proof of Marty’s theorem

(i) Fix $K\subseteq \mathrm{\Omega }$ compact. We have:

${\displaystyle \frac{2|f^{\prime }(z)|}{1+{|f(z)|}^2}}$$\le M_K\forall f\in ℱ,z\in K$($*$)

Let $V$ be a region with $K=\overline{V}$ and let $\gamma :[a,b]\to V$ be the $C^1$ curves connecting the points $P,Q\in \mathrm{\Omega }$. Then we have:

$d_{\sigma }(f(P),f(Q))$$=\underset{\gamma }{inf}{l}_{\sigma }(f\circ \gamma )=\underset{\gamma }{inf}{\displaystyle {\int }_a^b}{\parallel {(f\circ \gamma )}^{\prime }(t)\parallel }_{\sigma ,f\circ \gamma (t)}𝑑t$

$=\underset{\gamma }{inf}{\displaystyle {\int }_a^b}{\displaystyle \frac{2|f^{\prime }(\gamma (t))|}{1+{|f(\gamma (t))|}^2}}|{\gamma }^{\prime }(t)|𝑑t$

$\stackrel{\le }{\begin{array}{c}(\mathit{\text{<span class="ltx\_text ltx\_ref\_tag">(\$*\$)</span> ‣ <span class="ltx\_text ltx\_ref\_title">proof of Marty’s theorem</span>}})\end{array}}{M}_K\underset{\gamma }{inf}{\displaystyle {\int }_a^b}|{\gamma }^{\prime }(t)|𝑑t$

$=M_K\underset{\gamma }{inf}l(\gamma )=M_K|P-Q|$

Thus $f$ is Lipschitz continuous and thus $ℱ$ is equicontinuous. By the Ascoli-Arzelà Theorem we conclude that $ℱ$ is normal.

(ii) Now assume $ℱ$ to be normal. Define:

$f^{\mathrm{♯}}(z)$$:={\displaystyle \frac{2|f^{\prime }(z)|}{1+{|f(z)|}^2}}$

Let $K\subseteq \mathrm{\Omega }$ be compact. To obtain contradiction assume $\{f^{\mathrm{♯}}:f\in ℱ\}$ is not uniformly bounded on $K$.But then there exists a sequence $\{f_n\}\subset ℱ$ such that:

$\underset{z\in K}{\max }{f}_n^{\mathrm{♯}}(z)$$\to \mathrm{\infty }(n\to \mathrm{\infty })$

Since $ℱ$ is normal for each $P\in \mathrm{\Omega }$ let there be a neighbourhood $U_P$ such that $\{f_n\}$ converges normally to a meromorphic function $f$. But from ${(1/f_n)}^{\mathrm{♯}}=f_n^{\mathrm{♯}}$ we see that $\{f_n^{\mathrm{♯}}\}$ converges normally on $U_P$. Since $K$ is compact it can be covered by finitely many sets $U_P$. We conclude that $\{f_n^{\mathrm{♯}}\}$ must be bounded on $K$ and obtain a contradiction. ∎