### ...- Invarianten bei verallgemeinerten Carlesonmengen.

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Given a set of points in the complex plane, an incomplete polynomial is defined as the one which has these points as zeros except one of them. The classical result known as Gauss-Lucas theorem on the location of zeros of polynomials and their derivatives is extended to convex linear combinations of incomplete polynomials. An integral representation of convex linear combinations of incomplete polynomials is also given.

Let p(z) be a polynomial of the form $p\left(z\right)={\sum}_{j=0}^{n}{a}_{j}{z}^{j}$, ${a}_{j}\in -1,1$. We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.

It is shown that if A is a bounded linear operator on a complex Hilbert space, then $w\left(A\right)\le 1/2\left(\right|\left|A\right||+|\left|A\xb2\right|{|}^{1/2})$, where w(A) and ||A|| are the numerical radius and the usual operator norm of A, respectively. An application of this inequality is given to obtain a new estimate for the numerical radius of the Frobenius companion matrix. Bounds for the zeros of polynomials are also given.

Let $\Omega \subset {\u2102}^{n}$ be a bounded, simply connected $\u2102$-convex domain. Let $\alpha \in {\mathbb{Z}}_{+}^{n}$ and let $f$ be a function on $\Omega $ which is separately ${C}^{2{\alpha}_{j}-1}$-smooth with respect to ${z}_{j}$ (by which we mean jointly ${C}^{2{\alpha}_{j}-1}$-smooth with respect to $\mathrm{Re}{z}_{j}$, $\mathrm{Im}{z}_{j}$). If $f$ is $\alpha $-analytic on $\Omega \setminus {f}^{-1}\left(0\right)$, then $f$ is $\alpha $-analytic on $\Omega $. The result is well-known for the case ${\alpha}_{i}=1$, $1\le i\le n$, even when $f$ a priori is only known to be continuous.