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 $φ_p$-class. Let $ξ$ be any design on a compact set $X\subset\mathbb{R}^m$ with a nonsingular information matrix $\Mb(ξ)$, and let $δ$ be the maximum of the directional derivative $F_{φ_p}(ξ,x)$ over all $x\in X$. We show that any support point $x_*$ of a $φ_p$-optimal design satisfies the inequality $F_{φ_p}(ξ,x_*) \geq h_p[\Mb(ξ),δ]$, where the bound $h_p[\Mb(ξ),δ]$ 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 $φ_p$-optimal design and is illustrated on an example with $A$-optimal design.