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Jensen's Inequality

For a Real Continuous Concave Function

\begin{displaymath}
{\sum f(x_i)\over n} \leq f\left({{\sum x_i}\over n}\right)
\end{displaymath} (1)

if $f$ is concave down,
\begin{displaymath}
{\sum f(x_i)\over n} \geq f\left({{\sum x_i}\over n}\right)
\end{displaymath} (2)

if $f$ is concave up, and
\begin{displaymath}
{\sum f(x_i)\over n} = f\left({{\sum x_i}\over n}\right)
\end{displaymath} (3)

Iff $x_1=x_2=\ldots=x_n$. A special case is
\begin{displaymath}
{\root n\of {x_1x_2\cdots x_n}} \leq {x_1+x_2+\ldots+x_n\over n},
\end{displaymath} (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.




© 1996-9 Eric W. Weisstein
1999-05-25