## Abstract

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 RL(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 RL. 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(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.

Original language | English |
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Pages (from-to) | 23-31 |

Number of pages | 9 |

Journal | Open Mathematics |

Volume | 17 |

Issue number | 1 |

DOIs | |

State | Published - 17 Feb 2019 |

## Keywords

- Conformal mappings
- Extreme points
- Schlicht functions
- Support points

## ASJC Scopus subject areas

- Mathematics (all)