590 Chapter^8
and
r2 log(Ma/M1) r2
logM 2 :=:; -alog - +logM1 = l ( / ) log - +logM1
ri og ra ri ri
ra r2 ra (
log - log M 2 :=:; log - log M 3 +log - log Mi 8.14-3)
~ ~ ~
or,
Mlog(r3/r1) < Mlog(r3/r2) Mlog(r2/r1)
2 - 1 3
The reader will note that equality can occur only if F(z) is a constant in
the ring, i.e., only if f(z) is of the form f(z) = cz-<>. Inequality (8.14-3)
may also be written as
logM(r 2 ) :=:; logr^3 -logr2 logM(ri) + logr2 - logri logM(ra)
log r 3 - log ri log ra - log ri
which shows that Hadamard's three-circles theorem can be expressed as a
convexity property, namely, that log M( r) is a convex function of log r.
We recall that a real function e( x) is said to be convex· (downward) over
[x1, X2] if the graph of y = e(x) between X1 and X2 always lies below the
chord joining (x1, e(x1)) and (x2, e(x2)), i.e., if
(8.14-4)
for Xi < X < X2. If the < sign is replaced by :::; , then e( X) is said to be
convex in the wide sense.
Remark The three-circles tlieorem generalizes to multiply connected
regions bounded by n simple closed curves. See Bear [3].
Exercises 8.6
- Show that Hadamard's theorem can be expressed more compactly in
the form
M 2 -..Mi->.. 1 3
where >.. = log ( ra / r 2 ) / log ( ra / r 1 ) , or, alternatively,
logM2 :5 >..logM1 + (1->..)log Ma
where logr 2 = >..logr1 + (1 - >..)logr 3 •
- Also show that the theorem can be expressed in determinant form as
log ri log Mi 1
log r2 log M2 i' ;::: 0
log ra log M 3 1
