Previous |  Up |  Next

Article

Keywords:
Nevanlinna theorem; superharmonic function; $\delta $-subharmonic function; Riesz measure; mean value
Summary:
A generalization of Nevanlinna’s First Fundamental Theorem to superharmonic functions on Green balls is proved. This enables us to generalize many other theorems, on the behaviour of mean values of superharmonic functions over Green spheres, on the Hausdorff measures of certain sets, on the Riesz measures of superharmonic functions, and on differences of positive superharmonic functions.
References:
[1] D. H. Armitage: A Nevanlinna theorem for superharmonic functions in half-spaces, with applications. J. London Math. Soc. 23 (1981), 137–157. MR 0602246 | Zbl 0427.31003
[3] D. H. Armitage: Mean values and associated measures of superharmonic functions. Hiroshima Math. J. 13 (1983), 53–63. MR 0693550 | Zbl 0512.31009
[3] J. L. Doob: Classical Potential Theory and its Probabilistic Counterpart. Springer, New York, 1984. MR 0731258 | Zbl 0549.31001
[4] K. J. Falconer: The Geometry of Fractal Sets. Cambridge University Press, Cambridge, 1985. MR 0867284 | Zbl 0587.28004
[5] W. K. Hayman: Subharmonic Functions, Vol. 2. Academic Press, London, 1989. MR 1049148
[6] W. K. Hayman, P. B. Kennedy: Subharmonic Functions, Vol. 1. Academic Press, London, 1976.
[7] Ü. Kuran: On measures associated to superharmonic functions. Proc. Amer. Math. Soc. 36 (1972), 179–186. DOI 10.1090/S0002-9939-1972-0316728-0 | MR 0316728 | Zbl 0255.31004
[8] M. J. Parker: The fundamental function of the distance. Bull. London Math. Soc. 19 (1987), 337–342. DOI 10.1112/blms/19.4.337 | MR 0887772 | Zbl 0644.31005
[9] C. A. Rogers: Hausdorff Measures. Cambridge University Press, Cambridge, 1970. MR 0281862 | Zbl 0204.37601
[10] C. A. Rogers, S. J. Taylor: Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8 (1961), 1–31. DOI 10.1112/S0025579300002084 | MR 0130336
[11] N. A. Watson: Mean values of subharmonic functions over Green spheres. Math. Scand. 69 (1991), 307–319. MR 1156430 | Zbl 0769.31001
[12] N. A. Watson: Generalizations of the spherical mean convexity theorem on subharmonic functions. Ann. Acad. Sci. Fenn. Ser. A I Math. 17 (1992), 241–255. DOI 10.5186/aasfm.1992.1733 | MR 1190322 | Zbl 0770.31005
[13] N. A. Watson: Superharmonic extensions, mean values and Riesz measures. Potential Anal. 2 (1993), 269–294. DOI 10.1007/BF01048511 | MR 1245245 | Zbl 0785.31002
[14] N. A. Watson: Mean values and associated measures of $\delta $-subharmonic functions. Math. Bohem. 127 (2002), 83–102. MR 1895249 | Zbl 0998.31002
[15] N. A. Watson: A generalized Nevanlinna theorem for supertemperatures. Ann. Acad. Sci. Fenn. Math. 28 (2003), 35–54. MR 1976828 | Zbl 1035.31006
Partner of
EuDML logo