On the L 2 Markov Inequality with Laguerre Weight

Let w α (t) = t α e −t , α > −1, be the Laguerre weight function, and ∥ ⋅ ∥ w α $$\Vert \cdot \Vert _{w_{\alpha }}$$ denote the associated L 2-norm, i.e., ∥ f ∥ w α : = ∫ 0 ∞ w α ( t ) | f ( t ) | 2 d t ) 1 ∕ 2 . $$\displaystyle{ \Vert f\Vert _{w_{\alpha }}:=\Big (\int _{0}^{\infty }w_{\alpha }(t)\vert f(t)\vert ^{2}\,dt\Big)^{1/2}. }$$ Denote by 𝒫 n $$\mathcal{P}_{n}$$ the set of algebraic polynomials of degree not exceeding n. We study the best constant c n (α) in the Markov inequality in this norm, ∥ p ′ ∥ w α ≤ c n ( α ) ∥ p ∥ w α , p ∈ 𝒫 n , $$\displaystyle{ \Vert p^{{\prime}}\Vert _{ w_{\alpha }} \leq c_{n}(\alpha )\,\Vert p\Vert _{w_{\alpha }}\,,\quad p \in \mathcal{P}_{n}\,, }$$ namely the constant c n ( α ) = sup p ≠ 0 p ∈ 𝒫 n ∥ p ′ ∥ w α ∥ p ∥ w α , $$\displaystyle{ c_{n}(\alpha ) =\sup _{\mathop{}_{p\neq 0}^{p\in \mathcal{P}_{n}}}\frac{\Vert p^{{\prime}}\Vert _{w_{\alpha }}} {\Vert p\Vert _{w_{\alpha }}} \,, }$$ and we are also interested in its asymptotic value c ( α ) = lim n → ∞ c n ( α ) n . $$\displaystyle{ c(\alpha ) =\lim _{n\rightarrow \infty }\frac{c_{n}(\alpha )} {n} \,. }$$ In this paper we obtain lower and upper bounds for both c n (α) and c(α). Note that according to a result of P. Dörfler from 2002, c(α) = [j (α−1)∕2, 1]−1, with j ν, 1 being the first positive zero of the Bessel function J ν (z), hence our bounds for c(α) imply bounds for j (α−1)∕2, 1 as well.