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I am working on a problem and I got to this point. $$a_{nk} = p_k(\lambda_0)q_n(\lambda_0) - q_k(\lambda_0)p_n(\lambda_0).$$

The author now claim the following and I am not sure the inequality that was used. Please see the image belowenter image description here

Hints and suggestions will be appreciated.

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Hint: for the first inequality the author is using $$(a+b)^2\leq 2(a^2+b^2),$$ Or equivalently (just substitute $b$ eirh $-b$) $$(a-b)^2\leq 2(a^2+b^2).$$ (You can easily check these inequalities by expanding the bracket)

The second inequality is trivial, although a priori one shiuld put $\leq$ instead of $<$)

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