AM–GM inequality

For a geometrical interpretation, consider a rectangle with sides of length x and y; it has perimeter 2x + 2y and area xy.

Similarly, a square with all sides of length √xy has the perimeter 4√xy and the same area as the rectangle.

The simplest case is implicit in Euclid's Elements, Book 5, Proposition 25.

, xn is the sum of the numbers divided by n: The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division: If x1, x2, .

, xn > 0, this is equal to the exponential of the arithmetic mean of the natural logarithms of the numbers: Restating the inequality using mathematical notation, we have that for any list of n nonnegative real numbers x1, x2, .

Thus the AM–GM inequality states that only the n-cube has the smallest average length of edges connected to each vertex amongst all n-dimensional boxes with the same volume.

[citation needed] In financial mathematics, the AM-GM inequality shows that the annualized return, the geometric mean, is less than the average annual return, the arithmetic mean.

Jensen's inequality states that the value of a concave function of an arithmetic mean is greater than or equal to the arithmetic mean of the function's values.

Since the logarithm function is concave, we have Taking antilogs of the far left and far right sides, we have the AM–GM inequality.

For the following proof we apply mathematical induction and only well-known rules of arithmetic.

Induction hypothesis: Suppose that the AM–GM statement holds for all choices of n non-negative real numbers.

Induction hypothesis: Suppose that the statement is true for all natural numbers up to n – 1.

The following proof by cases relies directly on well-known rules of arithmetic but employs the rarely used technique of forward-backward-induction.

It is essentially from Augustin Louis Cauchy and can be found in his Cours d'analyse.

Induction hypothesis: Suppose that the AM–GM statement holds for all choices of n non-negative real numbers.

Induction step: In order to prove the statement for n + 1 non-negative real numbers x1, .

We consider the last number xn+1 as a variable and define the function Proving the induction step is equivalent to showing that f(t) ≥ 0 for all t > 0, with f(t) = 0 only if x1, .

Next we compute the value of the function at this global minimum: where the final inequality holds due to the induction hypothesis.

Observe that f(1) = 0, f′(1) = 0 and f′′(x) > 0 for all real x, hence f is strictly convex with the absolute minimum at x = 1.

By n-fold application of the above inequality, we obtain that with equality if and only if xi = α for every i ∈ {1, .

The argument of the exponential function can be simplified: Returning to (*), which produces x1 x2 · · · xn ≤ αn, hence the result[7] If any of the

Because the arithmetic and geometric means are homogeneous of degree 1, without loss of generality assume that

The method of Lagrange multipliers says that the global minimum is attained at a point

Then Using the finite form of Jensen's inequality for the natural logarithm, we can prove the inequality between the weighted arithmetic mean and the weighted geometric mean stated above.

Since the natural logarithm is strictly concave, the finite form of Jensen's inequality and the functional equations of the natural logarithm imply Since the natural logarithm is strictly increasing, Most matrix generalizations of the arithmetic geometric mean inequality apply on the level of unitarily invariant norms, since, even if the matrices

it is the case that Later, in [10] the same authors proved the stronger inequality that Finally, it is known for dimension

Indeed, he proved[11] In finance much research is concerned with accurately estimating the rate of return of an asset over multiple periods in the future.

is the variance of the observed asset returns This implicit equation for aN can be solved exactly as follows.

First, notice that by setting we obtain a polynomial equation of degree 2: Solving this equation for z and using the definition of z, we obtain 4 possible solutions for aN: However, notice that This implies that the only 2 possible solutions are (as asset returns are real numbers): Finally, we expect the derivative of aN with respect to gN to be non-negative as an increase in the geometric return should never cause a decrease in the arithmetic return.

Indeed, both measure the average growth of an asset's value and therefore should move in similar directions.