I am really bad when dealing with what I call "technique" exercises. In order to solve these exercises, we need to find a slick trick or well-known ones.
Today my teacher gave me these two exercises where I need to prove two inequalities.
Let's consider a sequence $(z_n)_{n \geq 1}$ where $\mid z_n \mid \leq 1, \forall n \geq 1$. Let $N, H \in \mathbb{N}$ such that $1 \leq H \leq N$.
Then I need to prove :
$$\mid \sum_{n = 1}^N z_n - \frac{1}{H} \sum_{n =1}^N \sum_{h = 1}^H z_{n+h} \mid \leq H+1$$
I think I found this one. Here is what I've done :
$$\mid \sum_{n = 1}^N z_n - \frac{1}{H} \sum_{n =1}^N \sum_{h = 1}^H z_{n+h} \mid = \mid\frac{1}{H} \sum_{n =1}^N \sum_{h = 1}^H (z_{n+h}-z_n) \mid = \mid\frac{1}{H} \sum_{h =1}^H \sum_{n = 1}^N (z_{n+h}-z_n) \mid $$
Most of the complex numbers telescope so we get : $$\mid \sum_{n = 1}^N z_n - \frac{1}{H} \sum_{n =1}^N \sum_{h = 1}^H z_{n+h} \mid = \frac{1}{H}\mid ( \sum_{h =1}^H (H-h+1)z_{N+h}+\sum_{n=1}^H (H-n+1)z_n) \mid \leq^{\text{triangle inequality}} \frac{2}{H} \sum_{i =1}^H i = H+1$$
If there is something incorrect or another approach do not hesitate to tell about it. For most of the users of this website this inequality is maybe trivial but for example, here it took me a while to get the right idea. At first, I was playing with the triangle inequality and I was getting really bad upper-bound like $2N$...
The second exercise is (using the same notation) :
Prove that :
$$\sum_{n=1}^N \mid \sum_{h =1}^H z_{n+h} \mid^2 \leq NH +2H\sum_{h =1}^H \mid\sum_{n =1}^N z_{n+h}\bar{z_n}\mid + H^2(H+1) $$
I didn't find this one. The square in the first expression seems to be inviting using C.S somewhere but I can't figure that out. Also, I tried developing what was inside the square and apply the triangle inequality, but I get really big upper-bound whereas this one seems sharp to me.
Any hints, ideas or complete detailed solutions are welcome.
I really want to progress in these kinds of exercises and I am looking for a baggage of tricks. So I am looking for files or books in which I can find exercises that asked to deal with sums in a tricky way (or integral) in order to prove an inequality or an equality (just as the two exercises above). If anyone knows about such a book or can provide a link where I can train working with sum or integral in every possible ways it will be very nice.
Thank you for taking your time reading this long post and understanding that I really want to improve my skills in these kind of exercises :)
Hint : \begin{align*} \sum_{n=1}^N \left\lvert \sum_{h =1}^H z_{n+h} \right\rvert^2&=\sum_{n=1}^N \left( \sum_{h_1 =1}^H z_{n+h_1} \right)\overline{\left( \sum_{h_2 =1}^H z_{n+h_2} \right)}\\ &=\sum_{n=1}^N \sum_{h_1 =1}^H\sum_{h_2 =1}^H z_{n+h_1} \bar{z}_{n+h_2}\\ &=\sum_{h_1 =1}^H\sum_{h_2 =1}^H \sum_{n=1}^N z_{n+h_1} \bar{z}_{n+h_2}\\ \end{align*}