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Jensen's Inequality -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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If p_1, ..., p_n are positive numbers which sum to 1 and f is a real continuous function that is convex, 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 of equal 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.

See also

Concave Function, Convex Function, Jensen's Formula

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References

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.

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Jensen's Inequality

Cite this as:

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

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