Voiv. 6, 1920 
MATHEMATICS: T. H. GRONWALL 
301 
ThKOREJm : When the analytic function 
w = z ae'^'z^ + a^z^ + . . . + anz"" + . . . 
(where aLO and y real) maps the circle \z\<l on the interior of a simple 
region D in the w-plane, then a ^2, and we have, for \z\ = r and 0<r<l ^ 
1 — 2f cos B -\- r^ 
1 - r' 
(1 + ar + f2)2 
< 
dw 
dz 
< 
[for a < 1 and r.^r{a) ], 
\ 1 + 2(a - l)f + 
I (1 -rY{\ + r) 
i [for a< \ and r^r{a), and for l^a52], 
^[(r^) ~0'^4(l-a)[^~(H^) ] 
1 + ar 
-<|w|< { 
4(1 + 
[for a<l], 
2 — a , 1 + f , a 
log ; + 
2 (1 - r)' 
4 ~ 1 
[for I5a<2], 
except that these bounds are reached for the functions w obtained upon re- 
placing r by ze'^^ and - ze^^ in the upper and lower bounds, respectively. When 
the region D is convex, then a^l, and we have 
1 
1 + 2ar + f 2 
< 
< 
1 
arctan 
V: 
dw 
dz 
a'\r 1 
(1 +r) 
1 + a 
Vi - a'^ 
[for a<l], 
r 
1 + ar 
\ <\w\< 
(1 +r)'- 
1 , 1 + r 
- log 
2 1 - r 
[for a = 0], 
1 
2a I 
[for 0<a<l 
1 - r 
[for a = 1], 
except for the functions w obtained as above. 
As an application, let us suppose that the function 
W = 1/z CiZ + . . CnZ^ + . . . 
(without constant term) maps 1^1 < 1 on a simple region in the w-plane 
(this region containing, of course, w = ^ m its interior). Inequalities 
of the form \w\ <k/r for 1^1 = r, 0<r<\, k = constant, were obtained by 
Fricke^ and the writer.^ From the assumption, it follows that 
W = \/w = z a^z' + . . . + ay + ... 
maps 1^1 < 1 on a simple region in the Vl^-plane, and the application of our 
theorem with a = 0 gives 
r I , r 
rx~2<^^1<^^ — '2' 
1 + 1 I 1 — f ^ 
or 
