I came across this inequalities that requires the conclusion of a prove. Please does this Inequalities hold? And how can I prove it?
Given a finite increasing sequence $\{x_i\}_{i=1}^n$.
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$$ x_1^2 k_1 + (x_2^2 – x_1^2) k_2 +\dots +(x_n^2 – x_{n-1}^2)k_n \geq \big (x_1 k_1 + (x_2 -x_1)k_2)+\dots+(x_n – x_{n-1})k_n \big)^2 $$ Where $0\leq k_i \leq k$ and $k_n, k_1 >0$
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$$k \left( \sum_i k_i {x_i}^2 + 2\sum_{i<j}k_j x_i x_j \right) \ge \left(
\sum_i k_i x_i \right)^2 $$
I am thinking of applying the Holder's inequality. But I don't really know how to approach this. Any help or hint is appreciated. Thanks
Best Answer
The second inequality is obviously true, and precisely because $ k \geq k_i$ as stated in the paper (from OP's comment).
$$k \left( \sum_i k_i {x_i}^2 + 2\sum_{i<j}k_j x_i x_j \right) = \sum k k_i x_i^2 + 2 \sum_{i<j} k k_j x_i x_j \geq \sum k_i^2 x_i^2 + 2 \sum_{i<j } k_i k_j x_i x_j = \left( \sum_i k_i x_i \right)^2. $$