PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 61, Number
1, November
1976
DEPENDENCE ON DIMENSION OF A CONSTANT
RELATED TO THE GROTZSCH RING
GLEN D. ANDERSON
Abstract.
For the constant A„ = lima_>0(mod Ron(a) + log a) associated
with the Grotzsch extremal ring RGn in euclidean n-space, we obtain the limit
linW^J7" = e1. Definitions and notation. By a ring R is meant a domain in euclidean nspace 7?" whose complement consists of two components C0 and Cx, where C0
is bounded. We let B0 = 9C0 and Bx = dCx be the boundary components of
7?. The conformal capacity (cf. [8]) of 7? is
cap 7? = inf j
\V<p\"du,
where V denotes the gradient, and where the infimum is taken over all realvalued C1 functions tp in R with boundary values 0 on B0 and 1 on Tit. Then
the modulus of the ring 7? is defined by
mod 7? = (on_x/cap R)l/{n~l),
where for each positive integer p we let a. denote the p-dimensional
of the unit sphere Sp = {{xx,... ,xp+x): 2/=i ■*; = 0- Then
measure
op = 2vA+1)/2r((p + i)/2)-'
(cf. [7], [9]), where T denotes the classical Gamma function;
relation
(!)
moreover, the
jfWt* -^
holds for each positive integer p.
2. Background of problem and statement of result. Let RGn{a) denote the ndimensional Grotzsch ring, that is, the ring whose complementary components
are
C0 = {{xx,.. .,xn): 0 < xx < a,Xj■= 0,2 < / < n)
and
Received by the editors June 2, 1975.
AMS (MOS) subject classifications (1970). Primary 30A60.
Copyright
77
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© 1977,
American
Mathematical
Society
78
G. D. ANDERSON
C, =\(xx,...,xn):
Jx?
> lj.
In [5]Gehring proved that mod AG3(a) + log a is monotone decreasing in the
interval 0 < a < 1 and he obtained bounds 4 < A3 < 12.4 • • •, where log A3
= lima_0(mod AG3(a) + log a). Using analogous methods in higher dimensions, Caraman [3] and Ikoma [7] have shown that the limit
log A„ = lim (mod AG (a) + log a)
a—>U
exists for each n > 3. Bounds for Xn have been obtained by Gehring [5], [6],
Caraman [3], Ikoma [7], and Anderson [2] (cf. [1]).
In the present paper we determine the order of growth of Xn for large n by
establishing the following result.
Theorem.
limn_>00An'" = e, where e is the base of Naperian
3. Proof of Theorem. First, by the work of Caraman
logarithms.
[3] and Ikoma [7] we
know that
p»
'-T<'.=r[(^f2V<"",,-]T-
Under the change of variable t = (r2 + l)/(r2
reduces to
fco t(n-2)/(n-l)
(3)
In=)x
Making
rewrite
exactly
easy to
- 1) the integral /„ in (2)
_ ,
-;y—i-dt.
use of the fact that (t2 - l)"1 = t~2 + t~2(t2 - 1)_1, one may
/„ as the sum of two integrals. The first of these is easy to evaluate
as n - 2, and by means of the Monotone Convergence Theorem it is
see that the second increases to the limit 1 - log 2. This procedure,
together with (2), yields the result
(4)
log ^ - n < lim (/„-«)
'r
n->oo
= -1 - log 2,
from which the upper bound \XJ" < e is easily derived (cf. [4, Lecture 9,
Theorem 1]).
On the other hand, it was determined in [2] that
(5)
iog h > £°
[(^/;/2
(sec2* + cschM(2-">/2^)1/(1~n) - l]dv.
By making the change of variable t = coth v in (5) and combining
resulting inequality with (1), (2), and (3), we achieve
(6)
where
0 > log^ - ln> f™ \(t)dt,
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the
DIMENSION OF A CONSTANT RELATED TO THE GROTZSCH RING
m
,/.
f(«-2)/(«-i)r/2a„_2
/-/2/t
tan2u\(2-"V2
\x«x-*
We consider first the integral of $„(/) over the finite interval (I,M).
using (1), it is convenient to rewrite (7) in the form
m «frt '(-2)/|-"r/
(8, *.W = ^yTT-[(||(l
79
1
First,
llcwlL-2
\(-W(->
-1
+ i_2tan2i(r,/2|u)
-ij.
where ||/|^, denotes the Lp[0,tr/2]-norm of /. Since the functions cos u and
(1 + t~2tan2u)~ ' are continuous on [0, vt/2], a well-known result in the
theory of Lp spaces gives
(9) hm _ll°°"4-2
™
=_^axp^^lcosul
||(1 + ?-2tan2M)-i/2)|n2
max0<u<V2|l
+ r2tan2M|-l/2
From (8) and (9) it follows that for each fixed t E {l,M),
(10)
v
lim <D„(f)= 0.
'
n—»oo
Next, in view of (8) it is easy to see that
0 > <S>n(t)> ,(»-2)/(«-l)(,(2-»)/(»-l)
_ ,)/(,2
This and (10) allow us to invoke the Dominated
_ jj > _1/(1 + ^
Convergence Theorem to
conclude that
rM
(11)
lim /
n—>oo-H
*_(/)<&= 0
for each fixed M E (1, oo).
Next, using the fact that t2/(t2 - 1) is decreasing
4>„(r) < 0, we may write
Jm "w
^m2-iLV
II1IU2/
for t > 1 and that
Jjm
Thus
lf°°,WW^
M2
lx
l\„1/(l-'<)r//2\1/("~1)||
and the same argument as that leading to (10) shows that
1 i"°0
lim
inf -nJM/
n-»x
$„{t)dt
> 0.
"w
But because of (11), (6), and (4), this implies the statement
(12)
liminf - log A_ > 1.
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\\(n-2)l(n-\)
,~|
80
G. D. ANDERSON
Finally, the Theorem follows from (2) and (12).
References
1.
G. D. Anderson, Symmetrization and extremal rings in space, Ann. Acad. Sci. Fenn. Ser. Al
No. 438 (1969), 1-24. MR 41 #465.
2.
-,
Extremal rings in n-space for fixed and varying n, Ann. Acad. Sci. Fenn. Ser. Al No.
575 (1974),1-21.
3.
P. Caraman,
On the equivalence of the definitions of the n-dimensional
quasiconformal
homeomorphisms
(QCfH), Rev. Roumaine Math. Pures Appl. 12 (1967),889-943. MR 37 #409.
4.
F. W. Gehring, Lectures on quasiconformal mappings, Institut Mittag-Leffler,
lished).
5. -,
24 #A2677.
6.
-,
1972 (unpub-
Symmetrization of rings in space, Trans. Amer. Math. Soc. 101 (1961), 499-519. MR
Inequalities
for condensers,
hyperbolic capacity,
and extremal
lengths, Michigan
Math. J. 18 (1971), 1-20. MR 44 #2915.
7.
K. Ikoma, An estimate for the modulus of the Grotzsch ring in n-space, Bull. Yamagata
Univ.
Natur. Sci. 6 (1967),no. 4, 395^100.MR 43 #508.
8. C. Loewner, On the conformal capacity in space, J. Math. Mech. 8 (1959), 411-414. MR 21
#3538.
9.
J. Vaisala, Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math., vol.
229, Springer-Verlag, Berlin, 1971.
Department of Mathematics,
Michigan State University, East Lansing, Michigan 48824
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