586 
Here follows a proof for the theorem of Picarp, which is founded 
on the theorem of Lanpau and which is for the rest elementary. 
1. Let f(z) be holomorphic for \z << # and let it there become 
1 
nowhere 0 or 1, while 0 <|f(0)| <u. Then according to the theorem 
of LANDAU: 
pw SlfE)|Splw, for |z| SA. 
By p(u) we can understand the upper limit of | f(z) |, when 
zi <4, for the functions which satisfy the conditions mentioned. 
Then p(u) is a monotonely increasing function of u. We shall prove 
that p (u) increases at a slower rate than a certain power of u. Let 
in the first place u —e?'*, where & is a positive integer, and con- 
sider the function 
1 eh SS 
(2k + 2) ni 
in which for z=O the numerator is equal to the principal value 
of Log f{O). 4(2) is uniform and holomorphic for |z)|< R and 
there nowhere zero or 1, because f(2) 40 and 41. Further 
| Log | f(0)| | + = 
Ee) eN 
From er < | f(0)| < e?* follows | Log | f(0)\| < 2hkz, so that 2(0) <1. 
Now the theorem of LANDAU gives 
AO |< 
R 
HOEP Her, for |2/ <>, 
so that 
R 
| Log f (2) | <C (2k + 2) ap, for |z| Sg 
and 
eHDap < | f(z) | << AAD ep, 
We have therefore: 
p (ec?) < e2k+2)=p for ka positive integer. 
If u be an arbitrary number >1, we can find a positive integer 
k for which 
UI u < e2kr, 
p (u) < — (er) SektDrp Set up. 
For u >1 we have therefore 
gua... «1-16.10. ae 
in which @=e*? and: p'— Pp (1,4): 
2. Let us now consider a function #'(z), holomorphic in a certain 
neighbourhood @ of O(z=0) with the exception of O. Let there 
exist a neigbourhood of 0, 2’ < 2 in which F(z) #0 and F 1. 
We describe a circle inside 2’ with radius 2 g. 
Then 
