Jensen's Inequality -- from Wolfram MathWorld

Jensen's Inequality

For a real continuous

(1)

if is concave function down,

(2)

if is concave function up, and

(3)

iff . A special case is

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

(4)

with equality iff .

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.

CITE THIS AS:

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