A basic inequality

**forumfann** · Nov5-09, 12:40 AM

Could anyone help me on this,
Is it true that for any given $LaTeX Code: r_{1},r_{2},r_{3},r_{4}>0$ and $LaTeX Code: t_{1},t_{2},t_{3},t_{4}\\in[0,2\\pi)$ if
$LaTeX Code: r_{1}\\left|\\cos(t-t_{1})\\right|+r_{2}\\left|\\cos(t-t_{2})\\right|$ $LaTeX Code: <r_{3}\\left|\\cos(t-t_{3})\\right|+r_{4}\\left|\\cos(t-t_{4})\\right|$ for all $LaTeX Code: t\\in[0,2\\pi)$
then $LaTeX Code: r_{1}+r_{2}<r_{3}+r_{4}$ ?

By the way, this is not a homework problem.

Any help will be highly appreciated!

**Kaimyn** · Nov5-09, 03:43 AM

Originally Posted by forumfann

Could anyone help me on this,
Is it true that for any given $LaTeX Code: r_{1},r_{2},r_{3},r_{4}>0$ and $LaTeX Code: t_{1},t_{2},t_{3},t_{4}\\in[0,2\\pi)$ if
$LaTeX Code: r_{1}\\left|\\cos(t-t_{1})\\right|+r_{2}\\left|\\cos(t-t_{2})\\right|$ $LaTeX Code: <r_{3}\\left|\\cos(t-t_{3})\\right|+r_{4}\\left|\\cos(t-t_{4})\\right|$ for all $LaTeX Code: t\\in[0,2\\pi)$
then $LaTeX Code: r_{1}+r_{2}<r_{3}+r_{4}$ ?

By the way, this is not a homework problem.

Any help will be highly appreciated!

I may be incorrect, but I would say this would be false.

What if $LaTeX Code: t-t_{1}$ and $LaTeX Code: t-t_{2}$ are equal to

,

or

? Then r₁ and r₂ can be anything, and don't have to satisfy the inequality!

**forumfann** · Nov5-09, 09:27 AM

If $LaTeX Code: t-t_{1}$ and $LaTeX Code: t-t_{2}$ are equal to

,

or

? Then the left hand side of the given inequality is

, which is less than the right hand side of the given inequality that is not larger than

. Thus the claim is automatically true.

I think what makes it possible to be true is "for all $LaTeX Code: x\\in[0,2\\pi]$ ", but I don't know how to prove it.

Again, any suggestion that can lead to the answer to the question will be greatly appreciated.

**Kaimyn** · Nov6-09, 06:40 AM

Ahh, sorry, I meant pi/2, meaning cos(pi/2) = 0. Then they do not have to be < r₃+ r₄

Besides, say they are both equal to pi anyway. Then, r₁ and r₂ can be greater than r₃ and r₄, yet still hold true in the first inequaility but not the second.