 Moderate growth

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 $x\ge 0$. 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.