Sharp Marchaud and converse inequalities in Orlicz spaces

For $L_p$ spaces on $\mathbb{T}^d$, $\mathbb{R}^d$ and $S^{d-1}$, sharp versions of the classical Marchaud inequality are known. These results are extended here to Orlicz spaces (on $\mathbb{T}^d$, $\mathbb{R}^d$ and $S^{d-1}$) for which $M(u^{1/q})$ is convex for some $q$, $1<q\le2$, where $M(u)$ is the Orlicz function. Sharp converse inequalities for such spaces are deduced.