QM–AM–GM–HM inequalities

There are three inequalities between means to prove. There are various methods to prove the inequalities, including mathematical induction, the Cauchy–Schwarz inequality, Lagrange multipliers, and Jensen's inequality. For several proofs that GM ≤ AM, see Inequality of arithmetic and geometric means. AM–QM inequality From the Cauchy–Schwarz inequality on real numbers, setting one vector to : :\left( \sum_{i=1}^n x_i \cdot 1 \right)^{\! 2} \leq \left( \sum_{i=1}^n x_i^2 \right) \left( \sum_{i=1}^n 1^2 \right) = n \,\sum_{i=1}^n x_i^2, hence \left( \frac{\sum_{i=1}^n x_i}{n} \right)^{\! 2} \leq \frac{\sum_{i=1}^n x_i^2}{n}. For positive x_i the square root of this gives the inequality. AM–GM inequality HM–GM inequality The reciprocal of the harmonic mean is the arithmetic mean of the reciprocals 1/x_1 , \dots, 1/x_n, and it exceeds 1/\sqrt[n]{x_1 \dots x_n} by the AM-GM inequality. x_i > 0 implies the inequality: : \frac{n}{\frac{1}{x_1} + \dots + \frac{1}{x_n}} \leq \sqrt[n]{x_1\dots x_n}. == The n = 2 case ==

When n = 2, the inequalities become :\frac {2x_1 x_2}{x_1+x_2} \leq \sqrt{x_1 x_2} \leq \frac{x_1+x_2}{2}\leq\sqrt{\frac{x_1^2+x_2^2}{2}} for all x_1, x_2 > 0, which can be visualized in a semi-circle whose diameter is x1+x2. Suppose C is a point on [AB] and let AC = x1 and BC = x2. Find the midpoint of [AB] as D and use as the center for the semi-circle from A to B. Construct perpendiculars to [AB] at D and C respectively, intersecting the circle at E and F respectively. Join [CE] and [DF] and further construct a perpendicular [CG] to [DF] at G. The length of DE is the arithmetic mean by the virtue of being the ray of the circle. CE can be calculated to be the quadratic mean from the Pythagorean theorem, CF to be the geometric mean from a combination of Thales's theorem (establishing that is a right triangle) and Geometric mean theorem, GF to be the harmonic mean from the similarity of triangle and (whose edge [DF]'s length can be calculated using the Pythagorean theorem and the two other known edges). == See also ==