A new characterization of Hermite polynomials (Q1088037)

Consider the polynomials on \((-\infty,\infty)\) equipped with the norm derived from the integral inner product with weight function \(\exp (-x^ 2):\) \[ \| P_ n\|^ 2=\int^{\infty}_{-\infty}P^ 2_ n(t)\exp (-t^ 2)dt \] \((P_ n:\) a general n-th degree polynomial). Based on the inequality \(\| P_ n'\| \leq a_ n\| P_ n\|^ 2+b_ n\| P_ n''\|^ 2\) with \(a_ n=(2n-1)\), \(b_ n=2n^ 2/(2n-1)\), which turns into an equality iff \(P_ n\) is - up to a multiplicative constant - the Hermite polynomial of degree n w.r.t. the weight \(\exp (-x^ 2)\) on \((-\infty,\infty)\), the author gives bounds for \(\| P_ n^{(q)}\|^ 2\) by a linear combination (coefficients depending on n) of \(\| P_ n\|^ 2\) and \(\| P_ n^{(r)}\|^ 2\) for \(1\leq q\leq r-1\) and \(r=2,3,4\). All inequalities characterize the Hermite polynomials: those are the only ones (up to a multiplicative constant) which lead to equality.