Abstract: Let $f$ be a holomorphic function in the unit disc with the property that the numbers of solutions of the equations $f(z)=0$ and $f(z)=1$ (counting multiplicity) are finite, non-zero and distinct. A. A. Goldberg considered in 1970 the following extremal problem in this class: to minimize the radius of the hyperbolic disc that contains all these solutions.

Later it was found that this and similar problems are relevant to control theory. A survey of recent results on this problem will be given.

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