Subharmonic functions on annuli

A.A.Kondratyuk; O. V.Stashyshyn

Nevanlinna characteristic of subharmonic functions in an annulus, which is invariant with respect to the inversion is introduced. A counterpart of Jensen's theorem for subharmonic functions in such annulus is proved. The best possible estimate of maximum of a subharmonic function via its Nevanlinna characteristic is established.

1. W. K. Hayman, P. B. Kennedy, Subharmonic functions. Vol. I. Academic Press London-New York, 1976.

2. Sheldon Axler, Harmonic functions from a complex analysis viewpoint, Amer. Math. Mont. 93 (1986), 246-258.

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

4. A. Kondratyuk, I. Laine, Meromorphic functions in multiply connected domains, Fourier series methods in complex analysis. Univ. Joensuu Dept. Math. Rep. Ser., (2006), № 10, 9-111.

5. 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.

6. G. af Hällström, Über meromorpher Functionen mit mehrfach zusammenhängenden Existenygebieten, Acta Acad. Math. et Phys., 12 (1940), № 8, 1-100.

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

8. R. Korhonen, Nevanlinna theory in an annulus, in: Value Distribution Theory and Related Topics, in: Adv. Complex Anal. Appl., V.3, Kluwer Acad. Publ., Boston, MA, 2004, 167-179.

9. M. Lund, Nevanlinna theory for annuli. PhD thesis, Northern Illinois University, DeKalb. IL, 2009.

10. G. Valiron, Lectures on the General Theory of Integral Functions, Chelsea Publishing Company, New York, 1949.

11. H. H. Mathevossian, On a factorization of meromorphic function in multiply connected domain and some of its applications. Izv. Akad. Nauk Arm. SSR IX (1974), № 5, 387-408.

12. N. Oguztöreli, Extension de la théorie de Nevanlinna aux domaines multiplement connexes. Rev. Fac. Sci. Univ. Istanbul. Sér A18 (1953), № 4, 384-419.

13. N. Oguztöreli, Représentations intégrales de la fonction caractéristique, de la fonction de nombre et de la forme sphérique normale généralisée et extension d'un théorème de Borel. Rev. Fac. Sci. Univ. Istanbul. Sér A19 (1954), № 2, 79-85.

14. V. A. Zmorovich, On certain classes of analytic functions in a circular ring. Math. Sb. 40 (1956), 225-238.

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

16. A. A. Kondratyuk, Representations of Green's functions in annuli, XVth Conference of analytic functions and related topics, Helm (Poland), July 5-9, 2009.

	Subharmonic functions on annuli
Author	A.A.Kondratyuk, O. V.Stashyshyn kond@franko.lviv.ua, ostap.stashyshyn@gmail.com Faculty of,Mechanics and Mathematics,Ivan Franko National University of L'viv
Abstract	Nevanlinna characteristic of subharmonic functions in an annulus, which is invariant with respect to the inversion is introduced. A counterpart of Jensen's theorem for subharmonic functions in such annulus is proved. The best possible estimate of maximum of a subharmonic function via its Nevanlinna characteristic is established.
Keywords	subharmonic function, Navanlinna characteristic, annulus, inversion, Jensen's theorem
DOI	doi:10.30970/ms.33.1.92-96
Reference	1. W. K. Hayman, P. B. Kennedy, Subharmonic functions. Vol. I. Academic Press London-New York, 1976. 2. Sheldon Axler, Harmonic functions from a complex analysis viewpoint, Amer. Math. Mont. 93 (1986), 246-258. 3. A. Ya. Khrystiyanyn, A. A. Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli, I. Mat. Stud. 23 (2005), № 1, 19-30. 4. A. Kondratyuk, I. Laine, Meromorphic functions in multiply connected domains, Fourier series methods in complex analysis. Univ. Joensuu Dept. Math. Rep. Ser., (2006), № 10, 9-111. 5. 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. 6. G. af Hällström, Über meromorpher Functionen mit mehrfach zusammenhängenden Existenygebieten, Acta Acad. Math. et Phys., 12 (1940), № 8, 1-100. 7. A. Ya. Khrystiyanyn, A. A. Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli, II. Mat. Stud. 24 (2005), № 2, 57-68. 8. R. Korhonen, Nevanlinna theory in an annulus, in: Value Distribution Theory and Related Topics, in: Adv. Complex Anal. Appl., V.3, Kluwer Acad. Publ., Boston, MA, 2004, 167-179. 9. M. Lund, Nevanlinna theory for annuli. PhD thesis, Northern Illinois University, DeKalb. IL, 2009. 10. G. Valiron, Lectures on the General Theory of Integral Functions, Chelsea Publishing Company, New York, 1949. 11. H. H. Mathevossian, On a factorization of meromorphic function in multiply connected domain and some of its applications. Izv. Akad. Nauk Arm. SSR IX (1974), № 5, 387-408. 12. N. Oguztöreli, Extension de la théorie de Nevanlinna aux domaines multiplement connexes. Rev. Fac. Sci. Univ. Istanbul. Sér A18 (1953), № 4, 384-419. 13. N. Oguztöreli, Représentations intégrales de la fonction caractéristique, de la fonction de nombre et de la forme sphérique normale généralisée et extension d'un théorème de Borel. Rev. Fac. Sci. Univ. Istanbul. Sér A19 (1954), № 2, 79-85. 14. V. A. Zmorovich, On certain classes of analytic functions in a circular ring. Math. Sb. 40 (1956), 225-238. 15. A. A. Kondratyuk, Meromorphic functions with several essential singularities. I. Mat. Stud. 30 (2008), № 2, 125-131. 16. A. A. Kondratyuk, Representations of Green's functions in annuli, XVth Conference of analytic functions and related topics, Helm (Poland), July 5-9, 2009.
Pages	92-96
Volume	33
Issue	1
Year	2010
Journal	Matematychni Studii
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