Proposition 4

Let $f$ be a continuous differentiable function of a complex variable defined on a convex open subset $A$ of the field $\mathbf{C}$, with values in a normed space $E$ over the field $\mathbf{C}$. If one has $\|f'(z)\| \leq m$ for all $z \in A$, then one has $\|f(b) - f(a)\| \leq m |b - a|$ for every pair of points $a, b$ of $A$.

We put $g(t) = \frac{1}{b - a} f(a + t(b - a))$ for $0 \leq t \leq 1$; since $g'(t) = f'(a + t(b - a))$, applying th. 2 to the function $g$ yields the proposition immediately.