TL;DR
This paper investigates two extremal problems in geometric function theory introduced by Goldberg, providing an exact solution for one and partial results for the other, with implications for control theory.
Contribution
It offers the first exact solution to one of Goldberg's extremal problems and advances understanding of hyperbolic geodesics in punctured planes.
Findings
Exact solution to one extremal problem
Partial results and conjecture on the second problem
New insights into hyperbolic geodesics in punctured planes
Abstract
We study two extremal problems of geometric function theory introduced by A. A. Goldberg in 1973. For one problem we find the exact solution, and for the second one we obtain partial results. In the process we study the lengths of hyperbolic geodesics in the twice punctured plane, prove several results about them and make a conjecture. Goldberg's problems have important applications to control theory.
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