A \(\Delta_2\)-equivalent condition (Q1852404)

A continuous increasing function \(\phi:[0,\infty)\to \mathbb R\) with \(\phi(0)=0\), \(\phi(x)>0\), \(x>0\) satisfies the condition \(\Delta_2\) if there is an \(a>0\) and a \(\delta>0\) such that \(\dfrac{\phi(2x)}{\phi(x)}\leq \delta\) for \(0<x\leq a\). It is shown that \(\phi\) satisfies the condition \(\Delta_2\) if and only if for any \(k\in (0,1)\) and \(x_n\downarrow 0\) we have \(\sum_{n=1}^\infty \dfrac{\phi(kx_n)}{\phi(x_n)}=\infty\). This equivalence is used to present an alternative proof of the Musielak-Orlicz theorem saying that the class \(\Phi\)BV of functions of bounded \(\phi\)-variation is linear if and only if \(\phi\) satisfies the condition \(\Delta_2\).