Let D be a division ring with the center F = Z(D). Suppose that N is a normal subgroup of D∗ which is radical over F, that is, for any element x ∈ N, there exists a positive integer nx, such that xnx ∈ F. In [5], Herstein conjectured that N is contained in F. In this paper, we show that the conjecture is true if there exists a positive integer d such that nx ≤ d for any x ∈ N.