ON SOME CONSTANTS OF QUASICONFORMAL DEFORMATION AND ZYGMUND CLASS

A real-valued function $f(x)$ on $ \Re$ belongs to Zygmund class $\Lambda_{*}(\Re)$ if its Zygmund norm $\|f\|_z=\underset{x,t}\to{\inf} \Bigl|\frac {f(x+t)-2f(x)+f(x-t)}t\Bigr|$ is finite. It is proved that when $f\in\Lambda_{*}(\Re)$, there exists an extension $F(z)$ of $f$ to $ H=\{\text{Im}z>0\}$ such that $$\aligned \|\overline {\partial}F\|_{\infty}\le\frac {\sqrt {1+53^2}}{72} \|f\|_z.\endaligned $$ It is also proved that if $f(0)=f(1)=0$, then $$\aligned \max_{x\in 0,1]}|f(x)|\le\!\frac 13\|f\|_z.\endaligned $$