Generalization of the Schwarz–Pick inequality and its application to the extreme problems of approximation of holomorphic functions
UDC 517.5 We propose a method for the pointwise estimation of the derivative \[\left(\frac{f(z)-S_n(f)(z)}{z^n}\right)',\quad n\in\mathbb Z_+,\] where $S_n(f)$ is a partial sum of the Taylor series of a bounded holomorphic function in the unit circle $\mathbb D$ in terms of the absolute val...
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| Datum: | 2025 |
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| Hauptverfasser: | , , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2025
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/8692 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | UDC 517.5
We propose a method for the pointwise estimation of the derivative \[\left(\frac{f(z)-S_n(f)(z)}{z^n}\right)',\quad n\in\mathbb Z_+,\] where $S_n(f)$ is a partial sum of the Taylor series of a bounded holomorphic function in the unit circle $\mathbb D$ in terms of the absolute value of a function or the quantities of the type of best approximations. Sharp inequalities are obtained and the corresponding extreme functions are described. As a result, we deduce the Schwartz–Pick inequality and obtain the solutions of several extreme problems for the pointwise approximations of bounded holomorphic functions by the Fejér means of the Taylor series. |
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| DOI: | 10.3842/umzh.v77i1.8692 |