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

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

with equality iff x1 = x2 = ... = xn.

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

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

Taking the exponential function of both sides,

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

Weighted AM-GM Inequality

Given n positive reals a1,a2,...,an, and n weights w1,w2,...,wn such that 0≤wi≤1 and wi=1,
w1a1+w2a2+...+wnan ≥ a1w1 a2w2 ... anwn 

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


The Quadratic Formula


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