Suppose $X$ is a measurable space, $\Omega$ is an open set in the complex plane, $\phi$ is a bounded function on $\Omega\times X$ such that $\phi(z,t)$ is a measurable function of $t$, for each $z\in\Omega$, and $\phi(z,t)$ is holomorphic in $\Omega$, for each $t\in X$.
Prove that to every compact $K\subset \Omega$ there corresponds a constant $M<\infty$ such that $\left|\frac{\phi(z,t)-\phi(z_{0},t)}{z-z_{0}}\right|
This question is really tough to me, and I have no idea how to prove it. Any hints will be appreciated. Thanks