In complex analysis, given initial data consisting of n {\displaystyle n} points λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} in the complex unit disk D {\displaystyle \mathbb {D} } and target data consisting of n {\displaystyle n} points z 1 , … , z n {\displaystyle z_{1},\ldots ,z_{n}} in D {\displaystyle \mathbb {D} } , the Nevanlinna–Pick interpolation problem is to find a holomorphic function φ {\displaystyle \varphi } that interpolates the data, that is for all i ∈ { 1 , . . . , n } {\displaystyle i\in \{1,...,n\}} ,
φ ( λ i ) = z i {\displaystyle \varphi (\lambda _{i})=z_{i}} , subject to the constraint | φ ( λ ) | ≤ 1 {\displaystyle \left\vert \varphi (\lambda )\right\vert \leq 1} for all λ ∈ D {\displaystyle \lambda \in \mathbb {D} } . Georg Pick and Rolf Nevanlinna solved the problem independently in 1916 and 1919 respectively, showing that an interpolating function exists if and only if a matrix defined in terms of the initial and target data is positive semi-definite.
Background The Nevanlinna–Pick theorem represents an n {\displaystyle n} -point generalization of the Schwarz lemma. The invariant form of the Schwarz lemma states that for a holomorphic function f : D → D {\displaystyle f:\mathbb {D} \to \mathbb {D} } , for all λ 1 , λ 2 ∈ D {\displaystyle \lambda _{1},\lambda _{2}\in \mathbb {D} } ,
| f ( λ 1 ) − f ( λ 2 ) 1 − f ( λ 2 ) ¯ f ( λ 1 ) | ≤ | λ 1 − λ 2 1 − λ 2 ¯ λ 1 | . {\displaystyle \left|{\frac {f(\lambda _{1})-f(\lambda _{2})}{1-{\overline {f(\lambda _{2})}}f(\lambda _{1})}}\right|\leq \left|{\frac {\lambda _{1}-\lambda _{2}}{1-{\overline {\lambda _{2}}}\lambda _{1}}}\right|.}
Setting f ( λ i ) = z i {\displaystyle f(\lambda _{i})=z_{i}} , this inequality is equivalent to the statement that the matrix given by
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