For a single-input/single-output (SISO) linear time-invariant dynamical system, the standard H-infinity-norm lower error bound of balanced truncation method is parallel to G(s) - G(r)(s)parallel to(H infinity) >= sigma(r+1), where sigma(i), i = 1,..., n, are the Hankel singular values of system in decreasing order. In this paper we provide a new estimation of the lower error, namely; parallel to G(s) - G(r)(s)parallel to(H infinity) >= max{sigma(d), 2 vertical bar Sigma(i is not an element of g) s(i)sigma(i)vertical bar},; where s(i) is the sign associated with the Hankel singular value sigma(i) in Ober's canonical form. The subset g and the index d in the above inequality will be introduced in the paper. We show by means of an example that the new bound may be relevant in deciding which states need to be kept in the balanced truncation method, and that using the standard result does not always yield the best approximation