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.
,
..., 
are positive numbers which sum to 1 and 
is a real continuous
function that is convex, then
is concave, then the inequality reverses, giving
with the concave function 
gives
![(x_1+x_2+...+x_n)/n>=RadicalBox[{{x, _, 1}, {x, _, 2}, ..., {x, _, n}}, n].](/images/equations/JensensInequality/NumberedEquation4.svg)
.
