Landau and Kolmogoroff type polynomial inequalities (Q1970041)

Given arbitrary integers \(j\), \(m\), \(n\) such that: \(0< j< m\leq n\) and positive constants \(A\) and \(B\). By considering the extremal problem: \[ \min\Biggl[{A\|f^{(m)}\|^2+ B\|f\|^2\over\|f^{(j)}\|^2}: f\in\pi_n,\;f(x)\neq 0\Biggr], \] where \(\pi_n\) denotes the set of real algebraic polynomials of degree at most \(n\) and \[ \|f\|^2= \int^{+\infty}_{-\infty} f^2(x)\exp(- x^2) dx, \] the authors obtain several sharp weighted polynomial inequalities of Landau-Kolmogoroff type. For instance it is proved the following: if \(-<{A\over B}< (4n(n- 1))^{-1}\), then \[ \|f'\|^2\leq {A\|f''\|^2+ B\|f\|^2\over 2A(n- 1)+ B(2n)^{-1}}. \] Moreover, equality is attained if and only if \(f(x)= cH_n(x)\), where \(H_n(x)\) is a Hermite polynomial and \(c\) is a constant.