Show $|\int f(z)\, dz|\leq4$

Solution 1:

The function $g(t):=f\bigl(e^{it}\bigr)\in[{-1},1]$ is $2\pi$-periodic. By definition of the line integral the parametrization $t\mapsto z(t):=e^{it}$ $(0\leq t\leq2\pi)$ gives $$\int_C f(z)\>dz=\int_0^{2\pi} g(t)\>ie^{it}\>dt=:i\rho \,e^{i\alpha}$$ for some $\rho\geq0$ and $\alpha\in{\mathbb R}$. We have to prove that $\rho\leq4$. To this end consider $$\int_0^{2\pi}g(t+\alpha)e^{it}\>dt=\int_0^{2\pi}g(\tau)e^{i(\tau-\alpha)}\>d\tau=\rho\ .$$ Since $g$ is real-valued we can conclude that in fact $$\rho=\int_0^{2\pi} g(t+\alpha)\,\cos t\>dt\leq\int_0^{2\pi}\bigl|\cos t\bigr|\>dt=4\ .$$