Here is a problem for solving it there is enough knowledge in the volume of 1-2 courses of mathematical specialties. I managed to do this, perhaps because the proof is not based on some kind of focus, but is short (a few lines), logical and consistent, although it requires a certain courage of thought. If a function $f(x)$ is convex on the segment from zero to two pi, then the inequality $$\forall n, n\in \mathbb N,\, \int_0^{2\pi}f(x)\cos(nx)\,dx \ge 0 $$ holds. In the above $0$ is not treated as a natural number.
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What is your question? – Kavi Rama Murthy Oct 03 '22 at 12:28
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How to prove it? Maybe, there are few ways. – user64494 Oct 03 '22 at 12:30
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2Does this answer your question? Prove that $\int_{0}^{2\pi}f(x)\cos(kx)dx \geq 0$ for every $k \geq 1$ given that $f$ is convex. – Mittens Oct 03 '22 at 13:05
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@OliverDiaz: Thank you for the reference. My proof is not tricky and the one is simpler in the case $f\in C^2[0,2\pi]$. – user64494 Oct 03 '22 at 13:23
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@OliverDiaz: I presented my answer in the thread inidcating by you. – user64494 Oct 03 '22 at 15:02