Uniform convexity and associate spaces

We prove that the associate space of a generalized Orlicz space \(L^{\varphi(\cdot)}\) is given by the conjugate modular \(\varphi^*\) even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling \(\Phi\)-function is equivalent to a doubling \(\Phi\)-function. As a consequence, we conclude that \(L^{\varphi(\cdot)}\) is uniformly convex if \(\varphi\) and \(\varphi^*\) are weakly doubling.