arXiv subjects
Publications and source records attributed to Robert Dalmasso.
The Sendov conjecture asserts that if $p(z) = \prod_{j=1}^{N}(z-z_j)$ is a polynomial with zeros $|z_j| \leq 1$, then each disk $|z-z_j| \leq 1$ contains a zero of $p'$. Our purpose is the following: Given a zero $z_j$ of order $n \geq 2$, determine whether there exists $ζ\not= z_j$ such that $p'(ζ) = 0$ and $|z_j - ζ| \leq 1$. In this paper we present some partial results on the problem.
Let $f$ be a nonnegative function of class $C^k$ ($k \geq 2$) such that $f^{(k)}$ is H\''older continuous with exponent $α$ in $(0,1]$. If $f'(x) = \cdots = f^{(k)}(x) = 0$ when $f(x) = 0$, we show that $f^μ$ is differentiable for $μ\in (1/(k+α), 1)$ and under an additional condition we show that $(f^μ)'$ is H\''older continuous with exponent $β= μ(1+α) - 1$ (if $β\leq 1$) at $x \in [0,T]$ when $f(x) = 0$. $(f^μ)'$ is Lipschitz continuous at $x$ if $f(x) > 0$.
The Sendov conjecture asserts that if all the zeros of a polynomial p lie in the closed unit disk then there must be a zero of p ' within unit distance of each zero. In this paper we give a partial result when p has simple zeros.