Remarks on the Obrechkoff inequality

Abstract:

Let $u$ be the logarithmic potential of a probability measure $\mu$ in the plane that satisfies \[ u(z)=u(\overline {z}),\quad u(z) \le u(|z|),\quad z\in \mathbb {C},\] and $m(t)=\mu \{ z\in \mathbb {C}^*:|\operatorname {Arg} z|\leq t\}$. Then \[ \frac {1}{a}\int _0^a m(t)dt\leq \frac {a}{2\pi },\] for every $a\in (0,\pi )$. This improves and generalizes a result of Obrechkoff on zeros of polynomials with positive coefficients.