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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 randomvariable and the value of the function at the expected value of the randomvariable. The bounds depend only on growth properties of the function andspecific moments of the random variable. The bounds are particularly useful fordistributions that are concentrated around the mean, a commonly occurringscenario such as the average of i.i.d. samples and in statistical mechanics.

Bounds on the Jensen Gap, and Implications for Mean-Concentrated Distributions