On the shape of the level loci of harmonic measure

ON THE SHAPE OF THE LEVEL LOCI

OF H A R M O N I C MEASURE*

By

J. L. WALSH** AND W . J . SCHNEIDER***

in College Park, M'd., U.S.A. in Syracuse, N.Y., U.S.A.

In the study of the shape of level loci of Green's function of a region in

the w-plane, a measure of the global nearness of the shape of a Jordan curve F

with origin 0 interior to F to the shape of a circle with center 0, may be called

[1] the circularity of F:

(1) x ( r ) = mini wl on F

< 1. max]w] on F =

We have x(F) = 1 when and only when F is a circle with center 0.

The object of the present note is to study this and other measures of the

shape of level loci of harmonic measure, for an annular region, a region of

higher connectivity, an infinite strip, an infinite sector, and a wedge. These

results are relatively easy to prove, but elegant and striking in application.

w Circulari ty for simply and doubly connected domains. We

have (loc. cit.)

Theorem 1.1. Let D be a bounded Jordan region of the w-plane con-

taining the origin O, whose boundary is F t . Let Fx denote the image in the

w-plane of the circle c :lzl = 2 1 when the interior of c , : l z l = 1 i,

mapped onto the interior of F x so that z = 0 corresponds to w = O. Then

x(Fa) varies monotonically with 2, and approaches unity when 2 ~ O.

* Abstract published in Amer. Math. Soc. Notices, vol. 16 (1969), p. 569. ** Research sponsored by the Air Force Office of Scientific Research, Office of Aero-

space Research, United States Air Force, under AFOSR Grant No. 1130-66. *** This research was partially supported by the National Science Foundation, Grant

G.P. 8787.

441

442 J. L. WALSH AND W. J. SCHNEIDER

Let the mapping function be z = ~b(w), ~b(0) = 0, and let mx and Mx denote

respectively min[w I and maxlw I for w on Cx.~.On the curve F 1 and hence throughout D we have

(1.1) [~w_) < lml, I~-~w) -<MI '

and on Fz

(1.2) 1 - ~ 1 = <'1-rex = < --ml'l ] ~-~w)l = < Mx2 = < M I '

(1.3) 1 ~ Mx _ _ _ 1 < M 1 _ 1 m~ ~:(Fx) = ml x(r l )"

This shows x(Fa) > x(F1). There follows the monotonic increasing property

of x(Fx) as '1 decreases; for we may replace F 1 by Fa~ and Fa by Fa,0 < '1 < '11 < 1.

Equality throughout (1.3) occurs when and only when F1 is a circle with center 0.

If we set (as we may) t k ( w ) = a w + a t w 2 + . . . , a # O , we have

ck(w)/w = a + alw . . . , so m~ and Ma approach a as ,l approaches zero, whence x(Fa) ~ 1 as '1 approaches zero.

Theorem 1.1 admits an extension to a region of higher connectivity:

Theo rem 1.2. Let D be a bounded annular region of the w-plane whose

boundary consists of two Jordan curves F 1 and Fp, with Fp interior to FI

and containing the origin w = O. Suppose D is mapped one-to-one and con-

formally onto the annulus 0 < p < [z[ < 1 by z = dp(w) (the number p is

uniquely determined) and let mx and M~ denote respectively min[w I and

max] w] for w on the image Fx of I zl = '1, p <_ '1 <__ 1. hen we have for

p < ' 1 < l

(1.4) ~r >__ min[x(Fp), ~(Fx)],

and the logarithm of ~(F~) is a concave function of log2.

We apply the Hadamard Three-Circle Theorem [5, p. 126] in the annulus

p < ]z ] < 1, in the z-plane, z = ~(w). We have then (p < ' 1 < 1)

LEVEL LOCI OF HARMONIC MEASURE 443

1 o g ~ . . . . logMa ~ 1-b-g-~gpmgmp + logp -- log).logM~ '

logp

l o g ~ . . log mx > l-ff~gpmg m p + logp - - log).log ml '

logp

log x(Fx) = log mx - log Mx

> log /rmP--~" + log / rml] 1-~' log), = ~M~! \ M d ' ~ ' = ~ '

(t.5) = \Mp] \M1] "

The conclusion of Theorem 1.2 follows at once from (1.5).

An interesting special case of Theorem 1.2 is

T h e o r e m 1.3. Under the conditions of Theorem 1.2, suppose Fp is the

circle Iwl = p, namely x(Fp) = 1. Then for arbitrary 2, p < 2 -< 1, we

have x(Fp) = 1 __> x(Fa) >__ x(F1). The relation lc(F~) = x(F1) occurs when

and only when F i is a circle with center 0, necessarily the circle I wl = 1.

Of course the region D can be mapped onto art annulus if and only if the

ratio of the radii of the two bounding circles of the annulus is 1 : p.

Another theorem is somewhat similar to Theorem 1.2, as we now indicate.

T h e o r e m 1.4. Let D be a bounded annular region of the w-plane whose

boundary consists of two Jordan curves FI and Fp, with Fp interior to F t

and containing the origin. Suppose D is mapped conformally onto the annulus

0 < p < Izl < 1 by z = c~(w), and let ma and M~ denote respectively minlw [

and maxlw I for w on the image Fa of ]z[ = 2, p < ;t < 1. Suppose the

function [c~(w)/ w[ takes its maximum and minimum in the closure of D

on F1. Then for arbitrary 2, p <- 2 < 1, we have ~c(Fx) __> x(Ft).

We write here

maxt 1 1 - - = , m i n = FI ml rt Mt '

max ck-~w)l = P min[~(wW--) -- P

444 J. L. WALSH AND W. jr SCHNEIDER

The hypothesis of Theorem 1.4 then yields

1 ~ p __1 < p m 1 mp M 1 M o

whence

ral < mp •(rl ) < ~(rp), M1 = Mo' =

and the conclusion follows from Theorem 1.2.

Theorem 1.4 can also be proved directly by the method of proof of Theorem

1.1.

In connection with Theorems 1.2-1.4, it is of interest to note that generically

x(F) is unchanged by an inversion with center 0.

Theorem 1.1 deals with the shape of the level loci of Green's function for

D with pole in 0. Theorems 1.2-1.4 deal with the shape of the level loci of

the harmonic measure og(z;F~,D), or of 1 - c o ( z ; F t , D ) = co(z;Fp,D). In

Theorems 1.2-l.4 the origin w = 0 may be chosen arbitrarily interior to Fp.

w Circularity for regions of higher connectivity.

Theorems 1.2 and 1.3 extend to the case of a multiply connected region D

of the w-plane bounded by a finite number ( > 2) of mutually disjoint Jordan

curves that are divided into two disjoint sets F1 and Fp, 0 < p < 1. Let

h(w) be a function harmonic in D, continuous in the closure of D, equal to

zero and log p ( < 0) on F~ and Fp respectively, and tet k(w) be the (possibly

multiple-valued) conjugate to h(w) in D. Then z = q~(w) = exp[-h(w) + ik(w)], maps D conformally but not simply onto the annulus p < l zl < 1, and

I zl -- expEhCw)] is single valued in D. Let Fx, p < 2. < 1, denote generically

the locus I cw)l = ). in D, the image of Izl -- )., and suppose the function

I ck(w)/w] assumes on F t both its maximum and minimum values in the

closure of D. Then for arbitrary )., p < ). < t , we have x(Fa) > x(F1).

The proof is as before, and the reasoning can be reapplied (if the hypothesis

is satisfied) to Fx and Fa,, p < ). < )'1 < 1. Compare here 1-11 and Green's

functions. The function ~:(Fx) is concave in log)..


We now show by means of an example that in an annular region the cir-

cularity ~c(F~) need not always vary monotonically with )`. Let the annulus

�89 < ] z[ < 1 be mapped onto an annular region D of the w-plane bounded

by an ellipse F�89 and the unit circle FI, where the ellipse has center w = 0

and is not a circle. Invert D in the unit circle in the w-plane, so that D is trans-

formed into some region D'. The annular region D + FI + D' is the image

of the annulus �89 < ]z I < 2 under the previous map (extended) of ~ < I z ] < 1.

We have ~c(F,)= x ( F 2 ) < l , to(F1)= 1. Indeed the circularity ~:(F~),

�89 ~ ]z I < 2 considered as a function of ). has a maximum for ), = 1.

Re ma r k . There exists an annular region D in the w-plane bounded

by analytic Jordan curves F1 and Fp such that the level loci F~ of the har-

monic measure of Fp with respect to D have a maximum circularity x(Fa)

in D for which that maximum is not unity. We choose F1 and Fp as disjoint

similar ellipses having the same center 0 and same orientation of axes, yet

so that no circle with center 0 separates them. It follows from the concavity

of ~c(F~) as a function of log), that for all ),, p < 2 < 1, we have

x(F~) > x(Fl) = x(Fp), so for some )`, p < ). < 1, x(F~) is a maximum but

< 1.

w E l l i p t i c i t y . Both Theorems 1.2 and 1.3 can be generalized by a

conformal map so as to apply to an arbitrary doubly connected region D

in the w-plane, and to the shape of the corresponding level loci of the harmonic

measure of one of the bounding continua with respect to the region, compared

with the shape of that bounding continuum itself as standard. If D is the

extended plane minus two disjoint continua F1 and F2, then we map the

complement of F2 onto the exterior D' of the circle C:lz [ = 1 in the z-plane.

We then compare by the methods already developed, the shape of the curves

in the z-plane (transforms of level loci in the w-plane) with the shape of the

level loci (i.e., circles) of Green's function for D' whose pole lies at infinity.

This comparison of shapes can be interpreted in the w-plane. Rather than

formulate a general result (which is left to the reader) we formulate explicitly

an interesting special case.

Denote by S the segment - 1 < u < 1 of the w(= u + iv)-plane, and let

446 J. L. WAI,,SH AND W. J. SCHNEIDER

F be an arbitrary Jordan curve containing S in its interior. We compare the

shape of F with that of the family of ellipses Fp whose common foci are

+ 1 and - 1, and define the ellipticity of F (nearness of the shape of F to

that of the ellipses Ep) as

(3.1) E(F)

sum of semi-axes of largest ellipse Ep having no point exterior to F

sum of semi-axis smallest ellipse Ep containing F in the closure of its interior

This quantity is less than unity unless F is an ellipse of the family Ep. The

family Ep is the set of ellipses u = �89 + p-1)cos0, v = �89 - p-1)sin 0,

p > 1, and the transformation

(3.2) w = �89 + z - l )

maps the w-plane onto [z I ___ 1, where z = pe ~~ This transformation carries

the ellipse Ep just mentioned (whose foci are + 1 and - 1 and whose semi-

axes are �89 + p-a) and �89 - p-x)) into the circumference I z] = p; the sum

of the semi-axes is p.

Another form of (3.1) is (z = z(w), I z I > 1)

E ( r ) = rain[z[ for w on F

max I z [ for w o n r = x(image of F in z-plane),

a form that emphasizes the analogy (indeed the identity under the transforma-

tion (3.2)) between ellipticity in the w-plane and circularity with respect to

z = 0 in the z-plane.

If we define E(F) generically by (3.1), we have by Theorem 1.3

T h e o r e m 3.1. Let the annular region D of the w-plane be bounded

by S : - 1 ~_ w ~_ 1 and by a Jordan curve F containing S in its interior.

Then each level locus L of the harmonic measure of S with respect to D is a

Jordan curve whose ellipticity E(L) is greater than or equal to that of F:

(3.3) E(L) >= E(F);


E(L) varies monotonically with L, and approaches unity as L approaches

S . The equality sign holds in (3 .3)for L ~ I ~ when and only when F is an

ellipse of the fami ly Ep.

The limiting case of Theorem 3.1 as F b~comes infinite is itself a limiting

case of the result of inverting the configuration of Theorem 1.1.

w A n o t h e r m e a s u r e o f c i r c u l a r i t y . In considering the shape of a

Jordan curve and the nearness of the shape to that of a circle, one may empha-

size global dimensions as in Theorems 1.1, 1.2, and 1.3, or one may emphasize

local infinitesimal properties. This contrast occurs for instance in comparing

the shape of an ellipse which is smooth but whose eccentricity is large, with

the shape of a gear wheel with small and sharp but numerous teeth. The latter

shape carl be described in terms of the angle ~k now to be mentioned, first

studied in this connection by H. Grunsky [4], by P. Davis and H. Pollak [3],

and later by Walsh [2].

If F is a smooth Jordan curve in the w-plane containing the origin in its

interior, the angle ~k is defined as the angle at w measured from the radius

vector extended through w, to the directed tangent to F at w in the counter-

clockwise sense on F , and ~ is to vary continuously with w. In particular

if F, , 0 < r < 1, is the image in the w-plane of the circle I z l = r in the z-plane,

when the interior of F is mapped onto I z I < 1 so that w = 0 and z = 0 cor-

respond to each other, then ~k is harmonic in w and z , and we have

[min ~, w on Fz] ___ [min ~ , w on F,] < [max ~ , w on F,] _~ [max ~k, w on Fz].

If F1 is star-shaped with respect to 0, so also is F,. Thus [min ~k, w on 1",]

and [max ~k, w on F,] give [2] a measure of the nearness of the shape of F,

to that of a circle, a measure that varies monotonically and approaches re/2

as r decreases and approaches zero. We now extend this property to level loci

of harmonic measure.

T h e o r e m 4.1. Let D be a doubly connected region of the w-plane whose

boundary consists of two smooth Jordan curves F z and Fp, with Fp interior

to F a and w = 0 interior to Fp. Let D be mapped by w = f ( z ) , z = ~(w)


onto the annulus p < I zl < 1 so that the curve I zl = r, p < r < 1, is carried

into a curve F, . Then we have

(4.1) min[O, w on F, + Fo] __< min[0, w on F,]

=< max[0, w on F,] __< max[0, w on F 1 + Fp].

These inequalit ies are strong unless both F~ and Fp are circles with center O.

It is especially remarkable here that the position of w = 0 interior to Fp

is completely arbitrary. Of course F, is a level locus of the harmonic measure

of F~ with respect to D.

We have now z = re io on [zl = r , dz = izdO, dw = f ' ( z ) d z = iz f ' (z)dO

and for w on F,

O = argdw - arg w = arg i z f ' ( z )dO lr [, z f ' ( z ) '~ S(z) - 2 + Im t i o g - ~ ( - - ~ - ) .

The function f ( z ) is single valued and analytic in p < ] z l < 1, and f ' ( z ) is

continuous in the closure of D. Also, since z ~ O, f ( z ) ~ O, l o g ( z f ' ( z ) / f ( z ) )

is locally harmonic in D. When z traces the circle ] z ] = r , arg z increases by

2rr, argf(z) increases by 2rr, and f ' ( z ) ~ O, so a r g ( z f ' ( z ) / f ( z ) ) d o e s not

change. Thus log ( z f ' ( z ) / f ( z ) ) is single valued and continuous for p < r < 1,

harmonic for p < r < 1. The maximum of 0 occurs on the boundary of D,

as does ra in0 , so (4.1) follows. Equality holds if and only if 0 ~ n/2.

On each curve F, we have min0 < zr/2 < m a x 0 , so if F1 or Fp is a circle

whose center is 0 (but not both) we may omit that curve in the inequality

(4.1); in that case, both min0 and m a x 0 vary monotonically on F, as r

varies. As measures of 0 in the large, one can use min 0 , m a x 0 , or indeed

the difference A = max0 - min 0 .

w Inf inite strips. In this section we shall be concerned with infinite

strip domains in the w-plane of the form D = {w = u + iv I - oo < u < + oo,

0 ~ g ( u ) < v < f ( u ) } where f ( u ) and g(u) are given continuous functions.

We shall denote by co(w) the harmonic measure of {w ] - oo < u < + oo,

v = f ( u ) } with respect to the domain D. For the level curves o~(w)= 2,


0 < ;t < 1, a natural analog of the concept of circularity is the flatness of

the 2 curve which we define as

inf Im w F~ -- ~ '~=x _ I~t (S~ ~ 0).

sup Im w S~

We note that flatness depends on the position of the axis v = 0.

The following theorem concerning the flatness of the 2-curves of an infinite

strip domain is analogous to our Theorem 1.2 concerning the circularity of

the 2-curves in an annular domain.

T h e o r e m 5.1. Let D be an infinite strip domain bounded by the con-

tinuous curves v = f ( u ) and v = g(u) ( < f ( u ) ) . Further suppose that

S r ( = s u p f (u) ) and S t ( = sup g(u)) aref ini te and that I~ ( = i n f g(u)) - - C O < u < CO - - O 0 < U < O 0 - - C O < U < CO

is positive. Under these conditions if F s ~ Fg then F x > F s .

Let w = c(z) be a conformal map from the domain

/ 3 = {z = x + iy[ - o o < x < + o o , 0 < y < l }

onto the domain D which takes the upper boundary o f / 3 onto the upper

boundary of D and the lower boundary o f /3 onto the lower boundary of D.

Let h(z) be the harmonic function which solves the Dirichlet problem o n / 3

with the constant boundary values Sy on y = 1 and S t on y --- 0. Since on

y = 1 we have Imc(z) < S.r and on y = 0 we have Imc(z) _~ S t we have

by the maximum principle that Im c(z) < h(z) throughout /3 . This inequality

combined with the facts i) c(z) takes the line y = 2 in the z-plane onto the

curve to = 2 in the w-plane and ii) on the line y = 2 we have h(z) = 2S s

+ ( 1 - 2)S t , leads us to the inequality

(5.1) Sx < 2S s + (1 - ;t)S t .

By a similar minorization argument we are led to

(5.2) I ~ > 2 I ~ + ( 1 - 2 ) I ~ .


Therefore we have

F4 = I--L > 2I f + ( 1 - 2)I e S 4 2S/ + ( 1 - 2 ) S e

>

We note that the following theorem is implicit in the proof of Theorem 5.1:

T h e o r e m 5.2. Let h(z) be a function positive and harmonic on

:D = {z = x + iy [ - oo < x < Go, 0 < y < 1} and continuous on the closure

of D. Let ~ x = sup h ( x + i2), 7 4 = inf h ( x + i2) (with 14> 0) and - - O 0 < X < O0 ~ ~ * < : X <: GO

and A4 = Iz/Sz.

Under these conditions if A 1 < A0 then A 4 => A 1.

We shall have cause to mention later this important ratio Aa, and shall

refer to it as the rain-max ratio.

Theorem 5.1 is the analog of Theorem 1.2. We proceed to prove the analog

of Theorem 1.4; here w = c(z) maps D onto p < y < 1 in the z-plane with

w = _ oo corresponding to z = _ oo:

T h e o r e m 5.3. Let D satisfy the conditions of Theorem 5.1, and suppose

v(w)/y(z), where w = c(z), takes on F 1 its supremum and infimum in the

closure D of D. Then we have Fz > F 1, 0 < 2 < 1.

By hypothesis we have supr~[V(W)-Sly(z)] < O, hence throughout 13

(5.3) v ( w ) - S l y ( z ) <= O, w =c(z);

hence there follows for w on Fp

(5.4) Sp - S ip < O.

In particular, (5.3) holds also for w on Fx, p _< 2 _< 1,

(5.5) S x - S1;t -<_ O.


Likewise by hypothesis we have infrl[V(W) - Ily(z)] >= O, hence throughout/3

(5.6) v ( w ) - I l y ( z ) > 0, w = c(z);

(5.7) I x - I12 > 0.

From (5.5) and (5.7) we now deduce

I~ I1 = F f , p ~ 2 ~ 1, =

provided Fx is defined.

C o r o l l a r y 5.1. I f we have g(v) - 0 in Theorem 5.3, and D is 0 < y < 1,

then Fi >-_ Ft without any hypothesis on v(w)/y(z).

Under the new hypothesis, (5.4) is obvious for p = 0, and (5.3) holds on F1,

so (5.5) follows. Also (5.6) is valid on F t , and clearly valid on Fo, hence valid

throughout D and on Fl . Thus (5.6) holds for 0 < 2 __< 1, namely (5.7), and

the Corollary follows.

If v(w) ~ S t on F t or v(W) ~ I t on F t , then the strong inequality holds

in (5.5), (5.7), and in Fa >- F1.

Two remarks are in order.

First, there are genuine difficulties in extending Theorem 5.1 to the case

where Sg (and hence Is, ) is equal to zero. In this case since g(v) - 0 , one might

be tempted to define Fg to be equal to one since Fg would equal one for g(v)

identically equal to any other non-negative constant. Also as we will see later

(Corollary 5.3) limz~ x Fx always exists. The problem is that lima_. 1Fa need

not equal one, as the following strip domain D shows. Let D be bounded

by the curve v - 0 and the curve v = f ( u ) , where 1 __<f(u) __< 2 and f (u) non-constant. Let og(z) be the harmonic measure of {w I - o o < u < + oo, v = f(u)} relative to the domain D. By ~o o we shall mean Oog/Ov. Since J(u) is non-constant, the function co o (which is well defined by the reflection principle) is non-constant on the u-axis. This follows from the fact that if co o _-- k on the u-axis then a~ - kv (otherwise the critical points of og-kv would be isolated). Let u 1 and u2 be such that ~oo(ul) > coo(u2). From the local behavior of o9 this implies for all 2 less than some fixed 2o we have Sa _-> [3oJo(ui) +

o~o(u2)] �9 2/4 and Ix < [3o)o(u2) + o)v(ul)] " 2/4. This implies that F a is bounded away from one.


Second, if we modify the hypothesis so that D is now bounded by more

general disjoint Jordan arcs C~ and C2 parametrized in the form C~(i = 1,2)

= {w= u + i v l u =ai(t), v=fl i ( t ) , - l < t < l , ill(t) > 0 , lim,_.• )

= -t- ~} the proof goes through essentially without change.

The following three corollaries are all almost immediate consequences of

Theorem 5.1.

Corollary 5.2. Under the hypothesis of Theorem 5.1, if F g = l then Fx

is a monotone decreasing function of), which approaches unity as 22 approaches

zero.

If Fg = 1 then from Theorem 5.1 it follows that

(5.8) F s < F,t < F , .

However the proof of that theorem applies if either of the boundary curves

(upper or lower) is an interior ).-curve; hence from (5.8) we have:

FaI<Fz<Fa2 if 0 < 2 1 < 2 2 < 2 2 2 < 1 .

To state Corollary 5.3 we need a definition, namely that a continuous

function a(t) is bi-monotone increasing in (0,1) if there exists a number

c(0 < c < 1) such that ~(t) is monotone increasing on (0,c] and 7(t) is mono-

tone decreasing [c , l ) .

Corollary 5.3. Under the hypotheses of Theorem 5.1, Fx is in (0,1)

either: i) monotone or ii) bi-monotone increasing.

We first note that the corollary follows immediately if we can show that

it holds in any arbitrary closed interval [L, R] contained in (0,1).

We know by the continuity of Fx that it has an absolute maximum on

[L ,R] .

Case I: The absolute maximum occurs at the right-hand end point R. In

this case F~ must be monotone increasing. Assume not, then there exist points

221 and 2 2 with 41 < 42 < R and Fa2 < Fal. By applying Theorem 5.1 to the

interval [221,R] we obtain a contradiction.


Case I[: The absolute maximum occurs at the left-hand end-point L. In

this case Fz must be monotone decreasing. The proof is essentially the same

as in Case I.

Case I11: The absolute maximum occurs at a point 2o with L < )-o < R.

In this case Fa is monotone increasing in [L,).o] and monotone decreasing

in [20,L ] . The proof is essentially the same as in Case I.

Finally as an immediate consequence of (5.1) and (5.2) we obtain

C o r o l l a r y 5.4. Under the hypotheses of Theorem 5.1 the curve

co(w) = 2 lies entirely within the closed strip

{w = u + i v I - o o < u < + co, 2 I ; + ( 1 - 2 ) I , < v < 2 S I + ( I - 2 ) S , } .

One may also obtain in an infinite strip domain theorems analogous to

our results in ~4. A sample theorem of this sort is the following

T h e o r e m 5.4. Let D be an infinite strip domain bounded by the con-

tinuously differentiable disjoint curves v = 0 and v = f ( u ) ( > 0 ) and let

co(w) be the harmonic measure of { w [ - o o < u < + oo, v = f ( u ) } with

respect to D. For each 2, 0 < 2 < I , let T(2) equal the supremum of the

values of the angles the tangents to the curve co(w) = 2 make with the line

v = O. Under these conditions T().) is a monotone increasing fi~nction of 2.

The proof follows immediately by applying the maximum principle to

the harmonic function argc '(z) where c(z) is the conformal map considered

in Theorem 5.1.

We conclude this section with two examples.

The condition that a function be "monotone or bi-monotone increasing" is quite reminiscent of the condition that a function be convex downward.

This is, in fact, a necessary condition for a function to be convex downward.

In w we shall show that the circularity is both "monotone or bi-monotone

increasing" and convex downward as a function of 2. The following example

shows this is not the case with flatness:

Example 5.1. There exists an infinite strip domain for which F~ is

not convex with respect to 2.


Let D be the domain consisting of the union of the two sets

{w=u+ivl- <u<oo, 1 < v < 4 9 }

and

{ w = u + i v I - N < u < N , 49 < v < 5 0 } .

By making N large we can make S~r as close as we like to 51/2. Independent

of our choice of N , I~ = 25. Therefore we have Fo = 1, F~ very close to

50/51 and F t = 49/50. However for F~ to be convex downward F~ would

have to be greater than or equal to 99/100, but 50/51 < 99/100.

If one has an infinite strip domain whose lower boundary is the line v = 0

and whose upper boundary is not necessarily a single valued function of u

one might wonder whether the 2-curves, other than 2 = 0, represent single-

valued functions of u. The following example shows that in some quite general

cases very good estimates can be made.

E x a m p l e 5.2. Let D be an infinite strip domain whose lower boundary

is the line v = 0 and whose upper boundary has a continuously turning

tangent which is parallel to the u-axis at one point and which never turns

more than rr radians (plus or minus) f rom zero. Under these conditions all

the 2-curves are single-valued functions of u for 2 < 1/2.

Let c(z) be the conformal map considered in the proof of Theorem 5.1.

Now all we need to do is note that we can strictly majorize [-rninorize] the

function argc ' (z) by the harmonic function h ( x , y ) = zryEh(x,y)= - n y ] .

Hence for 2 = 1/2 we have -7z/2 < argc ' (x + i2 )< 1r/2, which implies for

these values of 2 that the image of y = 2 under c(z) is single-valued.

~6. Wedge domains. In this section we shall consider domains in

the first quadrant of the W-plane ( W = U + iV) which are bounded by

two Jordan arcs J and K joining the origin to the point at infinity

with J c { W ] ~ 1 < a r g W < 0 t 2 } and K c { W I 0 t 3 < a r g W _ ~ 4 } where

0 < ~ < ct 2 < ~a < ~4 ~ re/2. Such domains will be called wedge domains.

By means of the conformal map w = log W we may interpret a number

of the results in the w-plane of w as theorems for wedge domains.


From Corollary 5.4 we obtain the following

Theorem 6.1. Let D be a wedge domain with notation as introduced

above. Also let o~(W) be the harmonic measure of K with respect to the do-

main D. Under these conditions the curve {WIco(W ) = 2) must lie in the

sector

{W[(I - 2)et + 2es N a r g W < ( l - 2)~2 + 2e4}.

A theorem closely related to Theorem 6.1 is the following theorem which

extends the Carath6odory Theorem on the behavior of a conformal map at

a corner [7, p. 104]:

Theorem 6.2. Let T = c(W) map the open upper half W-plane con-

formally onto a Jordan domain D o in the T-plane which has the origin as

a boundary point. Also suppose that e(0) = 0 under the extension of c(W)

to a continuous map on the closed upper half plane. In addition let there

exist some number Uo(> 0) such that the extension of c(W) takes the interval

{ W = U + i V [ O < U < U o, V=0} into {TIe , <:a rgT<~2} and the

interval { W = U + iV I - U o < i <= O, V=0} into {Z[e3_-<argT<e ,} .

Under these conditions the image of arg W = 2/2 is eventually (in some

neighborhood of T = O) contained in the sector

{r l ( l -2 /2n)a~ +(212~)a2 < a r g T < (l-A/2g)~3+(A/2~)a,}.

The argument follows the proof of Theorem 6.1; we note that the local

behavior of c(U + iV) in the neighborhood of the origin depends only on the

local behavior of the image of [ - U o , Uo] under T = c(U). (This follows

from the fact that our method reduces the problem to one concerning harmonic

functions and their local behavior).

For 2-1evel curves in a wedge the natural analog of circularity is angular

flatness which we define as

inf arg W A~ = ,~(w)=z _ I~ (Sz # 0).

sup arg W Sz

456 J .L . WALSlt AND W. J. SCHNEIDER

Angular flatness can be easily studied by the methods of w

w A n o t h e r f o r m o f m i n - m a x d e v i a t i o n . In previous sections we

have considered min-max deviations related to the geometry of the level

curves of harmonic measures. In this section we shall consider a rain-max

deviation for certain functions on the level curves of harmonic measures

of two particularly simple domains. More general cases will be considered

in the next section.

I f h(Z) is a bounded harmonic function in the strip {Z = X + i Y [ - ~ < X

< ~ , C1 < Y < Cz} we define the strip max-min difference to be

a r = sup h ( X + i ~ ) - inf h ( X + i y ) . - - o 0 < X < oo - o o < X < oo

I f k(Z)is a bounded harmonic function in the annulus {Z[R, < Iz[ < R2}

we define the max-rain difference to be

t t p = max k ( Z ) - min k(Z). [z I =ep Iz I =eP

The functions a~ and/ tp are convex functions of ), and p respectively. This

follows, in each case, by applying the two-constant theorem [-5, p. 126] to

each term in the difference.

As an application of the fact that/zp is convex we prove the following

T h e o r e m 7.1. Let A be an annular domain in the S-plane (S = P + iQ)

with inner boundary A 1 and outer boundary A2. Let the origin in the S-plane

be in the interior of the curve A 1. Denote by S = f (Z ) the conformal map

from {zll<lz I < Ro} onto A and by x(r) the circularity of the image

of the circle I zl--r under the map. Let generically z ( p ) = 1/x(e o)

(0 < p < logRo). Under these conditions ~(p) is a convex function of p.

Let k(Z)---loglr(z) I. Since f ( Z ) ~ O, k(Z) is harmonic and hence

max loglf<z)l- rain loglf</)l is a convex function of p. Elementary izl =ea /Zl =ep properties of the logarithm lead us to


max Iog l f (Z ) I - min l o g l f ( z ) [ IZl =e. IZl =e,

= log[ max [f(z)l]- og E rain If(z)t3 IZ I =ep IZl =ep

= l O g k ~ n If(Z) [ ] izl =ep

= log(l/r.(eP)) = logz(p) .

Therefore r(p) = e ~(p), where c(p) is a convex function of p. Since the ex-

ponential function is monotone and convex, it follows that r(p) is convex.

Let to b: the harmonic measure of A t with respect to A. Theorem 7.1 has

an immediate corollary concerning the circularity of the level curves o9 = 2

as a function of 2:

Corol lary 7.1. Let A and oJ be defined as above. Let if(2) be the cir-

cularity of the level curve to(S) = 2. Under these conditions the function

~_1(2) = 1/~(A) is a convex function of) . .

We know the r-circle in the Z-plane corresponds to the level curve

og(S) = log r/logR o. Therefore ~().) = x(e zt~176 and ~_1(2) = z(~.logRo).

Since convex functions remain such under a linear change of variable, it fol-

lows that ~(;t) is a convex function of 2.

w Further general izat ions . In this last section we shall consider

some generalizations of our previous work, first in the direction of theorems

on more general plane domains and second in the direction of theorems in

higher dimensions.

We will now outline how one can generalize Theorem 5.2 to more general

plane domains. It will then be clear, in principle, how to extend much of our

other work in this direction.

Theorem 8.1. Let G be a domain in the z-plane which is bounded by

finitely many mutually disjoint Jordan curves and let A consist of finitely

many subarcs of aG. Let to(z) be the harmonic measure of A with respect

to the domain G and let h(z) be a function positive and harmonic on G and

continuous on the closure of G. In addition let


Sx = sup h(z), 7a = inf h(z)

and rain-max ratio Ax = 714/S, ~. Under these conditions if A 1 ~ Ao then

A~ >= A 1 .

To start the proof of the theorem the following topological lemma is needed:

L e m m a 8,1. Under the hypotheses of Theorem 8.1 the set (z [ ~ < eg(z) <fl}

consists of the union of finitely many mutually disjoint domains, each of

which is bounded by the union of finitely many Jordan arcs or curves on

which co(z) = ~t or o~(z) = fl [except possibly for end points].

For domains bounded by analytic arcs the lemma follows from the fact

that the closure of G is compact and from applications of the theory of the

local behavior of the level curves of harmonic measures [6, w To prove

the lemma for domains bounded by arbitrary Jordan curves one uses the

fact that such domains are always conformally equivalent to ones bounded

by analytic Jordan curves.

The proof now follows by applying, together with the two-constant theorem

[5, p. 125], the methods of the proof of Theorem 5.1 to each of the domains

in {z < <

As an indication of how a number of our theorems can be generalized to

3-dimensions we conclude by proving a 3-dimensional analog of Theorem 1.1.

Theorem 8.2. Let S be a bounded, 3-dimensional, simply connected

domain containing the origin. Further suppose there exists a Green's func-

tion G(p) for S relative to the origin. Let to0. ) be the circularity relative to

the origin of the ).-level surface of G(p). Then to().) is a monotone decreasing

function of ) . ( - ~ < ). < 0).

By a Green's function for S relative to the origin we mean a function with

the following properties: i) G(p) + 1/![ p [l is harmonic throughout S (where

i[ P [] denotes the distance from p to the origin), ii) limp_,asG(p)=O. (In 3-di-

mensions some authors consider the Green's function to be the negative of

our Green's function.)

Let M4 be the maximum distance and m4 be the minimum distance from a

surface Sz: G(p) = 2 to the origin. The circularity of Sz is defined as x().)

= m~/Mx. We now consider the two functions


~n(p) = C(p) + [1/[[ p ![ - ~mo] , (8.1)

-1 tK(p) G(p) + [1/11 p H llmo] �9

Both H(p) and K(p) are harmonic in S, hence by the maximum and minimum

principles we have

(8.2) H(p) > 0 , K(p) <=0, p e S .

The equations (8.1) and (8.2) lead us to the following inequalities for p on

the ),-level surface:

it + 18pl I - 1/Mo >__ o,

it + 1/[1 P II- l/too _ o.

This implies in particular that

it + l i M a - 1/Mo > O, (8.3) { it + 1/rn z - 1/m o < O.

Rewriting (8.3) we obtain

This implies that

1/Ma > 1/Mo - it,

1/mz < 1 / m o - it,

x ( i t ) - ma > m o ( l - M o 2 ) Mx = M o(l - moit)"

Since it is negative we have ~:(2) > ml/M 1 (i.e. x(it) > x(0)). The same argu-

ment applied to any it~-surface instead of 8S leads to the inequality

x(it2) > x(itl) if it2 -~ itt. This completes the proof.

460 J. L. W A I ~ H AND W. J. SCHNEIDER

REFERENCES

1. J. L. Walsh, On the shape of level curves of Green's function, Amer. Math. Monthly, 44 (1937), 202-213.

2. J. L. Walsh, Note on the shape of level curves of Green's function, Amer. Math. Monthly, 60 (1953), 671-674.

3. P. Davis and H. Pollak, On the zeros of total sets of polynomials, Trans. Amer. Math. Soc., 72 (1952), 82-103.

4. H. Grunsky, Zwei Bemerkungen zur konformen Abbildung, Jahresber. d.d. Math. Vereinigung, 43 (1933), 140-143.

5. L. Bieberbach, Lehrbuch der Funktionentheorie II, CLeipzig, 1931). 6. J. L. Walsh, Interpolation and Approximation, Amer. Math. Soc. Coll., 20 (1935),

Providence, R.I. 7. C. Carath6odory, Theory of Functions of a Complex Variable II, Chelsea, New York

City, 1954.

UNIVERSITY OF MARYLAND COLLEGE PARK, MD., U.S.A. AND

SYRACUSE UNIVERSITY SYRACUSE, N. Y., U.S.A.

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On the shape of the level loci of harmonic measure