By Rainich G. Y.

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U0 ; v0 / ; 1; C 2 0 ; 0 denotes the hypotenuse of the triangle T where Ta;b a;b . ˝/. A 2=3/ . I /. Hence, u0 ; v0 exist, as was claimed. 0 is larger than the one of TA;b0 . c/ º. c/ is well deﬁned. c/ T . 11, it follows that T2u2 ;2v2 is a triangle of minimal area in T . a; 2u2 / 3 c 7 ! c/ / is strictly decreasing. ˝/ which contradicts the minimality of Ta;b . 13 (cf. [286]). 12. 0; R/ 2 @I , then . s h / DW E, where h WD hI . p Proof. , . E/. Then, A < a. , b R2 . ˝/: 38 Chapter 2 The Carathéodory pseudodistance and the Carathéodory–Reiffen .

K k 1 k 1 / jj k k 1/ j 1j k 1j kD1 n X " j Ä ck k k 1j C kD1 Ä"jz1 z2 j C M n X M 1 C jf . 1 C "// jz1 Letting " ! 0, we get the required estimate. 2 Some applications Finally, we are going to present the big Picard theorem. 7 (Big Picard Theorem). Let f W D that has an essential singularity at 0. Then, #¹w 2 C W #f 1 ! , the function f takes on all possible complex values, with at most a single exception, inﬁnitely often. Proof. wj / < C1, j D 1; 2. We may assume that w1 D 0, w2 D 1. z=2k /, z 2 D , k 2 N.

G Ä ıG ). z00 ; z000 / for some z00 , z000 2 G; z00 ¤ z000 , (resp. z0 I X0 / for some z0 2 G, X0 2 C n ; X0 ¤ 0) implies that F is biholomorphic (cf. z1 ; z2 / 7 ! z1 ; 0/ 2 C 2 ). This is not true even for D D G C n and even under more restrictive assumptions on z00 , z000 (resp. z1 ; z2 //, provided that jz1 j jz2 j (resp. X1 ; X2 //, provided that jX1 j jX2 j). 3. Let G0 , G, G G\G 1 z z z z G 6 G. G/ with f D f on G0 . G/, z and kfzk z Ä kf kG (cf. 12). G; z D/jG . G 24 Chapter 2 The Carathéodory pseudodistance and the Carathéodory–Reiffen .