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2 results

  • THE MEAN-VALUE PROPERTY AND (α,β)-HARMONICITY

  • Part of
  • DAN LI, CONGWEN LIU
  • Journal:
    Journal of the Australian Mathematical Society / Volume 91 / Issue 2 / October 2011
    Published online by Cambridge University Press:
    14 October 2011, pp. 189-206
    Print publication:
    October 2011
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  • In this article we consider whether, for integrable functions on the unit ball of ℂn, the mean-value property implies (α,β)-harmonicity. We find that the answer is affirmative when 0<n+α+β≤ρ0, but is negative when n+α+β>ρ0. Here ρ0 is a constant between 11.025 and 11.069.

  • MEAN-VALUE PROPERTY ON MANIFOLDS WITH MINIMAL HOROSPHERES

  • Part of
  • LEONARD TODJIHOUNDE
  • Journal:
    Journal of the Australian Mathematical Society / Volume 84 / Issue 2 / April 2008
    Published online by Cambridge University Press:
    01 April 2008, pp. 277-282
    Print publication:
    April 2008
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  • Let (M,g) be a non-compact and complete Riemannian manifold with minimal horospheres and infinite injectivity radius. In this paper we prove that bounded functions on (M,g) satisfying the mean-value property are constant. We thus extend a result of Ranjan and Shah [‘Harmonic manifolds with minimal horospheres’, J. Geom. Anal.12(4) (2002), 683–694] where they proved a similar result for bounded harmonic functions on harmonic manifolds with minimal horospheres.