as a counterpart of Jensen's Theorem are obtained. The relations between different growth characteristics of subharmonic functions in ball layers are established. The obtained results are relevant to some problems of electrostatics.

1. G. af Hällström , $\ddot{U}$ber meromorpher Functionen mit mehrfach zusammenh$\ddot{a}$ngenden Existenzgebieten, Acta Acad. Aboensis, Math. et Phys. 12, (1940), №8, 1--100.

2. A. Kondratyuk, I.Laine, Meromorphic functions in multiply connected domains, Fourier series methods in complex analysis, Univ. Joensuu Dept. Math. Rep. Ser. 10 (2006), 9--111.

3. R. Korhonen, Nevanlinna theory in an annulus, Adv. Complex Anal. Appl. Kluwer Acad. Publ. Boston, MA. 3, (2004), 167--179.

4. A.Ya. Khrystiyanyn, A.A Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli. I // Mat. Stud. 23 (2005), №1, 19--30.

5. A.Ya. Khrystiyanyn, A.A Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli. II // Mat. Stud. 24 (2005), №2, 57--68.

6. M.O. Hanyak, A.A Kondratyuk, Meromorphic functions in $m$-punctured complex planes // Mat. Stud. 27 (2007), №2, 53--69.

7. A.A. Kondratyuk, Meromorphic functions with several essential singularities. I // Mat. Stud. 30 (2008), №2, 125--131.

8. N. Oguztöreli, Extension de la théorie de Nevanlinna aux domaines multiplement connexes, Rev. Fac. Sci. Univ. Istanbul. S$\acute{e}$r A 18 (1953), №4, 384--419.

9. L.A. Rubel, B.A. Taylor, A Fourier series method for meromorphic and entire functions, Bull. Soc. Math. France. 96 (1968), 53--96.

10. T. Shimizu, On the theory of meromorphic functions, Japanese J. Math. 6 (1929), 119--171.

11. L. Villat, Le problem de Dirichlet dans aire annulaire, Rend. Circ. Mat. Palermo. 33 (1912).

12. M. Lund, Nevanlinna theory for annuli, PhD thesis, Northern Illinois University, Dekalb, IL, 2009.

13. A.A Kondratyuk, O.V. Stashyshyn, Subharmonic functions on annuli// Mat. Stud. 33 (2010), №1, 92--96.

14. D.G. Crowdy, J.S. Marshall, Green's functions for Laplace's equation in multiply connected domains, IMA. J. Appl. Math. 72 (2007), 278--301.

15. M. Embree, L.N. Trefethen, Green's functions for multiply connected domains via conformal mapping, SIAM Rev. 41 (1999), 745--761.

16. V.V. Mityushev, S.V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions, Monographs and Surveys in Pure and Applied Mathematics, Boca Raton, Fl: Chapman&Hall, 2000.

17. E.M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidian Spaces, Princeton University Press, 1971.

18. H. Berens, P. Butzer, S. Pavelke, Zimitirungsverhalten von Reihen mehrdimensionalen Kugelfuntkionen und deren Saturationsverhalten, Publs. Res. Inst. Math. Sci. 2 (1968), 201--268.

19. A.A. Kondratyuk, Spherical harmonics and subharmonics functions, Math. USSR Sbornic, 53 (1986), 147--166. (in Russian)

20. S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, Springer, N.Y., 1992.

21. W.K. Hayman, P.B. Kennedy, Subharmonic functions, Mir, Moskow, 1980. (in Russian)

22. O. Gnatiuk, O. Greshko, O. Stashyshyn, Two-parameter growth characteristics of nonnegative and meromorphic in annulus functions // Visnyk Lviv Univ, Ser. Mech. Math. 71 (2009), 62--70.

	Subharmonic functions and electric fields in ball layers. I
Author	O.P.Gnatiuk, A.A.Kondratyuk oksanka.gnatyuk@gmail.com, kond@franko.lviv.ua Faculty of Mechanics and Mathematics Lviv National University Universytets'ka 1, 79000, Lviv
Abstract	A two-parameter approach for investigation of subharmonic functions in ball layers is presented. The explicit forms of Green's function and of Poisson-Jensen's formula as well as a counterpart of Jensen's Theorem are obtained. The relations between different growth characteristics of subharmonic functions in ball layers are established. The obtained results are relevant to some problems of electrostatics.
Keywords	subharmonic function, ball layer, electric field, Green's function, Poisson-Jensen's formula
DOI	doi:10.30970/ms.34.2.180-192
Reference	1. G. af Hällström , $\ddot{U}$ber meromorpher Functionen mit mehrfach zusammenh$\ddot{a}$ngenden Existenzgebieten, Acta Acad. Aboensis, Math. et Phys. 12, (1940), №8, 1--100. 2. A. Kondratyuk, I.Laine, Meromorphic functions in multiply connected domains, Fourier series methods in complex analysis, Univ. Joensuu Dept. Math. Rep. Ser. 10 (2006), 9--111. 3. R. Korhonen, Nevanlinna theory in an annulus, Adv. Complex Anal. Appl. Kluwer Acad. Publ. Boston, MA. 3, (2004), 167--179. 4. A.Ya. Khrystiyanyn, A.A Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli. I // Mat. Stud. 23 (2005), №1, 19--30. 5. A.Ya. Khrystiyanyn, A.A Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli. II // Mat. Stud. 24 (2005), №2, 57--68. 6. M.O. Hanyak, A.A Kondratyuk, Meromorphic functions in $m$-punctured complex planes // Mat. Stud. 27 (2007), №2, 53--69. 7. A.A. Kondratyuk, Meromorphic functions with several essential singularities. I // Mat. Stud. 30 (2008), №2, 125--131. 8. N. Oguztöreli, Extension de la théorie de Nevanlinna aux domaines multiplement connexes, Rev. Fac. Sci. Univ. Istanbul. S$\acute{e}$r A 18 (1953), №4, 384--419. 9. L.A. Rubel, B.A. Taylor, A Fourier series method for meromorphic and entire functions, Bull. Soc. Math. France. 96 (1968), 53--96. 10. T. Shimizu, On the theory of meromorphic functions, Japanese J. Math. 6 (1929), 119--171. 11. L. Villat, Le problem de Dirichlet dans aire annulaire, Rend. Circ. Mat. Palermo. 33 (1912). 12. M. Lund, Nevanlinna theory for annuli, PhD thesis, Northern Illinois University, Dekalb, IL, 2009. 13. A.A Kondratyuk, O.V. Stashyshyn, Subharmonic functions on annuli// Mat. Stud. 33 (2010), №1, 92--96. 14. D.G. Crowdy, J.S. Marshall, Green's functions for Laplace's equation in multiply connected domains, IMA. J. Appl. Math. 72 (2007), 278--301. 15. M. Embree, L.N. Trefethen, Green's functions for multiply connected domains via conformal mapping, SIAM Rev. 41 (1999), 745--761. 16. V.V. Mityushev, S.V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions, Monographs and Surveys in Pure and Applied Mathematics, Boca Raton, Fl: Chapman&Hall, 2000. 17. E.M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidian Spaces, Princeton University Press, 1971. 18. H. Berens, P. Butzer, S. Pavelke, Zimitirungsverhalten von Reihen mehrdimensionalen Kugelfuntkionen und deren Saturationsverhalten, Publs. Res. Inst. Math. Sci. 2 (1968), 201--268. 19. A.A. Kondratyuk, Spherical harmonics and subharmonics functions, Math. USSR Sbornic, 53 (1986), 147--166. (in Russian) 20. S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, Springer, N.Y., 1992. 21. W.K. Hayman, P.B. Kennedy, Subharmonic functions, Mir, Moskow, 1980. (in Russian) 22. O. Gnatiuk, O. Greshko, O. Stashyshyn, Two-parameter growth characteristics of nonnegative and meromorphic in annulus functions // Visnyk Lviv Univ, Ser. Mech. Math. 71 (2009), 62--70.
Pages	180-192
Volume	34
Issue	2
Year	2010
Journal	Matematychni Studii
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