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In a previous paper the first author proved £/" ([(,edb) where e is a Brownian functional ¿M in absolute value and f is a convex function such that the right side is finite. We now prove a discrete analog of this inequality in which the integral is replaced by a martingale transform: EfÇyJ d,:yk)Ê f(M Tï/i). (The yfs are independent variables with mean zero, j^-d¡y¡ + -• ■+d1y, is a martingale, and 0^d,^M.) We further show that these inequalities are false if / or n is a stopping time, or if dj>0.doi:10.2307/2039197 fatcat:mymwbs67fnbj5ch6xjzoybn4su