Estimating the $L^1$ norm of the Dirichlet kernel

Question

Suppose $D_N(x)=\frac{\cos\frac{x}{2}-\cos(N+\frac{1}{2})x}{\sin\frac{x}{2}}$. How to prove the inequality below$$\int_{-\pi}^\pi|D_N(x)|\text{d}x\leq c\log N$$ for some constant $c>0$ ?

Answer 1

By recalling that $$D_n(x)=\sum_{k=-n}^n e^{ikx}=1+2\sum_{k=1}^n\cos(kx)=\frac{\sin\left(\left(n +1/2\right) x \right)}{\sin(x/2)}\tag{1}$$ it is not difficult to locate the stationary points of $D_n(x)$ in $(-\pi,\pi)$ and conclude that $$ \left|D_N(x)\right|\leq \min\left(2N+1,\frac{\pi}{|x|}\right)\tag{2}$$ from which: $$ \int_{-\pi}^{\pi}|D_N(x)|\,dx\leq \int_{-\frac{\pi}{2N+1}}^{\frac{\pi}{2N+1}}N\,dx+2\pi\int_{\frac{\pi}{2N+1}}^{\pi}\frac{dx}{x}\leq 10\log N\tag{3} $$ for any $N\geq 8$.