MATHEMATICS: T. H. GRONWALL 
249 
.... For the comparison of maps of different regions, two geometrical 
concepts are of fundamental importance: the distortion = \dw/dz\, or the ratio 
of the lengths of corresponding line elements in the w- and z-planes, and the 
twist = imaginary part of log {dw/dz), or the angle between corresponding 
line elements, this angle being always taken between — x(excl.) and tt (inch). 
Koebe^ has shown that on the circle \z\= r, 0<r<l, both \dw/dz\ and |2| lie 
between positive bounds which depend on r alone, and the writer^ has deter- 
mined the exact values of these bounds. 
It is the purpose of the present note to state the corresponding result in 
regard to the twist: 
When the analytic function 
w = z a2Z^ + . . . -\- a^z^ -\- . . . 
maps the circle \z\ <1 on the interior of a simple region D in the w-plane, the 
twist T satisfies the following inequalities for \z\= r and 0<r<2~' 
— 4 arcsin r<r< 4 arcsin r, . (1) 
except when 
2 (1 - cos ,8 • 
w = 
(1 - e("+^>«' 
(2) 
where a. and /3 are real, and cos ^ = r. In this case, r attains the upper or lower 
bound in (1) for z = re~^^ according as (3 = -\- arc cos r or ^ — —arc cos r. 
For ^r<l,we have only the inequality included in the definition of r 
-T<T^7r, (3) 
and no single class of functions analogous to (2) reaching the upper and lower 
bounds can be assigned. 
When the region D is convex, we have for \z\ = r in the whole interval 0<r<l 
— 2 arcsin r<r< 2 arcsin r (4) 
except when 
'^^ 1 _ /" + ^>'z ' 
the upper and lower bounds being then attained as above. 
The proof is similar to that of the distortion theorem outlined before.^ The 
region D is approximated by rectilinear polygons, for which we have the 
formula of Schwarz 
dw 
— = (1 - e^'^'zY' (1 - e^'^'zY' ... (1 - (6) 
whence, for z = re^\ 
r sin {6 + ^j,) 
T= — iXj, arctan r + Ikiri, 
1 1 — f cos + q:^) 
(7) 
