Extremal properties of derivative of algebraic polynomials
A problem of P. Turán runs: If \(\phi(x)\geq 0\) in \(I= [-1,1]\), let \(C\) be the class of all polynomials \(p\) of degree at most \(n\), such that \(| p(x)|\leq \phi(x)\) in \(I\). How large can \(\max_ I | p^{(k)}(x)|\) be if \(p\in C\)? The authors obtained some weighted \(L^ 2\)-inequalities in a similar spirit. For example, if \(p\) is a real algebraic polynomial of degree \((n+1)\), such that \(| p(x)|\leq (1- x^ 2)^{1/2}\) for \(x\in I\), then for \(k= 2,3,\dots\) we have \[ \int^ 1_{-1} | p^{(k)}(x)|^ 2(1- x^ 2)^{(2k- 3)/2} dx\leq \int^ 1_{-1} | f^{(k)}(x)|^ 2(1- x^ 2)^{(2k- 3)/2} dx, \] where \(f(x)= (1- x)^ 2 u_{n-1}(x)\) and \(u_ n\) denotes the Chebyshev polynomial of the second kind. Equality occurs if and only if \(p(x)= \pm f(x)\).
scientific article; zbMATH DE number 638212
scientific article