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Jensen's Inequality

Jensen's inequality states that, if p_1, ..., p_n are nonnegative real numbers with sum 1, f:I->R is a convex function on an interval I, and x_1,...,x_n in I, then

f(sum_(i=1)^np_ix_i)<=sum_(i=1)^np_if(x_i).
(1)

If f is concave, then the inequality reverses, giving

f(sum_(i=1)^np_ix_i)>=sum_(i=1)^np_if(x_i).
(2)

The special case x_i>0 and equal weights p_i=1/n with the concave function lnx gives

ln(1/nsum_(i=1)^nx_i)>=1/nsum_(i=1)^nlnx_i,
(3)

which can be exponentiated to give the arithmetic mean-geometric mean inequality

(x_1+x_2+...+x_n)/n>=RadicalBox[{{x, _, 1}, {x, _, 2}, ..., {x, _, n}}, n].
(4)

Here, equality holds iff x_1=x_2=...=x_n.

For an integrable real random variable X taking values in an interval I almost surely, let f:I->R be a convex function and assume f(X) is also integrable. The expectation form is

f(E[X])<=E[f(X)],
(5)

where E denotes the expectation value (Viaclovsky 2003, pp. 1-2; Boyd and Vandenberghe 2004, p. 77). The finite weighted form follows by taking P(X=x_i)=p_i for distinct x_i and combining weights when values coincide.

See also

Concave Function, Convex Function, Jensen's Formula

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References

Boyd, S. and Vandenberghe, L. Convex Optimization. Cambridge, England: Cambridge University Press, 2004. https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1101, 2000.Hardy, G. H.; Littlewood, J. E.; and Pólya, G. "Some Theorems Concerning Monotonic Functions." §3.14 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 83-84, 1988.Jensen, J. L. W. V. "Sur les fonctions convexes et les inégalités entre les valeurs moyennes." Acta Math. 30, 175-193, 1906.Krantz, S. G. "Jensen's Inequality." §9.1.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 118, 1999.Viaclovsky, J. "Measure and Integration: Lecture 14." Notes prepared by E. Brown. Cambridge, MA: Massachusetts Institute of Technology, Oct. 21, 2003. https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/6aa2562c8e9e459dd82775b0a4f01aea_18125_lec14.pdf.

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Jensen's Inequality

Cite this as:

Weisstein, Eric W. "Jensen's Inequality." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/JensensInequality.html

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