prove C-S ineq. Engel form
by considering (ax-by)^2>=0, or ortherwise,
prove Cauchy-Schwarz's inequality in the Engel form
state the condition for equality to hold
please use elementary proof as possible
(yet I don't think you can rely much on other theorems)
If you prove the first case, you can prove the inequality with self-generalizing property.
ie. prove (a+b)^2/(x+y)>=a^2/x+b^2/y
then sub b=b+c, y=y+z
in this case, can you tell then condition for the equality to hold?
(the generalized one)
very good you have solved the problem
I should have stated my aim clearly.
actually, the self-generalizing property can be used as
which gives a shorter proof
- 自由自在Lv 71 decade agoFavorite Answer
(ax - by)^2 >= 0
a^2x^2 - 2abxy + b^2y^2 >= 0
a^2x^2/y - 2abx + b^2y >= 0
a^2(Σx^2/y) - 2ab(Σx) + b^2(Σy) >= 0 (subscripts omitted for clarity in Yahoo)
Consider this as a quadratic in a, the above inequality just means there is at most one solution for a, therefore discriminant <= 0
[2b(Σx)]^2 - 4(Σx^2/y)(b^2)(Σy) <= 0
[(Σx)]^2 - (Σx^2/y)(Σy) <= 0
[(Σx)]^2/(Σy) <= (Σx^2/y) thus proved
In order for the equality to hold,
Σ(ax - by)^2 = 0
That means for each x and y, x/y = b/a is a constant, i.e. each pair of x and y should be of the same ratio.
2009-12-29 13:38:52 補充：
Second proof by Induction: