Difference between revisions of "Aczel's Inequality"
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(a_{i1}^{ p_i} - a_{i2}^{ p_i} - \dots - a_{in}^{ p_i})^\frac 1{ p_i}</math></center> | (a_{i1}^{ p_i} - a_{i2}^{ p_i} - \dots - a_{in}^{ p_i})^\frac 1{ p_i}</math></center> | ||
with equality if and only if all the sequences are proportional. | with equality if and only if all the sequences are proportional. | ||
+ | |||
+ | ==Examples== | ||
+ | |||
+ | '''Olympiad''' | ||
+ | Suppose <math>a_1, a_2, \ldots, a_n</math> and <math>b_1, b_2, \ldots, b_n</math> are real numbers such that <cmath> (a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 -1)(b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 - 1) > (a_1 b_1 + a_2 b_2 + \cdots + a_n b_n - 1)^2. </cmath> Prove that <math>a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 > 1</math> and <math>b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 > 1</math>. | ||
+ | |||
== References == | == References == |
Revision as of 15:24, 28 June 2014
Aczél's Inequality states that if or
, then
![$(a_1b_1-a_2b_2-\cdots -a_nb_n)^2\geq (a_1^2-a_2^2-\cdots -a_n^2)(b_1^2-b_2^2-\cdots -b_n^2).$](http://latex.artofproblemsolving.com/e/4/e/e4e586c1e0b1d22a10616d6ea1ada94e89009683.png)
Proof
Consider the function
.
We have , and from
we get
. Therefore,
must have at least one root,
.
General Form
Let such that
and let
![$(a_{11}, \dots,a_{1n}),$](http://latex.artofproblemsolving.com/6/1/9/6197e49bdf70b32797f481c1e4e8e3a0fc36432a.png)
![$(a_{m1}, \dots , a_{mn})$](http://latex.artofproblemsolving.com/3/9/6/396e03df5674b31d12ef1d8ced59c780966d25d4.png)
be sequences of positive real numbers such that
for
. Then
![$\prod_{i=1}^m a_{i1} - \prod_{i=1}^m a_{i2} -\dots- \prod_{i=1}^m a_{in} \ge\prod_{i=1}^m (a_{i1}^{ p_i} - a_{i2}^{ p_i} - \dots - a_{in}^{ p_i})^\frac 1{ p_i}$](http://latex.artofproblemsolving.com/a/7/0/a70d08ed51a4cb24591f28fb9bf54801251fc6a2.png)
with equality if and only if all the sequences are proportional.
Examples
Olympiad
Suppose and
are real numbers such that
Prove that
and
.
References
- Mascioni, Vania, A note on Aczél-type inequalities, JIPAM volume 3 (2002), issue 5, article 69.
- Popoviciu, T., Sur quelques inégalités, Gaz. Mat. Fiz. Ser. A, 11 (64) (1959) 451–461
See also
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