Vol. 22, No. 2, 1967

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Estimates for the transfinite diameter with applications to confomral mapping

Melvyn Klein

Vol. 22 (1967), No. 2, 267–279
Abstract

Let f(z) be a member of the family S of functions regular and univalent in the open unit disk whose Taylor expansion is of the form: f(z) = z + a2z2 + . Let Dw be the image of the unit disk under the mapping: w = f(z). An inequality for the transfinite diameter of n compact sets in the plane {Ti}1n is established, generalizing a result of Renngli:

d(T1 ∩ T2)⋅d(T1 ∪T2) ≦ d(T1) ⋅d(T2).

This inequality is applied to derive covering theorems for Dw relative to a class of curves issuing from w = 0, arcs on the circle: |w| = R as well as other point sets.

Mathematical Subject Classification
Primary: 30.41
Milestones
Received: 22 August 1966
Published: 1 August 1967
Authors
Melvyn Klein