In my old high school notebook (20 years ago), the following inequality appears with its proof:
$$1+\cos x + \frac{1}{2}\cos 2x + \cdots + \frac{1}{n}\cos nx \geq 0$$ for any real $x$ and positive integer $n$.
I am not the one that created this inequality. So the my question is where references for this inequality can be found.
According to the last sentence on page 16 of this paper, this inequality was proved by
W. H. Young, On certain series of Fourier, Proc. London Math. Soc. (2) 11 (1912), 357–366.
A long comment: the derivative is not positive for all $x$, $n$. Thus there is the need for some more sophisticated argument than monotonicity.
The derivative is $$ - \Im \left( e^{ix}\cdot \frac{e^{inx} -1}{e^{ix}-1} \right).$$ The derivative being positive corresponds to $$\textrm{arg}\left( e^{ix}\cdot \frac{e^{inx} -1}{e^{ix}-1} \right) \le 0 .$$ Using additivity of arg, this reformulates as $$ \textrm{arg}(e^{ix}) -\textrm{arg}(e^{ix} -1 ) + \textrm{arg}(e^{inx} -1) \pmod{2 \pi} \in [0, \pi] .$$ Note that $y\equiv nx \pmod{2 \pi} $ can be almost anything if $x$ is irrational, thus in particular we would have $$ \textrm{arg}(e^{ix}) -\textrm{arg}(e^{ix} -1 ) + \textrm{arg}(e^{iy} -1) \pmod{2 \pi} \in [0, \pi] .$$ The $y$-term ranges over $[3\pi/4, 5\pi/4]$, thus we should have $$\textrm{arg}(e^{ix}) -\textrm{arg}(e^{ix} -1 ) \in [5/4\pi, 7/4\pi].$$
If $x$ is very small and positive, the expression is close to $3/4 \pi$, thus it is false.
