Convexity of the bounds induced by Markov's inequality

X is a nonnegative random variable such that EXt < [infinity] for 0<= t < [lambda] <= [infinity]. The (l-[epsilon]) quantile of the distribution of X is bounded above by [[epsilon]-1 EXt]1[+45 degree rule]t. We show that there exist positive [epsilon]1 >= [epsilon]2 such that for all 0 <[epsilon]<=[epsilon]1 the function g(t) = [[epsilon]-1EXt]1[+45 degree rule]t is log-convex in [0, c] and such that for all 0 < [epsilon] <= [epsilon]2 the function log g(t) is nonincreasing in [0, c].