Definition:

A reald-valued function f is concave on an interval I if

for all x,y  I.  f is stictly concave on an interval I if
 
 

for all x,y  I, xy.
 
 
 

Jensen's Inequality:
 

Suppose f is a continuous strictly concave function on the interval I,
 

and a_i > 0, 1 <= i <= n. Then

where x_i I, 1 <= i <= n.  Further, equality occurs if and only if x₁ = ... = x_n .