On inequality connected with Laguerre weight (Q1850636)

The authors prove the following theorem: for any algebraic polynomial \(p_n(x)\) of degree at most \(n\) the relation \[ \sup_{0\leq x<\infty}\sqrt{x}|p_n(x)|e^{-x/2}=\max_{x\in I_n}\sqrt{x}|p_n(x)|e^{-x/2}, \] holds, where \[ I_n= \left[\frac{\pi^2}{4(4n+5)},4n+5 \right]. \] The left end of the interval \(I_n\) is exact, namely for the polynomial \[ P_n(x)=H_{2n}(\sqrt{x})-\frac{2\sqrt{n}}{\sqrt{x}}H_{2n-1}(\sqrt{x}), \] where \(H_n(x)\) denotes the Hermite polynomial of degree \(n\), the supremum norm \[ \|\sqrt{x}P_n(x)e^{-x/2}\|_{C(0,\infty)} \] is attained on the finite interval \([x_{1,2n+2}^2,x_{1,2n}^2],\) with \(x_{1,k}\) being the smallest positive zero of \(H_k(x),\) and \[ \lim_{n\to\infty}\frac{x_{1,2n+2}}{\frac{\pi}{2\sqrt{4n+5}}}=1. \]