Optimal Stopping Inequalities for the Integral of Brownian Paths

for all stopping times for B , and all p > 0 , where Ap and Bp are numerical constants. Although the best values for the constants Ap and Bp in (1.1) are found below too, in most of the cases it is much easier to evaluate E( ) rather than E( 1+p=2) . In this paper we shall answer the question on how the inequality (1.1) can be optimally modified if the quantity E( 1+p=2) is replaced by a function of E( ) . It turns out that the left-hand inequality in (1.1) admits such a modification. In Theorem 2.2 below we prove that: