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My real problem is about graph theory, but I've boiled it down to this statement. The issue is, I can't figure out if it's true or false. Can anyone shed some light on the matter? Thanks!

Let $a_1$, $a_2$, ... $a_n$, $b_1$, $b_2$, ... $b_n$ be distinct, positive, real numbers. If $$\sum_{i=1}^na_i < \sum_{j=1}^nb_j$$ then $$\sum_{i=1}^na_i^2 < \sum_{j=1}^nb_j^2$$ Any guidance is appreciated!

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    $\begingroup$ There are issues already with integers. For example, $1+10\lt 6+6$, but $1^2+10^2 \gt 6^2+6^2$. $\endgroup$ – André Nicolas Feb 27 '13 at 5:46
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For example, $\displaystyle 1+\frac{1}{9}< \frac{4}{9}+ \frac{8}{9}$ but $\displaystyle 1+ \frac{1}{9^2}> \frac{4^2}{9^2}+\frac{8^2}{9^2}$.

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It's not necessarily true. For example, 1+1.00001>1.5+0.0001, But$(1.5)^2+(0.00001)^2>1^2+(1.0001)^2$

The reason it's not true is that if we have a few large no.s in the $a_i$ Then those large no.s are increased significantly by squaring.

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  • $\begingroup$ distinct positive numbers? $\endgroup$ – Inquest Feb 27 '13 at 5:43
  • $\begingroup$ 0 is not positive. $\endgroup$ – p.koch Feb 27 '13 at 5:43
  • $\begingroup$ There, they're distinct now(and positive). $\endgroup$ – Ishan Banerjee Feb 27 '13 at 5:45
  • $\begingroup$ $0$ is very close to being positive. $\endgroup$ – Gerry Myerson Feb 27 '13 at 5:47
  • $\begingroup$ @GerryMyerson but 0.00001 is definitely positive $\endgroup$ – Ishan Banerjee Feb 27 '13 at 5:48
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No. If one $a$ is large and all the rest are small, while the $b$'s are about the same size it will fail. A specific example is $n=10, a_1=1000, \text {else } a_i=i, b_i=i+100$

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