On the Location of the Critical Points of Harmonic MeasureAuthor(s): J. L. WalshSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 33, No. 1 (Jan. 15, 1947), pp. 18-20Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/87461 .

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18 MA THE MA TICS: J. L. WALSH PROC. N. A. S.

ON THE LOCATION OF THE CRITICAL POINTS OF HARMONIC MEASURE

BY J. L. WALSI

DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITrY

Communicated November 30, 1946

The object of this note is the proof of THEOREM 1. Let C be the unit circle in the z-plane, and let closed arcs

ak (k = 1,2, . . , n) of C be mutually disjoint. Denote by z = a and z =

bk the initial and terminal points of akc, with notation so chosen that the points ak and bk are ordered positively. (i.e., counter-clockwise) on C as a,, bl, a2, b2,

a... , a, = a+,+ , ba = b&+,. Denote by u(z) the harmonic measure with

respect to the interior of C of the set a = ai + a2 + ... + an at a point z interior to C; thus u(z) is harmonic and bounded interior to C, and approaches unity as z approaches an interior point of an arc ak, and approaches zero as z approaches an interior point of the arcs complementary to the set a. Then no critical points of u(z) lie in the region Rk [or Sk] interior to C bounded by the arc A ,: akbka- 1 [or Bk:bkak lbk+ 1] of C and the arc A ':aka+ 1 [or Bk': bkb+,l] of a circle orthogonal to C. No critical point of u(z) lies on A' [or Bk'] unless n = 2, in which case the unique critical point of u(z) lies at the intersection of Al' and B1'.

Thus a point z interior to C cannot be a critical point of u(z) if a circle orthogonal to C separates z and one of the points a,, bk from all the other points aj, bj.

In the case n > 2, Theorem 1 enables us readily to construct a polygon of 2n sides interior to C bounded by arcs of non-euclidean straight lines which contains in its interior all the n- I critical points of u(z) interior to C.

It is to be expected that the arcs Ak' and B,' should play similar roles in Theorem 1, for the harmonic measure with respect to the interior of C of the complement of the set a in the point z is 1 - u(z), which has the same critical points as u(z). In the proof of Theorem 1, we restrict our- selves to the case of Rk and the arcs Ak and Ak', as is sufficient.

The harmonic measure of the arc ak with respect to C in the point z is readily computed to be

-[arg(z - bk) - arg(z - a) - (I/2)ak], (1) 7r

where the arguments are suitably chosen, and where a, represents the

angular measure of the arc cak. The function (1) is the real part of the

analytic function

l log(- b) -log(z - a) - (2) 7bL 2 J

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VOL. 33, 1947 MATHEMATICS: J. L. WALSH 19

The critical points of u(z) are then the zeros of the function q'(z), where <p(z) is the sum of the functions (2) for all k; we have

f()= 1 Ib-z )-a (3)

r_ k= z-

b

-- akJ

In (3) we take the conjugate of each term. The quantity 1/(2 - bk) rep- resents a vector of magnitude 1/[ z - b, I whose direction and sense are those of the vector (z - bJ). It follows that the zeros of 9'(z) are the positions of equilibrium in thefield offorce due to unit positive particles situated at each point b,, and unit negative particles situated at each point ak, where each positive particle repels and each negative particle attracts, with a force equal to the inverse distance.

The total force at the center 0 of C can be interpreted in magnitude, direction and sense as the sum of all the vectors bkO and Oak, or as the sum of all the vectors bkak. Moreover, in any conformal map of the interior of C onto itself the function u(z) is invariant, as are the critical points of u(z), so we proceed to study those critical points by transforming C into itself so that the point to be studied is transformed into 0.

Let zo denote an arbitrary point interior to Rk or on the arc Ak,; we transform zo into 0, 1/fo into the point at infinity, and b- into the point z = 1, by a linear transformation of z; we retain the original notation. The point z0 = 0 lies in R, or on Ak' so the arc Ak has angular measure at least 7r, and has angular measure 7r if and only if zo lies on Ak'. The positive arc (a +l, -1) is not greater than the positive arc (ak, +1).

If zo = 0 is a position of equilibrium in the field of force previously con- sidered, the algebraic sum of the vectors b-aj must vanish. However, the point ak+l cannot lie interior to the positive arc (bk, -ak), and the points bk+ , ak+2, . .., n, al, . . ., bk-1 lie on the positive arc ak+lak, from which it follows that the sum of the horizontal components in the positive horizontal direction of the vectors bk+lak+1, bk+2ak+2, ..., b,an , bal ... b,-lak-- is

algebraically less than the magnitude of the horizontal component of the vector bkak (which is negative), except in the special case n = 2, a, =, -a2, b = -b2. Thus the total force at z0 = 0 cannot be zero, and zo = 0 cannot be a critical point of u(z) except in this special case, and the theorem is established.

Theorem 1 is remarkable in the fact that for n > 2 it exhibits a polygonal region containing all critical points, and yet a degenerate case of the region is the unique critical point in a non-trivial situation (n = 2).

Theorem 1 can be interpreted as referring to the zeros of the derivative (or logarithmic derivative) of an arbitrary rational function

k=1 z - ak

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20 MA THEMA TICS: KASNER AND DE CICCO PROC. N. A. S.

whose poles and zeros are all simple and finite, and which alternate (i.e., are interlaced) on an arbitrary circle C of the extended plane. From this standpoint, the finite zeros of the derivative lie in two curvilinear polygons; the polygons lie one in each of the two regions into which C separates the plane; polygons and finite zeros are symmetric in C. The point at infinity is also a zero of the derivative.

Theorem 1 can obviously be extended by conformal mapping: THEOREM 2. Let C be an arbitrary Jordan curve, and let closed arcs

cak (k = 1, 2, ..., n) of C be mutually disjoint. We denote by a, and bk the initial and terminal points of ak, with notation so chosen that the points ak and b, are ordered counter-clockwise on C as al, bl, a2, b2, ..., a,, b,, a =

an+ , bi = bn+ . Denote by u(z) the harmonic measure with respect to the interior of C of the set al + 0a2 + ... + n,, at a point z interior to C. Then no criti- cal points of u(z) lie in the region Rk [or Sk] bounded by the arc Ak: akbkak,l [or Bk:bka,+lb,+ ] of C and the arc A': aakak+i [or Bk':bkbk+ ] of a non- euclidean straight line for the interior of C. No critical point of u(z) lies on Ak' unless n = 2, in which case the unique critical point of u(z) lies at the intersection of A,' and Bi'.

It may be noted that in Theorems 1 and 2 the arc Ak' [or Bk'] is the locus of points z at which the harmonic measure of the arc Ak [or Bk] of C with respect to the interior of C has the value one-half. Thus the conclusion of Theorems 1 and 2 may be expressed by asserting that at a critical point z of u(z) interior to C, the harmonic measure of every arc Ak [or Bk] with respect to the interior of C has a value less than one-half, except in the case n = 2, when this value equals one-half.

Theorem 1 extends to the case of an infinite number of arcs ak and as thus extended applies in the study of critical points of harmonic measures in multiply connected regions, by a conformal map of their universal

covering surfaces.

THEORY OF HARMONIC TRANSFORMATIONS*

BY EDWARD KASNER AND JOHN DE CICCO

DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY, NEW YORK

Communicated November 14, 1946

1. Consider a transformation T of the plane

X = (x, y), Y = (x, y), (1)

with the Jacobian J = -,V - 4,x 4 0, such that the components satisfy the Laplace equation

20 MA THEMA TICS: KASNER AND DE CICCO PROC. N. A. S.

whose poles and zeros are all simple and finite, and which alternate (i.e., are interlaced) on an arbitrary circle C of the extended plane. From this standpoint, the finite zeros of the derivative lie in two curvilinear polygons; the polygons lie one in each of the two regions into which C separates the plane; polygons and finite zeros are symmetric in C. The point at infinity is also a zero of the derivative.

Theorem 1 can obviously be extended by conformal mapping: THEOREM 2. Let C be an arbitrary Jordan curve, and let closed arcs

cak (k = 1, 2, ..., n) of C be mutually disjoint. We denote by a, and bk the initial and terminal points of ak, with notation so chosen that the points ak and b, are ordered counter-clockwise on C as al, bl, a2, b2, ..., a,, b,, a =

an+ , bi = bn+ . Denote by u(z) the harmonic measure with respect to the interior of C of the set al + 0a2 + ... + n,, at a point z interior to C. Then no criti- cal points of u(z) lie in the region Rk [or Sk] bounded by the arc Ak: akbkak,l [or Bk:bka,+lb,+ ] of C and the arc A': aakak+i [or Bk':bkbk+ ] of a non- euclidean straight line for the interior of C. No critical point of u(z) lies on Ak' unless n = 2, in which case the unique critical point of u(z) lies at the intersection of A,' and Bi'.

It may be noted that in Theorems 1 and 2 the arc Ak' [or Bk'] is the locus of points z at which the harmonic measure of the arc Ak [or Bk] of C with respect to the interior of C has the value one-half. Thus the conclusion of Theorems 1 and 2 may be expressed by asserting that at a critical point z of u(z) interior to C, the harmonic measure of every arc Ak [or Bk] with respect to the interior of C has a value less than one-half, except in the case n = 2, when this value equals one-half.

Theorem 1 extends to the case of an infinite number of arcs ak and as thus extended applies in the study of critical points of harmonic measures in multiply connected regions, by a conformal map of their universal

covering surfaces.

THEORY OF HARMONIC TRANSFORMATIONS*

BY EDWARD KASNER AND JOHN DE CICCO

DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY, NEW YORK

Communicated November 14, 1946

1. Consider a transformation T of the plane

X = (x, y), Y = (x, y), (1)

with the Jacobian J = -,V - 4,x 4 0, such that the components satisfy the Laplace equation

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