By Nils Fridolf Valdemar Svartholm
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Additional resources for Abdus Salam - Weak and Electromagnetic Interactions. Published in Elementary Particle Theory: Proceedings of the 8th Nobel Symposium
For another generalization see Bromwich [1;2 pp. 57-58, 242, 418-420, 473-4741. 3. Some Interpretations and Applications. (a) A2 , G2 The means and H2 , with equal weights, have a simple geometrie interpretations. Let IJ ABCD be a trapezium with is parallel to AB and divides AB ABCD and bisects AB = b, oe = a, being the parallel sides; AD, and, of course, into two similar trapezia; EKF BC; GH is parallel to is also parallel to AB. The above diagram, that can be justified in a few numerical cases by the reader, suggests that (9) with equality only if a = b ; this is an important result that we will return to in section 2 below.
3) Clearly (1) does not depend on numbers and many of the properties of ~ consisting of positive real A can be deduced without this assumption, however unless otherwise mentioned n-tples will be taken to be positive. The question of more general n-tples will be discussed later. Some elementary properties of Aare listed in Theorem 2. :) n with equality iff a = b. A(~) h~O (g) Proof. (a) n- Easy applications of simple properties of positive numbers. :). 2, and is known as Chebyshev's Inequality.
Iab. Since (4) is homogeneous there is no loss in generality in assuming Hence Lemma 3 is equivalent to the following: Lemma 3'. If rb-ra )2 2 - a *b If a > 0 and b > 0, inequality iff a - b. assume, a < 1 < b, and ab - 1 then a + b ~ 2 with which implies that (a-1 )(b-1) < 0 , or (5) a+b>1+ab - 2. *4 Using the idea of putting ab· 1 , or equivalently in the previous proof, Lemma 3 is equivalent to the following: b - ä ' introduced 45 THE ARITHMETIC, GEOMETRIC AND HARMONIC MEANS Lemma 3". 2, (6) with equality iff a = 1.
Abdus Salam - Weak and Electromagnetic Interactions. Published in Elementary Particle Theory: Proceedings of the 8th Nobel Symposium by Nils Fridolf Valdemar Svartholm