For a holomorphic function, does Newton's difference quotient converge uniformly on compact sets?

For a holomorphic function, does Newton's difference quotient converge
uniformly on compact sets?

Let $K\subset A\subset C$, where $K$ is compact, $A$ is open, and $C$ is
the complex numbers. Let $f:C\rightarrow C$ be holomorphic on $A$
(holomorphic = complex derivative exists for all points in $A$). Is it
true that on $K$, $[f(z+\Delta z) - f(z)]/\Delta z$ converges uniformly as
$\Delta z\rightarrow 0$?