## Currently displaying 1 – 5 of 5

Showing per page

Order by Relevance | Title | Year of publication

### Estimates for polynomials in the unit disk with varying constant terms

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

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

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

### Gauss curvature estimates for minimal graphs

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

We estimate the Gauss curvature of nonparametric minimal surfaces over the two-slit plane $ℂ\setminus \left(\left(-\infty ,-1\right]\cup \left[1,\infty \right)\right)$ at points above the interval $\left(-1,1\right)$.

### Harmonic mappings onto parallel slit domains

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.

Page 1