送交者: 自如 于 2005-8-11, 16:42:14:
回答: Some discrete and specialized form of Holder's inequality? 由 mangolasi 于 2005-8-11, 14:28:30:
Holder's (aka Minkowski's) is an important inequality for measure theory. One of its collaries, i.e., when p=q=2, is the famous Cauchy-Schwarz inequality. This latter one is used a lot in math competition problems, more agressive problems even expect the employment of the discrete form of Holder's.
The inequality that could be used to solve the given problem states that the arithmetic mean of n positive reals is no less than the geometric mean of them, and the equality holds iff the n numbers are equal. For math competition, this is often taught in early days while the Cauchy one is considered more "advanced". :)
The proof of this inequality is not easy. I could only think about two ways right now: one uses the Lagrange multiplier directly; the other slightly simpler one uses induction so it effectively reduces the problem to one-variable extremal problem. I could not figure out an elementary proof w/o using calculus.
It's fairly legitimate to use such a problem in competitions for this inequality is considered a common sense in that community. But it's a bit too much as an ordinary primary-school homework problem.