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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2010, Volume 16, Number 4, Pages 254–263
(Mi timm659)
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This article is cited in 2 scientific papers (total in 2 papers)
Exact inequality between uniform norms of an algebraic polynomial and its real part on concentric circles in the complex plane
A. V. Parfenenkov Ural State University
Abstract:
In the class $\mathcal P_n^*$ of algebraic polynomials of a complex variable of degree at most n with complex coefficients and a real constant term, we estimate the uniform norm of a polynomial $P_n\in\mathcal P_n^*$ on the circle $\Gamma_r=\{z\in\mathbb C\colon|z|=r\}$ of radius $r>1$ in terms of the norm of its real part on the unit circle $\Gamma_1$. More precisely, we study the best constant $\mu(r,n)$ in the inequality $\|P_n\|_{C(\Gamma_r)}\leq\mu(r,n)\|\operatorname{Re}P_n\|_{C(\Gamma_1)}$. Necessary and sufficient conditions for the equality $\mu(r,n)=r^n$ are found.
Keywords:
inequalities for algebraic polynomials, uniform norm, circle in the complex plane.
Received: 23.05.2010
Citation:
A. V. Parfenenkov, “Exact inequality between uniform norms of an algebraic polynomial and its real part on concentric circles in the complex plane”, Trudy Inst. Mat. i Mekh. UrO RAN, 16, no. 4, 2010, 254–263
Linking options:
https://www.mathnet.ru/eng/timm659 https://www.mathnet.ru/eng/timm/v16/i4/p254
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Abstract page: | 318 | Full-text PDF : | 74 | References: | 51 |
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