Uniform domains and -domains in Carnot groups.
Let a < 0, Ω = ℂ -(-∞, a] and U = z: |z| < 1. We consider the class of functions f which are univalent, harmonic and sense preserving with f(U) = Ω and satisfy f(0) = 0, and . We describe the closure of and determine the extreme points of .
Let a < 0 < b and Ω(a,b) = ℂ - ((-∞, a] ∪ [b,+∞)) and U= z: |z| < 1. We consider the class of functions f which are univalent, harmonic and sense-preserving with f(U) = Ω and satisfying f(0) = 0, and .
We prove universal overconvergence phenomena for harmonic functions on the real hyperbolic space.
A holomorphic function on a simply connected domain is said to possess a universal Taylor series about a point in if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta outside (provided only that has connected complement). This paper shows that this property is not conformally invariant, and, in the case where is the unit disc, that such functions have extreme angular boundary behaviour.