125. I like very much this Darij Grinberg's message.
by Virgil Nicula, Sep 14, 2010, 9:04 PM
Darij Grinberg wrote:
Quote:
The sum of the minima of 
functions is less or equal to the minimum of the sum of these functions.
functions is less or equal to the minimum of the sum of these functions.This fact, which was taught me by Arthur Engel (but which he didn't mention in his Problem-Solving Strategies book), is a trivial consequence of the definition of a minimum, but it turns out to be a helpful principle for proving inequalities.
For instance, the quadratic function
has the minimum 
. Now let 
, 
, ..., 
be arbitrary positive numbers, and let 
, 
, ..., 
be arbitrary real numbers. Consider the quadratic functions 
, for all natural 
with 
. These quadratic functions have the respective minima 
, while their sum is the quadratic function 
and has the minimum 
. Hence, since the sum of the minima of our n functions is less or equal to the minimum of the sum, we get 
. Thus, 
. This is the so-called Cauchy-Schwarz inequality in the Engel form, also called Andreescu Lemma. It is equivalent to what is usually called the Cauchy-Schwarz inequality. Finally I should mention that the minimum thesis of Arthur Engel can be applied some more times to rather complicated functions, but mostly the minima must be found by calculus then.NOTE: Some time ago it turned out that all of the above facts together with their proofs were known in Russia long before Engel and before Andreescu as well. This neither surprised nor disappointed me. What actually keeps annoying me is when I write "by the Cauchy-Schwarz inequality in Engel form, we have..." and somebody replies to me that the inequality should be called differently and that it was known before Engel. Indeed, I do know that the inequality was known before Engel, and I don't claim it is Engel's invention; I just use the name "Cauchy-Schwarz inequality in Engel form" since this name is widely used (at least among the German olympiad participants) and people understand me when I call it this way. Same holds for the Karamata inequality.
(Just a historic excursus: You probably know that the terminus "Simson's line" is apparently ahistoric, as Simson didn't have anything to do with this line, at least no mention of it was located in his works. The first discoverer of the line seemed to be William Wallace. When some historian noted this, he proposed to rename the Simson's line into Wallace's line. Yet this new name didn't become mainstream, it caused a lot of trouble; still, I sometimes talk about Simson's lines and people don't understand me since they are used to calling them Wallace's lines, and conversely.)
Darij
This post has been edited 2 times. Last edited by Virgil Nicula, Nov 23, 2015, 7:34 AM
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