MATHEMATICS: T. H. GRONWALL 23 
It should be noted that no assumption is made regarding the convergence 
of the power series at z = 1. For the proof, we write wiz) = Wi{z) + Wiiz), 
where 
^'iW = "^^fe) = 2/iv^«^"- 
First, we have 
= (2 - 20 2r'^» + 2' + • • • +2""') 
and since | z - z' | = p I e^* - 6^'* 1 ^ 2p, | z | < 1 and | z' | < 1, it follows 
that 
In the second place, 
and applying Lagrange's inequality 
Denoting by the z the conjugate of z, we have 
I 2« - I 2 = (z« - (i« - i7«) = (2 2)« + iz' - (Z Z'Y - {Z' z)\ 
and consequently, since |z|<l,|z'|<l, 
Z"-Z'^P , (1 -Z^) (1 -Z'2) 
Z« - Z'" P 
z" - z'" M 
2- z'-z"n 
1 — ^ = 
(1 - zz) (1 - z'^O* ' 
Introducing z = l — pe^*, z' = l — pg^', a simple calculation shows that 
for p < sin € and observing the limitations governing B and 6' , 
(1 -zz') (1 - z'z) ^ 2 + 2 cos 4- ^0 - 2 p (cos B + cos 6') + P^ 
(1 - z z) (1 - z' z') (2 cos ^ - p) (2 cos 9' - p) 
2+2 + 2(1 + 1) + ! _ 9 
< 
(2 sin € — sin e) (2 sin e — sin e) sin^ e 
so that finally 
g(S>U»lf (log ^-iJ-j' (2) 
for p < sin €, 
