Summary: The Classical Inequalities
Putnam Practice
September 28, 2005
Arithmetic mean - Geometric mean inequality says that for any
n non-negative real numbers a1, ...an we have:
a1 + a2 + ... + an
n
n

a1a2...an
and equality holds a1 = a2 = ... = an. Define the Harmonic mean by
H =
n
1/a1 + 1/a2 + ... + 1/an
then it is not hard to check that
HM GM AM.
Theorem 1 (Power mean inequality) Let a1, ...an be positive real num-
bers, and let be real. Let
M(a1, ...an) = (
a