let $b>0$, $y:[0,b]\rightarrow\mathbb{C}$ and $y(0)=0$, where $\mathbb{C}$ denotes the space of complex numbers. Is the following inequality true or not? $$|y(x)|\le \max_{\tau\in[0,b]} |y'(\tau)| \cdot b$$
Please notes that $y(x)$ is not a real function. Thank you very much.
It is true. Since Wikipedia does a good job of explaining it in the general case (here $\Bbb C$ isn't much different from $\Bbb R^2$), I'll contend myself to linking to the relevant part on the Mean Value theorem page (see equation (**)).