Remarks on the Obrechkoff inequality (Q2789867)

Obrechkoff proved that if \(P\) is a polynomial of degree \(d\) with non-negative coefficients, then the number of zeros of \(P\) in the sector \(\{z:|\arg z|\leq \alpha\}\) is at most \(2\alpha d/\pi\). Here the authors obtain a generalization of this result. For a probability measure \(\mu\) in \(\mathbb{C}\), which is symmetric to the real axis, the associated potential \(\mu\) is defined by NEWLINE\[NEWLINEu_\mu(z)\int_{|\zeta|\leq 1}\log|z-\zeta|d\mu+\int_{|z|>1}\log|1-z/\zeta|d\mu.NEWLINE\]NEWLINE Suppose that \(u_\mu(z)\leq u_\mu(|z|)\) for all \(z\) and define \(m(\alpha)=\mu(\{z:|\arg z|\leq \alpha\})\). It is shown that NEWLINE\[NEWLINE\frac{1}{a}\int ^a_0 m(t) dt\leq \frac{a}{2\pi}NEWLINE\]NEWLINE for \(0\leq a \leq \pi\).