In this paper the following two theorems are shown: if $U, V$ are Burkholder type operators on martingales and if the inequality $E\lbrack U(X) \rbrack \leqq c \cdot E\lbrack V(X) \rbrack$ holds for every martingale $X$, then the inequality $E\lbrack F \circ U(X) \rbrack \leqq C \cdot E\lbrack F \circ V(X) \rbrack$ holds, for $F$ concave if $V$ is "predictable," for $F$ convex if $U$ is "predictable."
@article{1176996409,
author = {Pratelli, Maurizio},
title = {Deux Inegalites Concernant Les Operateurs de Burkholder Sur Les Martingales},
journal = {Ann. Probab.},
volume = {3},
number = {6},
year = {1975},
pages = { 365-370},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996409}
}
Pratelli, Maurizio. Deux Inegalites Concernant Les Operateurs de Burkholder Sur Les Martingales. Ann. Probab., Tome 3 (1975) no. 6, pp. 365-370. http://gdmltest.u-ga.fr/item/1176996409/