Right tail from exponential-generating function --- better than exponential Chebyshev? For any random variable $X$, and any real number $\theta \geq 0$, you have the inequality
$$
P(X \geq x) \leq E[e^{\theta X}]/e^{\theta x} = e^{-\theta x} M(\theta)
$$
Here $M(\theta)$ is the exponential-generating function for $X$.
Consequently you have Chebyshev's exponential inequality
$$
P(X \geq x) \leq \inf_\theta e^{-\theta x} M(\theta)
$$
How tight is this formula? While it gives the right asymptotic bounds on $P(X \geq x)$, what if one wants more exact (non-asymptotic) information?
Are there any other functions of $M$ that give tighter bounds on $P(X \geq x)$?

One such alternative bound is the moment bound:
$$P(X \geq x) \leq \inf_{n} \frac{E[X_*^n]}{x^n},$$
where $X_*=\max\{X,0\}$.  This is in general tighter than the exponential inequality (see T. Phillips and R. Nelson's The Moment Bound Is Tighter Than Chernoff's Bound for Positive Tail Probabilities (The American Statistician, 1995, 175-178); in Problem 5.1 of Motvani/Raghavan's Randomized Algorithms textbook a similar observation for two-sided tail bounds is credited to J. Naor without a reference given), though it can be more difficult to compute (e.g. if $X$ is the sum of a number of independent variables $E[e^{\theta X}]$ factors nicely, but $E[X^n]$ does not).  
