An exposition of some elementary inequalities

by ayushkhaitan3437

Short, clever proofs have played an immensely important role in the development of Mathematics. However, I can’t say I am a big fan of them. Mostly because I can never imagine myself coming up with those. Moreover, they don’t give a clear glimpse of the structure of what they prove. They just somehow manage to corroborate the conclusion of the proof with the axiomatic setup, which is not why people do Mathematics.

Today, I will try and decode some basic inequalities like a^2+b^2\geq 2ab (1) and (a^2+b^2)(c^2+d^2)\geq (ac+bd)^2 (2).

(1) a^2+b^2\geq 2ab can also be written as a^2+b^2\geq ab+ab. How do we change ab+ab to get a^2+b^2? We add a(a-b)+b(b-a) to it. We’re taking the same number (b-a), multiplying it by both -a and b, and then adding the two products. One easy way to end this discussion right here is to note that a(a-b)+b(b-a)= (a-b)^2\geq 0. However, I’d prefer to make it even more elementary. Although I could write paragraphs on it, I’d suggest that the reader do the needful, considering all possibilities like a<0, a<b, etc.

(2) (ac+bd)^2=(ac+bd)(ac+bd). One adds (ac+bd)[(a(a-c)+c(c-a)+b(b-d)+d(d-b)] to it to make it equal to (a^2+b^2)(c^2+d^2). The conclusion follows directly from here, using the same argument that was given for (1).

(1) is obviously a form of the AP>GP inequality and (2) is the Cauchy-Schwartz inequality. These proofs may seem less beautiful and clever than those given in textbooks. But the hope is you’d find these proofs more helpful.