Moderate growth
= Moderate growth
{title2=$\Phi(ax)\leq K\Phi(x)$}
A continuous nondecreasing function $\Phi:[0,\infty)\to[0,\infty)$ with $\Phi(0)=0$ has moderate growth if the displayed inequality holds for some $a>1$ and finite $K$, uniformly in $x\geq0$. Iteration gives a bound for any fixed positive dilation, and at most polynomial growth at infinity. The powers $x^p$, $p>0$, satisfy the condition. It allows a good-lambda tail comparison to be integrated into a moment comparison.