2018/06/10 by Peretz, Ronen
#Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1806.03616
There are three types of results in this paper. The first, extending a representation theorem on a conformal mapping that omits two values of equal modulus. This was due to Brickman and Wilken. They constructed a representation as a convex combination with two terms. Our representation constructs convex combinations with unlimited number of terms. In the limit one can think of it as an integration over a probability space with the uniform distribution. The second result determines the sign of \Re L(z0(f(z))2) up to a remainder term which is expressed using a certain integral that involves the Löwner chain induced by f(z), for a support point f(z) which maximizes \Re L. Here L is a continuous linear functional on H(U), the topological vector space of the holomorphic functions in the unit disk U=\z∈ℂ | |z|<1\. Such a support point is known to be a slit mapping and f(z0) is the tip of the slit ℂ-f(U). The third demonstrates some properties of support points of the subspace Sn of S. Sn contains all the polynomials in S of degree n or less. For instance such a support point p(z) has a zero of its derivative p'(z) on ∂ U.