On some F\'ejer-type trigonometric sums

We examine the four F\'ejer-type trigonometric sums of the form $$S_n(x)=\sum_{k=1}^n \frac{f(g(kx))}{k}\qquad (0<x<\pi)$$ where $f(x)$, $g(x)$ are chosen to be either $\sin x$ or $\cos x$. The analysis of the sums with $f(x)=g(x)=\cos x$, $f(x)=\cos x$, $g(x)=\sin x$ and $f(x)=\sin x$, $g(x)=\cos x$ is reasonably straightforward. It is shown that these sums exhibit unbounded growth as $n\to\infty$ and also present `spikes' in their graphs at certain $x$ values for which we give an explanation. The main effort is devoted to the case $f(x)=g(x)=\sin x$, where we present arguments that strongly support the conjecture made by H. Alzer that $S_n(x)>0$ in $0<x<\pi$. The graph of the sum in this case presents a jump in the neighbourhood of $x=2\pi/3$. This jump is explained and is quantitatively estimated when $n\to\infty$.