Expectations and estimates for some conformal invariants

This dissertation explores how certain conformal invariants of domains behave when randomness or uncertainty of the related points or domains is introduced. By `conformal invariant' it is meant that the quantity is either preserved or follows some transformation rule when the domain is mapped conformally to a second domain.

In Chapter 2 we first examine the coefficients $ak$ of the power series centered at 0 for conformal maps from the unit disk $D$ to a simply connected domain $D$. Briefly we look to find all moments of all coefficients of automorphisms of $D$ when the image of 0 is distributed uniformly in $D$. We then examine in depth the expected value of the coefficient $a1$, the conformal radius, for mappings from $D$ onto arbitrary domains. We start with some properties and transformation rules and will conclude with calculations for specific classes of domains. These calculations provide evidence which supports many conjectures which are stated and remain unproven.

In Chapter 3 the product of conformal radii of points lying in disjoint domains is explored. In a deterministic setting this problem is well studied and a tight upper bound is known for the product involving only two points. In this chapter we examine the situation when the base points are distributed randomly in a single set. In this situation there is no good way to randomly distribute the domains containing these points, so the best that can be hoped for is an estimate on the expected value. Through examples we demonstrate how the upper bound for two points in the deterministic case can be used to find an upper bound for moments in a random setting. We then examine the product for three points. To this end we derive a tight upper bound for the product of conformal radii for three points in a deterministic setting and then use it to find upper bounds for the expected values of the product of conformal radii.

For Chapter 4 another conformal invariant, harmonic measure, is studied. In one problem we find the expected value of the harmonic measure of a boundary set for a domain when the boundary set is randomly distributed. In another problem we make use of symmetry to find the expected value of the harmonic measure when the base point is randomly distributed along a line segment.