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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 ℜ  L (  z   0 ( 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 ℜ  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 | &lt; 1}. Such a support point is known to be a slit mapping and  f ( z  0 ) is the tip of the slit ℂ −  f ( U ). The third demonstrates some properties of support points of the subspace  S n   of  S .  S n   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 .

Extreme points and support points of conformal mappings