 Contraction constant

A contraction constant for a function $T$ on a metric space is a single $q<1$ that bounds the ratio $d(Tx,Ty)/d(x,y)$ for every distinct pair. Pointwise strict inequalities $d(Tx,Ty)<d(x,y)$ do not imply such a common constant. For a differentiable function on an interval, the mean value theorem shows that $|T^{\prime }|\le q<1$ supplies a contraction constant.