Inequalities for the derivatives of polynomials with real zeros
The following sharp inequalities are established. Let \(P_ n\) be a polynomial of degree \(n\geq 2\) with all its zeros on \([-1,1]\). Then \(\| P_ n''\| \geq \min\{n,(n-1)n/4\}\| P_ n\|\) (sup norms). For a trigonometric polynomial \(T_ n\) of order n with 2n zeros on \([0,2\pi)\), the corresponding result is \(\| T_ n''\| \geq n\| T_ n\|.\) If also \(0<h<\min \{h(T_ n')\), \(2\cos^{-1}(1/2)^{{1/2}n}\}\), where (T) denotes the minimum distance between the zeros of T, then \[ \max_ x |\Delta^ 2_ h T_ n(x)\geq 2(1-(\cos h/2)^{2n})\| T_ n\|. \]
scientific article; zbMATH DE number 4044416