Monotonicity of zeros of Laguerre polynomials

Abstract

Denote by $x_{nk}\left(\alpha \right)$, $k=1,\dots ,n$, the zeros of the Laguerre polynomial $L_n^{\left(\alpha \right)}\left(x\right)$. We establish monotonicity with respect to the parameter $\alpha$ of certain functions involving $x_{nk}\left(\alpha \right)$. As a consequence we obtain sharp upper bounds for the largest zero of $L_n^{\left(\alpha \right)}\left(x\right)$.