The AM-GM Inequality

For positive numbers, the arithmetic mean is greater than or equal to the geometric mean

Proof using Jensen's Inequality:

Jensen's Inequality is

∑(f(x_i))/n ≤ f(∑(x_i)/n) if f is concave (i.e. f''(x)<0, which is sometimes called "concave down"),

∑(f(x_i))/n ≥ f(∑(x_i)/n) if f is convex (i.e. f''(x)>0, which is sometimes called "concave up"),

with equality iff x₁ = x₂ = ... = x_n.

ln(x) is concave, so by Jensen's Inequality,

(ln(a₁) + ln(a₂) + ... + ln(a_n))/n ≤ ln((a₁+a₂+...+a_n)/n),

Taking the exponential function of both sides,

geometric mean(a₁,a₂,...,a_n) ≤ arithmetic mean(a₁,a₂,...,a_n)

Weighted AM-GM Inequality

Given n positive reals a₁,a₂,...,a_n, and n weights w₁,w₂,...,w_n such that 0≤w_i≤1 and ∑w_i=1,
w₁a₁+w₂a₂+...+w_na_n ≥ a₁^w1 a₂^w2 ... a_n^wn 

Proof Outline: same as above, using the weighted version of Jensen's Inequality; or use the ordinary AM-GM inequality for rational weights (put all the weights over a common denominator, d, and then show that the AM of the d numbers (many identical) is never less than the GM of those same numbers); and then extend to irrational weights by continuity.

Internet references

Mathworld -- Jensen's Inequality

Related pages in this website

The Cauchy-Schwarz inequality

The Triangle Inequality

The Chebyshev Sum Inequality

Puzzles

The Quadratic Formula

The webmaster and author of this Math Help site is Graeme McRae.