PlanetMath: proof of Marty's theorem

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proof of Marty's theorem

(Proof)

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

$\displaystyle \frac{2 \vert f'(z)\vert}{1 + \vert f(z)\vert^2}$

$\displaystyle \leq M_K \ \forall \ f \in \mathcal{F}, z \in K$

(

)

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

$\displaystyle d_{\sigma}(f(P), f(Q))$	$\displaystyle = \inf_{\gamma} l_{\sigma}(f \circ \gamma) = \inf_{\gamma} \int_{a}^{b} \Vert(f \circ \gamma)'(t)\Vert _{\sigma, f \circ \gamma(t)} \,dt$
	$\displaystyle = \inf_{\gamma} \int_{a}^{b} \frac{2 \vert f'(\gamma(t))\vert}{1 + \vert f(\gamma(t))\vert^2} \vert\gamma'(t)\vert \,dt$
	$\displaystyle \stackrel{\leq}{\substack{\eqref{1}}} M_K \inf_{\gamma} \int_{a}^{b} \vert\gamma'(t)\vert \,dt$
	$\displaystyle = M_K \inf_{\gamma} l(\gamma) = M_K \vert P - Q\vert$

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

(ii) Now assume $\mathcal{F}$ to be normal. Define:

$\displaystyle f^{\sharp}(z)$

$\displaystyle := \frac{2 \vert f'(z)\vert}{1 + \vert f(z)\vert^2}$

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

$\displaystyle \max_{z \in K} f_n^{\sharp}(z)$

$\displaystyle \to \infty \ (n \to \infty)$

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

"proof of Marty's theorem" is owned by karstenb.

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Cross-references: function, meromorphic, converges, neighbourhood, sequence, bounded, contradiction, normal, theorem, equicontinuous, Lipschitz continuous, points, curves, region, compact, fix

This is version 1 of proof of Marty's theorem, born on 2008-09-15.
Object id is 11030, canonical name is ProofOfMartysTheorem.
Accessed 436 times total.

Classification:

AMS MSC:

30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory)

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