We prove an inequality of the form ‖f‖ ≤ A‖f‖ +B‖f‖ for polynomials of degree n and any fixed 0 < j < m ≤ n. Here ‖ · ‖ is the L2-norm on (−∞,∞) with a weight e−t 2 . The coefficients A and B are given explicitly and depend on j,m and n only. The equality is attained for the Hermite orthogonal polynomials Hn(t).