  This paper gives upper and lower bounds on the gap in Jensen's inequality,
i.e., the difference between the expected value of a function of a random
variable and the value of the function at the expected value of the random
variable. The bounds depend only on growth properties of the function and
specific moments of the random variable. The bounds are particularly useful for
distributions that are concentrated around the mean, a commonly occurring
scenario such as the average of i.i.d. samples and in statistical mechanics.
