The M\"{o}bius transformation of continued fractions with bounded upper and lower partial quotients

Let $h$: $x\mapsto \frac{ax+b}{cx+d}$ be the nondegenerate M"{o}bius transformation with integer entries. We get a bound of the continued fraction of $h(x)$ by the upper and lower bound of continued fraction of $x$, which extends a result of Stambul \cite{S2}.