Reich–Strebel inequality ($K_0\le {\int }_X\frac{|1+{\mu }_{f}q/|q|{|}^2}{1-|{\mu }_f{|}^2}|q|$)

On closed Riemann surfaces of genus at least two, if a Teichmüller map $f_0:X\to Y$ has unit-area source differential $q$ and dilatation $K_0$, any quasiconformal map $f:X\to Y$ in the same homotopy class satisfies $K_0\le {\int }_X|q||1+{\mu }_{f}q/|q|{|}^2/(1-|{\mu }_f{|}^2)$. The plus sign corresponds to a Teichmüller map with ${\mu }_{f_0}=k_0|q|/q$ and $K_0=(1+k_0)/(1-k_0)$. This fundamental inequality yields Teichmüller's uniqueness theorem by the pointwise triangle inequality and its equality case. See Gardiner and Hu, §5, equation (11).