I have some random variable with a complicated but known analytical CDF and I need an upper bound on the MGF. Direct MGF computation is intractable due to the messy nature of the CDF so I was trying to use Markov inequality to try to get some bounds on the MGF but I can only get lower bounds, exactly opposite of what I need!
PS - the random variable is almost surely in $[0,1]$. Is there any other inequality I can use to get an upper bound on the MGF?