Let \(h_{a,b}\) be a smooth transition function . Then

  1. \(\lvert h'_{0,1}(x)\rvert\le 2\) for all \(x\in \mathbb{R}\), and thus, \(\lvert h'_{a,b}(x)\rvert\le \frac{2}{b-a}\) for \(b>a\).
  2. for higher derivatives, the following estimate holds \[ \lVert h_{a,b}^{(n)}\rVert_{\infty } \le \frac{1}{(b-a)^n} \lVert h_{0,1}^{(n)}\rVert_{\infty }, \] where \(n\in \mathbb{N}\).