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Let P be a homogeneous polynomial on RN
and let be the partial differential operator associated to P. Bounds for Dph(0) are obtained, where h is a positive harmonic function on the unit ball of RN
, or h is the difference of two such function. For certain polynomials P the bounds are shown to be sharp and extremal harmonic functions are identified. The results extend a theorem of Goldstein and Kuran [7] which gives sharp bounds for
.
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