Inequalities

(a-b)^2 >=0 -> ab<=(a^2+b^2)/2

root(xy) <= (x+y)/2 

Geometric mean <= arithmetic mean

n-th root of x1x2x3...xn <= (x1+x2+x3...+xn)/n

Cauchy-Schwarz inequality:

　　a1,a2,a3,a4...an, b1,b2...bn are real numbers    

　　abs(a1b1+a2b2+a3b3+...+anbn) <= root(a1^2+a2^2+...+an^2)*root(b1^2+b2^2+...+bn^2)

　　if and only if there is a real number