Please tell me wether the following is true or not. $$\sum_{i=1}^{n} a_{i} b_{i}\leq \sqrt{\left(\sum_{i=1}^{n} a_{i}\right)\left(\sum_{i=1}^{n} b_{i}\right)}$$
Is it true that $\sum_{i=1}^{n} a_{i} b_{i}\leq \sqrt{(\sum_{i=1}^{n} a_{i})(\sum_{i=1}^{n} b_{i})}$?
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inequality
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0Not. Such Cauchy-Schwarz-inequality-gone-wrong is not even homogeneous. – 2017-01-08
1 Answers
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Observe if $a=(2, 2)$ and $b= (2, 1)$, then we see that \begin{align} a\cdot b = 6 \ \ \text{ and } \ \ \sqrt{4\cdot 3}. \end{align}
Additional. Observe if the inequality holds, then we see \begin{align} \sum^n_{i=1}\lambda^2 a_ib_i \leq \lambda \sqrt{\sum^n_{i=1}a_i\sum^n_{i=1}b_i} \end{align} if we rescale $a=(a_1, \ldots, a_n)$ and $b=(b_1, \ldots, b_n)$ by $\lambda>0$.
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0if all $a_{i}$ and $b_{i}$ are nonnegative number what about this inequlity? – 2017-01-08
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0Thanks, Jacky Chong. Very useful examples. – 2017-01-08
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0In words, the good explanation given by @Jacky Chong says that this inequation doesn't resist to the uneven _homogeneity_ between its LHS and RHS. – 2017-01-08