Find maximum of a complex function $f(z)$ I am  trying to find the following maximum, whose existence is justified by the compactness of the close ball $\Delta$  of $\mathbb C$  and continuity of the function  
$$f(z)=\sum_{k=1}^\infty (-1)^k \frac{2z}{k^2 \pi^2-z^2}\cos kt$$
from $\mathbb C$   to  $\mathbb C$ 
$$\large { \displaystyle \max_{z \in {\mathbb C},|z| \leq \frac{\pi}{4}} \left|f(z)\right|}$$
any suggestions please? I try to use Maximum Modulus Principle:
$$\large { \displaystyle \max_{z \in {\mathbb C},|z| \leq r} \left|\sum_{k=1}^\infty (-1)^k \frac{2z}{k^2 \pi^2-z^2}\cos kt\right|}=\large { \displaystyle \max_{z \in {\mathbb C},|z| = r} \left|\sum_{k=1}^\infty (-1)^k \frac{2z}{k^2 \pi^2-z^2}\cos kt\right|}=\large { \displaystyle \max_{\theta\in \mathbb R} \left|\sum_{k=1}^\infty (-1)^k \frac{2re^{i\theta}}{k^2 \pi^2-r^2e^{i2\theta}}\cos kt\right|}$$. How can I go on?

Hint: holomorphic functions on some bounded open area attaint their maximum on the boundary of the area.
