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Aczel's Inequality

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Aczel's Inequality states that if a_1^2>a_2^2+\cdots +a_n^2 or b_1^2>b_2^2+\cdots +b_n^2, then

(a_1b_1-a_2b_2-\cdots -a_nb_n)^2\geq (a_1^2-a_2^2-\cdots -a_n^2)(b_1^2-b_2^2-\cdots -b_n^2).

Proof

Let us get the function f(x)=(a_1 x - b_1)^2-\sum_{i=2}^n(a_i x - b_i)^2= (a_1^2-a_2^2-\cdots -a_n^2)x^2-2(a_1b_1-a_2b_2-\cdots -a_nb_n)x+(b_1^2-b_2^2-\cdots -b_n^2).

f\left( \frac{b_1}{a_1} \right)=-\sum_{i=2}^n\left(a_i \frac{b_1}{a_1} - b_i\right)^2\leq 0 and since a_1^2>a_2^2+\cdots +a_n^2, then \lim_{x\rightarrow \infty}f(x)\rightarrow \infty. Therefore, f(x) has to have at least one root, \Leftrightarrow D=(a_1b_1-a_2b_2-\cdots -a_nb_n)^2- (a_1^2-a_2^2-\cdots -a_n^2)(b_1^2-b_2^2-\cdots -b_n^2)\geq 0.

See also

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