Sharp growth conditions for boundedness of maximal function in generalized Orlicz spaces

We study sharp growth conditions for the boundedness of Hardy-Littlewood maximal function in generalized Orlicz spaces. assume that \(\phi(x,t)\) satisfies standard continuity properties (A0), (A1) and (A2). show if is bounded from space to itself then \(\phi(x,t)/t^p\) almost increasing large \(t\) some \(p>1\). Moreover we \(L^\phi(\mathbb{R}^n)\) only \(\phi\) weakly equivalent a \(\psi\) satisfying (A2) which \(\psi(x,t)/t^p\) all \(t>0\) \(p>1\).