Inequalities of Rafalson type for algebraic polynomials (Q596811)

The authors study the constant \[ \gamma_n(d\nu; d\mu)=\sup\limits_{\pi\in\mathcal P_n(0)} \frac{\int^{+\infty}_{-\infty}\pi^2(x) d\nu(x)} {\int^{+\infty}_{-\infty}\pi^2(x)d\mu(x)} \] where \(d\mu\) is a positive Borel measure. They prove that the above constant can be represented by the zeros of orthogonal polynomials corresponding to \(d\mu\) in the following cases: (i) \(d\nu (x)=(A+Bx)d\mu (x)\) where \(A+Bx\) is nonnegative on the support of \(d\mu\); (ii) \(d\nu(x)=(A+Bx^2)d\mu(x)\) where \(d\mu\) is symmetric and \(A+Bx^2\) is nonnegative on the support of \(d\mu\). The results obtained generalize the results obtained by \textit{S. Z. Rafalson} [J. Approximation Theory 95, 161--177 (1998; Zbl 0924.41013)].