The paper extends the result of Harman and Pronzato [Stat. \& Prob. Lett., 77:90--94, 2007], which corresponds to p=0, to all strictly concave criteria in Kiefer's phi_p-class. Let xi be any design on a compact set X C R^m with a nonsingular information matrix M(xi), and let delta be the maximum of the directional derivative F_phi_p(xi,x) over all x in X. We show that any support point x_* of a phi_p-optimal design satisfies the inequality F_phi_p(xi,x_*) >= h_p[M(xi),delta], where the bound h_p[Mb(xi),delta] is easily computed: it requires the determination of the unique root of a simple univariate equation (polynomial when p is integer) in a given interval. The construction can be used to accelerate algorithms for phi_p-optimal design and is illustrated on an example with A-optimal design.