A peaking and interpolation problem for univalent functions (Q1070053)

The main result is the following: If \(\alpha_ 1<\alpha_ 2<\cdots <\alpha_ n<\alpha_ 1+2\pi\) and \(\beta_ 1<\beta_ 2<\cdots <\beta_ n<\beta_ 1+2\pi\), then there is a function f, analytic and univalent in\(\{\) \(| z| \leq 1\}\), for which \(| f(z)| <1\) when \(| z| \leq 1\) and \(z\neq z_ k=e^{i\alpha_ k}\) \((k=1,2,\ldots,n)\), while \(f(z_ k)=e^{i\beta_ k}=w_ k\) \((k=1,2,\ldots,n)\). It is also shown that f may be taken to be a polynomial. The paper also contains simple proofs of some less general results, e.g. the existence of functions f which may fail to be univalent or which do not interpolate given values.