Denote by RnG the remainder functional of the Gaussian quadrature formula involving n nodes. Using an elementary method, we derive the asymptotically best possible inequalities | RnG [ f ] | < \frac1N3 ( \fracp4 512 + \frac1.45?N ) || f""¡é ||£¤ \left| {R_n^G \left[ f \right]} \right| \underline{\underline