Abstract
This note presents maximum principles for subharmonic functions, based on asymptotic behaviour, in both the Euclidean and Martin boundary settings. Several known results are generalized.
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References
D.H. Armitage, A strong type of regularity for the Dirichlet problem, Proc. Amer. Math. Soc. 61 (1976), 285–289.
D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer, London, 2001.
K. F. Barth and P. J. Rippon, Minimal fine limits of subharmonic functions of slow growth, J. London Math. Soc. (2) 52 (1995), 517–528.
K. F. Barth and P. J. Rippon, Extensions of a theorem of Valiron, Bull. London Math. Soc. 38 (2006), 815–824.
C. Huaihui and P. M. Gauthier, A maximum principle for subharmonic and plurisubharmonic functions, Canad. Math. Bull. 35 (1992), 34–39.
B. E. J. Dahlberg, Estimates for harmonic measure, Arch. Rational Mech. Anal. 65 (1977), 275–288.
B. Fuglede, Finely Harmonic Functions, Lecture Notes in Math. 289, Springer, Berlin, 1972.
B. Fuglede, Asymptotic paths for subharmonic functions, Math. Ann. 213 (1975), 261–274.
S. J. Gardiner, Harmonic Approximation, London Math. Soc. Lecture Note Series, 221, Cambridge Univ. Press, 1995.
P. M. Gauthier, M. Goldstein and W. H. Ow, Uniform approximation on unbounded sets by harmonic functions with logarithmic singularities, Trans. Amer. Math. Soc. 261 (1980), 169–183.
P. M. Gauthier, M. Goldstein and W. H. Ow, Uniform approximation on closed sets by harmonic functions with Newtonian singularities, J. London. Math. Soc. (2) 28 (1983), 71–82.
P. M. Gauthier, R. Grothmann and W. Hengartner, Asymptotic maximum principles for subharmonic and plurisubharmonic functions, Canad. J. Math. 40 (1988), 477–486.
T. Lyons, Finely holomorphic functions, J. Functional Anal. 37 (1980), 1–18.
G. R. MacLane, Asymptotic values of holomorphic functions, Rice University Studies 49 (1963), 83pp.
J. E. McMillan, Asymptotic values of functions holomorphic in the unit disc, Michigan Math. J. 12 (1965), 141–154.
L. Naïm, Sur le rôle de la frontiére de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier (Grenoble) 7 (1957), 183–281.
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Dedicated to Walter Hayman on the occasion of his 80th birthday
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Gardiner, S.J. Asymptotic Maximum Principles for Subharmonic Functions. Comput. Methods Funct. Theory 8, 167–172 (2008). https://doi.org/10.1007/BF03321680
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DOI: https://doi.org/10.1007/BF03321680