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Estimates for polynomials in the unit disk with varying constant terms

Stephan RuscheweyhMagdalena Wołoszkiewicz — 2011

Annales UMCS, Mathematica

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

Estimates for polynomials in the unit disk with varying constant terms

Stephan RuscheweyhMagdalena Wołoszkiewicz — 2011

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

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

Harmonic mappings onto parallel slit domains

Michael DorffMaria NowakMagdalena Wołoszkiewicz — 2011

Annales Polonici Mathematici

We consider typically real harmonic univalent functions in the unit disk 𝔻 whose range is the complex plane slit along infinite intervals on each of the lines x ± ib, b > 0. They are obtained via the shear construction of conformal mappings of 𝔻 onto the plane without two or four half-lines symmetric with respect to the real axis.

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