The author obtains a new sufficient condition for an infinite Hankel matrix \((a_{i+ j})_{i, j\geq 0}\) to determine a bounded linear operator on a Hilbert space. One form of this condition is that we can write \(a_ k= \lambda_ k \alpha_ k\), with \(\{\lambda_ k\}\) a decreasing sequence in \(\ell^ 2\) and \(\{\alpha_ k\}\) satisfying, for some constant \(K\), \(\left|\sum^ N_{k= m+ 1} \alpha_ k z^ k\right|\leq K\sqrt{N- M}\) for \(| z|= 1\) and \(N> M\geq 0\). The author remarks that the \textit{W. Rudin} and \textit{J. H. Shapiro} polynomials [Proc. Am. Math. Soc. 10, 855-869 (1959; Zbl 0091.057)] can be used to obtain a sequence \(\{\alpha_ k\}\) satisfying this necessary condition and for which \(\{|\alpha_ k|\}\) does not satisfy the well-known Fefferman condition for boundedness. The author also remarks that the question of necessity for his condition remains open.