不等式讲义-the uvw method.pdfVIP

The uvw method - Tejs The uvw method by Mathias Bæk Tejs Knudsen 1 The uvw method - Tejs 1 BASIC CONCEPTS 1 Basic Concepts The basic concept of the method is this: With an inequality in the numbers a, b, c ∈ R, we write everything in terms of 3u = a + b + c, 3v2 = ab + bc + ca, w3 = abc. The condition a, b, c ≥ 0 is given in many inequalities. 2 3 2 3 In this case, we see that u, v , w ≥ 0. But if this is not the case, u, v , w might be negative. E.g. a = −1, b = −2, c = 3 we get 3v2 = −7. So don’t be confused! Sometimes v2 can be negative! The Idiot-Theorem: u ≥ v ≥ w , when a, b, c ≥ 0. Proof: 9u2 − 9v2 = a2 + b2 + c2 − ab − bc − ca = 1 cyc (a − b)2 ≥ 0. And hence u ≥ v . 2 √ 2 ab+bc+ca 3 2 2 2 2 From the AM-GM inequality we see that v = 3 ≥ a b c = w , and hence v ≥ w , which concludes u ≥ v ≥ w . The idiot-theorem is usually of little use alone, but it is important to know. 2 2 Sometimes we introduce the ratio t = u : v . If u 0, v = 0 we define t = ∞, and if u 0, v = 0 we define t = −∞. If u = v = 0 we set t = 1. When a, b, c ≥ 0, we see that t ≥ 1. And we always see that |t | ≥ 1. 2 3 The UVW-Theorem: Given u, v , w ∈ R : ∃a, b, c ∈ R s