Estimates for polynomials in the unit disk with varying constant terms

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

Stable functions and Vietoris' theorem

AbstractAn analytic function f(z) in the unit disc D is called stable if sn(f,·)/f≺1/f holds for all for n∈N0. Here sn stands for the nth partial sum of the Taylor expansion about the origin of f, and ≺ denotes the subordination of analytic functions in D. We prove that (1−z)λ, λ∈[−1,1], are stable. The stability of (1+z)/(1−z) turns out to be equivalent to a famous result of Vietoris on non-negative trigonometric sums. We discuss some generalizations of these results, and related conjectures, always with an eye on applications to positivity results for trigonometric and other polynomials

Inequalities for Cyclic Functions

AbstractThe nth cyclic function is defined byϕn(z)=∑ν=0∞znν(nν)!(z∈C,2⩽n∈N). We prove that if k is an integer with 1⩽k⩽n−1, then(n−k)!ϕ(k)n(x)xn−kα<ϕn(x)<(n−k)!ϕ(k)n(x)xn−kβ holds for all positive real numbers x with the best possible constantsα=1andβ=2n-k over n