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 $X\subset\mathbb{R}^m$ with a nonsingular information matrix $\Mb(\xi)$, 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 $F_{\phi_p}(\xi,x_*) \geq h_p[\Mb(\xi),\delta]$, 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.