I'm having trouble with an exercise in the Cauchy Schwarz Master Class by Steele. Exercise 1.3b asks to prove or disprove this generalization of the Cauchy-Schwarz inequality:
The following is the solution at the end of the book:
After struggling to understand the solution for a few hours, I still cannot see why the substitution $c_k^2 / (c_1^2 + \ldots + c_n^2)$ would bring the target inequality to the solvable inequality. Neither do I understand what the $n^2 < n^3$ bound has to do with anything or how it allows us to take a "cheap shot".
Thanks!
Edit: I'm also wondering, is there a name for this generalization of Cauchy-Schwarz? Any known results in this direction?
Best Answer
I have reason to believe the text has a typo; maybe someone can correct me on this point. Because, to my mind, the definition of the $\hat{c}_i$'s would apply to the sum $\sum |a_k b_k c_k^2|$. I suspect it should read
$$\hat{c}_i=\frac{c_i}{\sqrt{c_1^2+c_2^2+\cdots+c_n^2}}.$$
If my hunch is correct, we would argue as follows (using the $\hat{c}_i$'s defined right above):
$$\left|\sum_{k=1}^n a_kb_k\hat{c}_k \right| \color{Red}\le \sum_{k=1}^n |a_kb_k \hat{c}_k| \color{Green}\le \sum_{k=1}^n |a_kb_k| \color{Blue}=\left|\sum_{k=1}^n |a_k|\cdot|b_k|\right| \color{Purple}\le \left(\sum_{k=1}^n |a_k|^2\right)^{1/2}\left(\sum_{k=1}^n |b_k|^2\right)^{1/2} $$
Above:
Now take the far left and far right side of this, square, and multiply by $c_1^2+c_2^2\cdots+c_n^2$ (apply to $\hat{c}_i$).