On a polynomial inequality of Turán (Q757763)

P. Turán proved the inequality \(\prod_{| \alpha_ k| \geq 1}| \alpha_ k| \leq 2^{n-1}\| P\|_{L^{\infty}[- 1,1]}\) for a polynomial \(P(x)=\prod^{n}_{k=1}(x-\alpha_ k).\) Although this inequality is exact for Chebyshev polynomials, it is not sharp if there are several large \(\alpha_ k's\) in the definition of P(x). In this note a weighted \(L^ P\) inequality is given that performs better in such cases.