LetPbe a homogeneous polynomial onRNand letbe the partial differential operator associated toP. Bounds forDph(0) are obtained, wherehis a positive harmonic function on the unit ball ofRN, orhis the difference of two such function. For certain polynomialsPthe 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.