(10 pts) Let U ⊂ C be an open-set. A function f : U → C satisfies a L

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(10 pts) Let U ⊂ C be an open-set. A function f : U → C satisfies a Lipschitz condition of order α(0 < α ≤ 1) in U if |f(z2) − f(z1)| ≤ m|z2 − z1| α for all points z1, z2 ∈ U, where m is a constant. If f is analytic in U, and an open disk Δ(z0, r) = {z:|z − z0| < r} is contained in U , then prove that |f′(z0)| ≤ mr α − 1 .

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(10 pts) Let $U \subset C$ be an open-set. A function $f : U \to C$ satisfies a Lipschitz condition of order $α (0 < α \leq 1)$ in $U$ if $| f (z_{2}) - f (z_{1}) | \leq m {| z_{2} - z_{1} |}^{α}$ for all points $z_{1}, z_{2} \in U$ , where $m$ is a constant. If $f$ is analytic in $U$ , and an open disk $Δ (z_{0}, r) = {z$ : $| z - z_{0} | < r}$ is contained in $U$ , then prove that $| f^{'} (z_{0}) | \leq m r^{α - 1}$ .