Radial limiting behaviour of harmonic functions in cones
D. H. Armitage, Myron Goldstein · Complex Variables Theory and Application An International Journal · 1993
Let E be a second category subset of the sphere in and let C be an open cone in , with vertex at the origin, such that E⊂C. Extending results of W.J. Schneider, who worked with in place of C, we show that (i) if h is harmonic in C and h(rx) = O(exp(-r μ)) for all μ>0 and all x∊E, then h=0 and (ii) there is no harmonic function g in C such that r -μ g(rx)→∞ as r→∞ for all μ>0 and ail x∊E. Examples are given to show the sharpness of these theorems.