$L^p$ norms of a sine sum

George Pólya and Gábor Szegő, Problems and theorems in analysis, Volume II, volume 216 of Die Grundlehren der mathematischen Wissenschaften, Springer, 1976, translated from the German by C. E. Billigheimer.

The following is an estimate of the $L^p$ norms of a sum with nonnegative terms (p. 77, no. 38).

Let ${\mathrm{\Gamma }}_n(t)={\sum }_{k=1}^n\frac{|\sin kt|}{k}$. As $|\sin kt|\le 1$, we have ${\mathrm{\Gamma }}_n(t)\le {\sum }_{k=1}^n\frac{1}{k}$, but we can give a sharper upper bound for ${\mathrm{\Gamma }}_n(t)$ using the following two results. First, if $B_n(t)={\sum }_{k=1}^n\frac{\cos kt}{k}$, then $B_n(t)\ge -1$ for all $t$ (p. 75, no. 28). Second , for all $t$ (p. 76, no. 34),

Therefore

using

Thus ${\parallel {\mathrm{\Gamma }}_n\parallel }_p\le {\parallel {\mathrm{\Gamma }}_n\parallel }_{\mathrm{\infty }}\le \frac{2}{\pi }{\sum }_{k=1}^n\frac{1}{k}+\frac{2}{\pi }$. On the other hand,

Therefore ${\parallel {\mathrm{\Gamma }}_n\parallel }_p\ge {\parallel {\mathrm{\Gamma }}_n\parallel }_1=\frac{2}{\pi }{\sum }_{k=1}^n\frac{1}{k}$.

Hence for $1\le p\le \mathrm{\infty }$,

So,