Estimates for the Dirichlet kernel: $\frac{4}{\pi^2} \ln(n) \leq\frac{1}{\pi} \int_{-\pi}^\pi \big| D_{n}(t) \big| \, dt\leq 3 + \ln(n)$

Question

Let's define $$A_n := \frac{1}{\pi} \int_{-\pi}^\pi \big| D_{n}(t) \big| \, dt,$$ where $D_{n}(t)$ is Dirichlet Kernel

$$D_n(t):=\frac{1}{2} + \sum_{k=1}^{n} \cos(kt)= \frac{\sin \left(t\left(n+\frac{1}{2}\right)\right)}{2\sin \left(\frac{t}{2}\right)}.$$

I need to prove that $$\frac{4}{\pi^2} \ln(n) \leq A_n \leq 3 + \ln(n).$$

Any ideas, clues or hints on how to prove it? Something to start from?

Answer 1

All you need to know is that $3 < \pi < 4$ and $e > 2$.

As $n \ge 1$ you have $1 + \dfrac 1{2n} \le \dfrac 32$ so that $\dfrac 2\pi \left( 1 + \dfrac 1{2n} \le \dfrac 32 \right) \le \dfrac 3 \pi < 1$,

and $\ln \pi < \log_2 4 = 2$.

Answer 2

What I managed to find out is: $$A := \frac{1}{\pi} \int_{-\pi}^\pi \big| D_{n}(t) \big| \, dt \leq \frac{2}{\pi} \int_{0}^{\frac{1}{n}} (n+\frac{1}{2}) dt + \frac{2}{\pi} \int_{\frac{1}{n}}^{\pi} \frac{\pi}{2t}dt.$$ After integration I get: $$\frac{2}{\pi}(nt+\frac{t}{2}) + \frac{2}{\pi} \frac{\pi}{2} \ln|t| = \frac{2}{\pi}(1+\frac{1}{2n}) + \ln \pi + \ln n.$$

Any idea how to get 3 in this?