On some propertis of functions of generalized variation (Q1091490)

We show that if \(\phi\) is a convex \(\phi\)-function then the space \(V^*_{\phi}\), generated by a \(\phi\)-variation, forms a commutative Banach algebra with respect to the pointwise multiplication under the norms \[ \| x\| =\sup_{t\in [a,b]}| x(t)| +\| x\|^ v_{\phi}\quad or\quad \| x\| =2\phi^{-1}(1)\| x\|^ v_{\phi}, \] where \[ \| x\|^ v_{\phi}=\inf \{r>0: var_{\phi}(x/r):=\sup_{\pi}\sum^{n}_{k=1}\phi (| x(t_ k)- x(t_{k-1})| /r)\leq 1\}. \] However, if \(\phi (u)=u^ p\), \(0<p\leq 1\), then \(var_ p(xy)\leq var_ p(x)var_ p(y).\) Moreover, for any convex \(\phi\)-function \(\phi\) \(var(\phi (| x|))\leq \phi (var(| x|)).\) The above inequality is used in a new proof of the generalized Opial's inequality.