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Proof without words of the AM–GM inequality: PR is the diameter of a circle centered on O; its radius AO is the arithmetic mean of a and b. Using the geometric mean theorem, triangle PGR's altitude GQ is the geometric mean. For any ratio a:b, AO ≥ GQ. Visual proof that (x + y) 2 ≥ 4xy. Taking square roots and dividing by two gives the AM ...
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.
Proof without words of the inequality of arithmetic and geometric means, drawn by CMG Lee. PR is a diameter of a circle centred on O; its radius AO is the arithmetic mean of a and b . Using the geometric mean theorem, right triangle PGR can be split into two similar triangles PQG and GQR; GQ / a = b / GQ, hence GQ = √( ab ), the geometric mean.
The proof follows from Jensen's inequality, making use of the fact the logarithm is concave: = = = =. By applying the exponential function to both sides and observing that as a strictly increasing function it preserves the sign of the inequality, we get ∏ i = 1 n x i w i ≤ ∑ i = 1 n w i x i . {\displaystyle \prod _{i=1}^{n}x ...
Bernoulli's inequality can be proved for case 2, in which is a non-negative integer and , using mathematical induction in the following form: we prove the inequality for {,}, from validity for some r we deduce validity for +.
1.1 First proof: AM-HM inequality. 1.2 Second proof: Rearrangement. 1.3 Third proof: Sum of Squares. 1.4 Fourth proof: Cauchy–Schwarz. ... Fifth proof: AM-GM
The proof follows from the arithmetic–geometric mean inequality, , and reciprocal duality (and are also reciprocal dual to each other). The study of the Pythagorean means is closely related to the study of majorization and Schur-convex functions. The harmonic and geometric means are concave symmetric functions of their arguments, and hence ...
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