Estimates for polynomials in the unit disk with varying constant terms

Stephan Ruscheweyh, Magdalena Wołoszkiewicz

Abstract


Let \(\| \cdot\|\) be the uniform norm in the unit disk. We study the quantities \(M_n(\alpha) := \inf(\|zP(z) + \alpha\|-\alpha)\) where the infimum is taken over all polynomials \(P\) of degree \(n-1\) with \(\|P(z)\| = 1\) and \(\alpha> 0\). In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that \(\inf_{\alpha> 0} M_n(\alpha) = 1/n\). We find the exact values of \(M_n(\alpha)\) and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.

Keywords


Bernstein-type inequalities for complex polynomials; maximal ranges for polynomials

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References


Andrievskii, V., Ruscheweyh, S., Complex polynomials and maximal ranges: background and applications, Recent progress in inequalities (Nis, 1996), Math. Appl.,

, Kluwer Acad. Publ., Dordrecht, 1998, 31-54.

Córdova, A., Ruscheweyh, S., On maximal polynomial ranges in circular domains, Complex Variables Theory Appl. 10 (1988), 295-309.

Córdova, A., Ruscheweyh, S., On maximal ranges of polynomial spaces in the unit disk, Constr. Approx. 5 (1989), 309-327.

Fournier, R., Letac, G. and Ruscheweyh, S., Estimates for the uniform norm of complex polynomials in the unit disk, Math. Nachr. 283 (2010), 193-199.

Ruscheweyh, S., Varga, R., On the minimum moduli of normalized polynomials with two prescribed values, Constr. Approx. 2 (1986), 349-368.




DOI: http://dx.doi.org/10.2478/v10062-011-0022-5
Date of publication: 2016-07-27 21:54:12
Date of submission: 2016-07-27 14:26:52


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Copyright (c) 2011 Stephan Ruscheweyh, Magdalena Wołoszkiewicz