= A superlinear function has a convex superlinear minorant
{title2=$H\geq0,\quad H(t)/|t|\to\infty\implies\Psi(|t|)\leq H(t)$}
Choose increasing $R_m\to\infty$ so $H(t)\geq m^2|t|$ whenever $|t|\geq R_m$. Put $\Psi(s)=\sum_{m\geq1}(s-R_m)_+$ for $s\geq0$. This is nonnegative, increasing and convex. If $m$ terms are active, then $\Psi(s)\leq ms\leq H(t)$ for $s=|t|$. For every fixed $N$, $s\geq2R_N$ gives $\Psi(s)\geq Ns/2$, proving superlinearity. Thus a nonconvex superlinear energy lower bound still supplies the <convex function> needed in the <De la Vallée-Poussin criterion>. A continuous superlinear <function> not initially nonnegative can first have its finite <global minimum> subtracted.
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