The AM-QM (Arithmetic Mean-Quadratic Mean) inequality, also known as the Cauchy-Schwarz inequality, is a fundamental inequality in mathematics. It establishes a relationship between the arithmetic mean and the quadratic mean (also known as the root mean square) of a set of real numbers.
The inequality states that for any set of real numbers x1,x2,…,xnx_1, x_2, ldots, x_nx1,x2,…,xn, the following inequality holds:
(x1+x2+…+xnn)2≤x12+x22+…+xn2nleft(frac{x_1 + x_2 + ldots + x_n}{n}
ight)^2 leq frac{x_1^2 + x_2^2 + ldots + x_n^2}{n}(nx1+x2+…+xn)2≤nx12+x22+…+xn2
In other words, the square of the arithmetic mean of a set of numbers is always less than or equal to the arithmetic mean of their squares. Equality occurs if and only if all the numbers are proportional to each other.
The AM-QM inequality has important applications in various areas of mathematics and other fields. It is often used in probability theory, analysis, and linear algebra, as well as in physics, particularly in quantum mechanics, where it has connections to uncertainty principles and the concept of wavefunctions.