I am trying to show that
$$
\left\lvert\int_\gamma \frac{\cos(z)}z \,dz\right\rvert \le 2\pi e
$$
if $\gamma$ is a path that traces the unit circle once.
The book recommends that I show that $\lvert \cos(z) \rvert \le e$ if $\lvert z \rvert = 1$ to help prove this. I know that if $\lvert f(z) \rvert \le M$ for all $z \in \gamma (I)$ then
$$
\left\lvert \int_\gamma f(z) \,dz \right\rvert \le M\ell (\gamma),
$$ where $\ell (\gamma)$ is the length of the path, which in this case is $2\pi$. So I can see why I would need to show $\lvert cos(z) \rvert \le e$ if $\lvert z \rvert = 1$ to prove the inequality, but I am not sure where to go from here to show that.
Best Answer
With the series expansion of $\cos$ we get for $|z|=1$:
$$| \cos z| \le \sum_{n=0}^{\infty}\frac{1}{(2n)!} \le \sum_{n=0}^{\infty}\frac{1}{n!}=e.$$