Difference between revisions of "Aczel's Inequality"

Aczél'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

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

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

General Form

Let $p_1,\dots,p_m \ge1$ such that $\sum_{i=1}^m\frac1{p_i} = 1$ and let $(a_{11}, \dots,a_{1n}),$ $$\vdots$$ $(a_{m1}, \dots , a_{mn})$

be $m$ sequences of positive real numbers such that $a_{i1}^{ p_i} - a_{i2}^{ p_i} - \dots - a_{in}^{ p_i} > 0$ for $i=1,\dots,m$. Then $\prod_{i=1}^m a_{i1} - \prod_{i=1}^m a_{i2} -\dots- \prod_{i=1}^m a_{in} \ge\prod_{i=1}^m (a_{i1}^{ p_i} - a_{i2}^{ p_i} - \dots - a_{in}^{ p_i})^\frac 1{ p_i}$

with equality if and only if all the sequences are proportional.