A simple proof for Schur's theorem (Q2731912)
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scientific article; zbMATH DE number 1626787
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A simple proof for Schur's theorem |
scientific article; zbMATH DE number 1626787 |
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30 July 2001
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A simple proof for Schur's theorem (English)
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The author presents a simple, elementary proof of the following Schur theorem: Let \(f(z)=\sum_{k=0}^\infty a_kz^k\) be a function holomorphic in the unit disc \(\Delta\). Then \(f(\Delta)\subset\overline\Delta\) iff NEWLINE\[NEWLINE \sum_{k=0}^N\biggr|\sum_{n=k}^Na_{n-k}\lambda_n\biggl|^2 \leq\sum_{k=0}^N|\lambda_k|^2,\quad N\in\mathbb N, \lambda_0,\dots,\lambda_N\in\mathbb C .NEWLINE\]
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