Mathematics. — “On the Theorem of Picard.” By Prof. J. Worer. 
(Communicated by Prof. L. E. J. Brouwer.) 
(Communicated at the meeting of June 26, 1920). 
The theorem of Picard on the conduct of a uniform analytical 
function in the neighbourhood of an isolated essentially singular 
point was proved in 1896 by Bore. without the use of the modular 
function.) By this a series of elementary proofs was opened for 
the celebrated theorem. In 1904 ScHorrky made the demonstration 
of Bore considerably stricter.®) He found an important theorem 
on holomorphic and meromorphic functions which are nowhere 
zero and nowhere 1, and on this he founded the elementary proof 
for the theorem of Picarv.*) After this Lanpavu discovered an ex- 
tension of the theorem used by Scuortky‘). The remarkable result 
is as follows: if f(z) ts holomorphic for |z| < R, if tt is there 
nowhere zero or 1, if further | f(O)| <p, then for |z| SOR, in which 
0 <1, we have | f(z)|<P(u, 4), where ® only depends on 6 and u. 
As Scrorrky did not possess this proposition, his reason- 
ing is here and there subtile. Elegant proofs of the theorem of 
Picarp were given in 1912 and 1913 by Monrrer, but they are 
founded on the consideration of the so-called normal families of 
functions.*) BerNays, who in 1911 quite simply brought forth the 
theorem of Lanpau out of that of Scnorrky’), gave in 1913 new 
derivations of LANDAU’s theorem, and investigated at the same time 
the function p (a), the upper limit for the radius of a circle, where 
the series f= a-+z-+a,z*-+.... converges and nowhere becomes 
zero or 1‘). 
1) Comptes rendus, May 11 1896, part 122, p. 1045—1048. 
2) Sitzungsber. der K. Pr. Ak. d. Wiss., 1904, p. 1244—1262. 
5) l. ce. p. 1255 sqq. 
4) Göttinger Nachrichten 1910, p. 309— 312. 
5) Annales de l'école normale, part 29 (1912), p. 512 and part 33 (1916) p. 251. 
Monte. gives here at the same time a simple proof of the theorem of LaANpau 
(part 33 (1916) p. 517.) 
6) See e.g. Sitz. ber. der K. Pr. Ak. d. Wiss. 1911, p. 597. 
Levy made this derivation still more simple in the Bulletin de la Soc. Math. de 
Fr., part 40 (1912) p. 25—39. It deserves to be mentioned that in 1907 Scuortxy 
(Sitz Ber. der K. Pr. Ak. d. Wiss. p. 823—840) gave two new proofs of the 
theorem of Picarp. They are, however, no more simple than the one of 1904. 
7) Vierteljahrschrift der Naturf. Ges., Zürich, 58 (1914), part 3, p. 203—238. 
