Upper bounds for superquantiles of martingales

Let $(M_n)_n$ be a discrete martingale in $L^p$ for $p$ in $]1,2]$ or $p=3$. In this note, we give upper bounds on the superquantiles of $M_n$ and the quantiles and superquantiles of $M_n^* = \max (M_0,M_1,\,\ldots ,\,M_n)$