Jensen's Inequality

For a Real Continuous Concave Function

$${\sum f(x_i)\over n} \leq f\left({{\sum x_i}\over n}\right)$$ (1)

if $f$ is concave down,

$${\sum f(x_i)\over n} \geq f\left({{\sum x_i}\over n}\right)$$ (2)

if $f$ is concave up, and

$${\sum f(x_i)\over n} = f\left({{\sum x_i}\over n}\right)$$ (3)

Iff $x_1=x_2=\ldots=x_n$. A special case is

$${\root n\of {x_1x_2\cdots x_n}} \leq {x_1+x_2+\ldots+x_n\over n},$$ (4)

with equality Iff $x_1=x_2=\ldots=x_n$.

See also Concave Function, Mahler's Measure

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1101, 1979.