A NOTE ON THE NOSHIRO-WARSCHAWSKI THEOREM* By A . W . GOODMAN in Tampa, Florida, U.S.A. 1. I n t r o d u c t i o n .

We recall the well-known theorem due to Noshiro [6]

and Warschawski [8] independently. T h e o r e m A.

Let f ( z ) be regular in a convex domain D. I f Ref'(z) > 0

in D, then f ( z ) is univalent in D. It is convenient to replace the condition R e f ' ( z ) by the equivalent condition l arg f'(z) I < n/2, and to search for a suitable generalization of Theorem A. Indeed, suppose that (1)

Ia r g f ' ( z ) [ < B

for z in D. Can we find an upper bound for the valence o f f ( z ) in D, as a function of the upper bound B? The answer to this natural question is no. In fact we will prove T h e o r e m 1.

Let e > 0 be given, and let p be an arbitrary positive

integer. Then there is a function f ( z ) , regular in the unit circle E, such that

(2)

[argf'(z)[ = 2 + 8

and f ( z ) is p-valent in E. A similar, but less precise result holds for an arbitrary convex domain. 2. The e x a m p l e f u n c t i o n .

The function in question is one that maps

E onto a surface S composed of a square R, and p triangles T k ( k = 1, 2, ..., p). * This research was supported by the National Science Foundation Grant GP-18558. 401

402

A.W. GOODMAN

Let R be the square in the upper half w-plane with vertices at _+ A, and

+_A +2Ai, where A is a large positive number. We select p intervals Ik, k = 1,2,--',p on the u-axis (in the w-plane) so that they all lie in the open interval (0,1), and such that their closures are pairwise disjoint. Let Pa~ and

P3k+2be

the end points Oflk, and let uj be the coordinate of Py. To be

specific we set U3k = ( 2 k - 1 ) Q and u3t,+2 = 2kQ, (k = 1,2,..-,p) where

O < Q < l/2p. We complete the construction of S by attaching to R, along each interval lk, a triangle Tk with vertices P3k, P3k+l, P3k+2 such that each Tk contains in its interior the point w = - M i , where M > 0. In the plane these triangles will have common points but in our construction each triangle lies in a separate sheet so that S is a simply-connected surface that has no branch points. Consequently there is an f(z), regular in E , that maps E onto S. Further f(z) is p-valent in E , and f'(z) ~ 0 in E . By selecting the p triangles so that they are very small and thin, and at the same time using a suitable normalization for f(z), we can show that argf'(z) satisfies the inequality (2). We now give the details. Let ilk, 7k, and 6k be the interior angles of Tk at the points

P3k, P3k+l, and

Pak+2 respectively. We select 6k SO that 0 < 6k < 7r/2 and M tan 6k > ~g~U----~'

(3) Similarly we select

(4)

flk

SO

k = 1, 2,..., p .

that re/2 < flk < ZCand

(2k M - 1) Q > [ tan

flk I > tan 6k,

k = 1, 2, ..., p "

The second inequality in (4) assures us that the two sides of the triangle will meet below the u-axis. The inequalities (3) and (4) together assure us that the point - M i lies in the triangle Tk. If we express the angles of Tk as multiples of n we find that

(5)

/ ~ = (89 + ~/k)7c,

r~ = (ek - r/k)rc,

6~ = (89 - ~k)n,

where ek > qk > 0, (k = 1 , 2 , . " , p ) . Further by selecting M and Q so that M/Q is sufficiently large we can make ek and qk arbitrarily small.

A NOTE ON THE NOSHIRO-WARSCHAWSKI THEOREM

403

We now use the Sehwarz-Christoffel transformation to give a formula for f ( z ) . As usually presented in textbooks [51, this transformation is applied to plane domains. However, both Schwarz I7"1 and Christoffel [1] mention that their transformation is also valid for multi-sheeted polygons. More details on this extension and some examples can be found in [2, 3]. Thus if the parameters are selected properly the function that maps E onto S taking the origin into w = A i , with f'(0) > 0 is

f ( z ) = C f 3 p + 3I I (1 - ~.k()~"d( + A i .

(6)

0

km0

Here Zk = e i~ is the preimage of the vertex Pk of S, and the notation is such that 0 __< 0o < 0~ < -.. < 03p+3 < 2zr. The exponents ek are real numbers that are related to the angle at the vertex Pk in a well-known manner. The points P0, P1, P2, and Pap+3 are the vertices of the square with coordinates and A respectively.

A + 2Ai, - A + 2 A i , - A ,

It is convenient to decompose the integrand into a product of terms

(7)

to ~'~ H (1--Zk~) -1/2'

dl'~ {0,1,2,3p+3}

keJ

for the square, and p terms of the form (8)

tk --=

(1 -- Z3k~)l/2+t/~:(1 -- Z3k+2~) 1/2-ek (1 -- Z3k + 1~) 1 - ek+r/k

one for each triangle Tk, (k = 1,2,...,p). We select that branch of argf'(z) for which argf'(0) = 0, and make the same determination for each term arg(1 - 5kz). Then C > 0, and p

(9)

argf'(z) =

Y~argt k k=O

where (10)

argto = - 8 9 Y~ arg(1 - e-~~ keJ

and for k = 1,2,...,p

404 (11)

A.W. GOODMAN

argtk = (89 + qk)arg( 1 -- e-~~

+ (3 -- eg)arg(1 -- e-i~

- (1 - ek + r/k)arg(1 -- e -i~

'z).

Now a r g f ' ( z ) is harmonic in E and, as we will soon see, it is continuous in/? except for a finite number of jump discontinuities on the boundary. Hence it is sufficient to examine argf'(e~~

(12)

arg(1

ei(O-O~))

For this purpose we need the formula

~ 89

if 0 - < 0 < 0 1 ,

(3(0--0 i-Tr),

if O j < O < 2 z c .

We first consider argtk(e i~ for k > 0. In equation (11) the sum of the coefficients is zero. Hence when we use (12) in (11) we find that argt k is constant on each interval that does not contain 03k, Oak+ 1, or 03k+2. With a little labor we find that if 0 < 0 < 03k, (k = 1, 2, ...,p) then

(13)

arg tk(e i~ = (3 + rlk)

03k+ 1 - 03k 2

(3 - ek)

Oak § 2 -- Oat + 1 2 -- dpk"

We note in passing that ~bk can be made arbitrarily small if 03k+2 is sufficiently close to Oat. For the remaining intervals we find that

(14)

arg tk(e ~~ =

(

~ + (~ - ~)~

if

03k < 0 < 03k+l ,

(~k "]- (5 -- ~'k)7~, if Oak+l < 0 < 03k+2 , ~bk,

if 03k+2 <( 0 <~ 2n.

Consequently with a suitable choice of the parameters, [arg t k l < ~Z/2+ ~k*. Further if j ~ k , then argtj is small whenever arg tk is near + re/2. Hence (with suitable selection of ek*)

(15)

[Pk~larg_ tk I < ~ + k=l•" e*.

Next we apply (12) to the terms in t o . Here it is worth noting from (12) that each term in (10) is linear in 0 so that the sum is piecewise linear with

405

A NOTE ON THE NOSHIRO-WARSCHAWSKI THEOREM

slope - 1 , and that at each Ok with k ~ J , arg 0o has a finite jump zr/2. For convenience we introduce the constant tr = (0o + 0~ + 05 + 03p+3)/4. Then f-0-~+~,

-0-~+~, (16)

argto=

~ -O+a,

if 0 < 0 < 0 o , if 0o < 0 < 0 x , if 0 1 < 0 < 0 2 ,

-0+~+~,

if 02 < 0

--0+~+~,

if 03p+a < 0 < 2Z.

<

03p+3 ,

If we make the special selection

(17)

7~ 3z~ 5~z 7~ 0o = ,~:~' 01 =-~- 02 = ~ - , 03p+3 = ~-,

then argto is periodic with period n/2 and ]argto[ < n/4. Consequently if

Ok is near to these special values for k ~ J , then [ arg to ] < ~/4 + %. Finally if 03,04,"',03p+2 are sufficiently near to 3n/2, where argto is near to zero, then the term (15) will be the crucial one in (9) and we will obtain (2). We now indicate why all of the conditions on the parameters can be satisfied simultaneously. Given an arbitrarily small b > 0 we can select M and Q < 1/2p so that ek < b, qk < b (k = 1,2,...,p) and the conditions (3), (4) and (5) are met. Next we let Q ~ 0 and M ~ 0 while keeping the ratio M/Q fixed so that we can also keep ek and r/k fixed. Consequently we have I PkPk+l [ ~ 0 for k = 3 , 4 , . . . , 3 p + 1. It is reasonably clear that for the corresponding

Ok we have Ok+l -- Ok -~ 0 as Q ~ 0. However, to prove this we will use Theorem 2 (below) which is an easy consequence of Loewner's Lemma.

[4] Let F(z) be regular in E,

with F ( 0 ) = 0

and [F(z)[ < 1 in E. Suppose that on an arc A of length ~ lying on ]z[ -- 1, we have [F(z) ] = 1 and that the image of A under F(z) is an arc B of length ft. Then ~ < fl, and the equality sign occurs if and only if F(z) = cz, with

!cl=l.

406

A.W. GOODMAN

Under the conditions of the lemma F(z)/z is regular in E , and on the are A, we have F(e~~ t~ = e~r where V is a nondecreasing function of 0. This gives ~
Let R be a plane domain bounded by a piecewise smooth

Jordan curve. Let W 1 and W2 be two points on the boundary of R , and let C 1 and C2 be the two components of the boundary with W1 and W2 as end points. Let S be a (hyperbolic type) Riemann surface formed by adjoining to R along C2, a simply-connected domain T (which may be multisheeted or unbounded) so that R ~ S - R U T U C 2 . Further suppose that fl(z) and fz(z) are regular in E , fl(z) maps E onto R , f2(z) maps E onto S , and fl(0) = f 2 ( 0 ) . Finally let F be a subarc of C l, where A = f ~ l ( F ) a n d B = f 2 ~ ( F ) . l f ~ is the length of A and fl is the length of B , then ~ < ft. Equality occurs if and only if S = R . Proof.

By hypothesis the inverse f u n c t i o n f f f I is regular in R and takes

R into E . Then the composite function F(z) =-f2 l(fl(z)) satisfies the conditions of Loewner's Lemma and takes the arc A into B. Hence ct < ft. It is clear from the proof that Theorem 2 can be generalized to the case where R is unbounded, has multiple points, and inaccessible boundary points. Returning now to our example function we let R be the square, we let C2 be that arc of the boundary from P3 to P3p+2 along which the triangles were added, and we let F = C 1 , the complementary arc. Since the function fi(z) that maps the unit circle onto the square is continuous on the boundary, we see that ~ ~ 27r as P3p+2 ~ P3, that is, as Q ~ 0. Consequently, by Theorem 2, fl--. 2~, and this means that 0ap+2 ~ 02. Hence ~bk, defined by equation (13), approaches zero as Q ~ 0. Finally we apply Theorem 2 when F is any one of the line segments P3p+ aPo, POP1, PIP2, P2Pa, and P3p+2Pap+3. Again fl(z) is the function that maps E onto R with fl(0) --- A i . If fl denotes the length of the corresponding arc on

l zl

= 1, under f Z 1, then fl > re/2, for the first three segments and

> r~/4 - e for the last two segments. Since the sum of these lengths cannot exceed 2zc, it follows that as Q ~ 0 we have fl ~ zt/2 and fl ~ re/4 respectively. This does not quite prove that Ok approaches the limits given in (17) for k e J ,

A NOTE ON THE NOSHIRO-WARSCHAWSKI THEOREM

407

but it does prove that for small Q, the values of 0~ are close to a set that is congruent to the set (17). This additive constant can be handled (if necessary) by selecting C in (6) to be a suitable complex constant and incorporating arg C in the expression (10) for argto. 3. Arbitrary convex domains.

Let D be an arbitrary convex domain

and let Z = r E conformally onto D, with r 0. The domain D can be approximated by a convex polygon, and consequently tk(z) can be approximated by a Schwarz-Christoffel transformation of the form (6) in which ek < 0 and ~ ek = --2. It follows from this representation that

(18)

[argr

< 2arcsinlz I < zc.

Now let f ( z ) be the p-valent function constructed in w2 and let z = r be the inverse function of r Then F(Z) ~- f ( ~ ( Z ) ) is p-valent in D. Further (19) (20)

F'(Z) = f'(z)~P'(Z) = f'(z)/dp'(z).

l a r g F ' ( Z ) [ - - - [ a r g f ' ( z ) - argr

< [argf'(z)[ + / a r g r

From (18) and (20) we have Theorem 3. Let D be an arbitrary convex domain. Let ~ > 0 be given, and let p be an arbitrary positive integer. Then there is a function F(Z) that is regular and p-valent in D and such that in D

(21)

37Z

I arg F'(Z) I < -)- + e .

The inequality (2) in Theorem 1 is of course best possible, but there may be some room for improving the right side of (21). Suppose that we replace p-valence by mean-p-valence. Can one find a bound for p in terms of B? Here again the answer is essentially no. Let f ( z ) be the example function constructed in section 2 and let a be such that f ( a ) = - M i . Then

408

A.W. GOODMAN

+ Mi

F(z)

(22)

\ l---~z,] f ' ( a ) ( 1 - l a [ 2)

= z+...

has p-zeros in I z [ < 1 and hence is mean-p-valent (in addition to being p-valent). Further a brief computation shows that in E ,

3n 2

3n ~ - argf'(a) < argF'(z) < ff + e - argf'(a).

Consequently if the total variation o f a r g F ' ( z ) exceeds 3n, then F(z) m a y have an arbitrarily high mean-valence. Let C be a condition on f ( z ) that implies that f ( z ) is p-valent (for example a condition on the coefficients). Then C also implies that f ( z ) i s mean q-valent for some q < p . Aside f r o m conditions o f this type, I do not k n o w any condition on f ( z ) that gives a b o u n d for the mean-valence of f ( z ) . REFERENCES 1. E. Christoffel, Ueber die Abbildung einer n-bl/ittrigen einfach zusammenhangenden FJ/iche auf einem Kreise, G6ttingen Nachrichten, 1870, 359-369. 2. A. W. Goodman, On the Schwarz-Christoffel transformation and p-valent functions, Trans. Amer. Math. Soc., 68 (1950), 204-223. 3. G. Julia, Lemons sur la repr6sentation conforme des aires simplementconnexes, Gauthier-Villars, 1931. 4. C. Loewner, Untersuchungen iiber schlichte konforme Abbildungen des Einheitskreises, Math. Ann., 89 (1923), 103-121. 5. Z. Nehari, Conformal mapping, McGraw-Hill, New York, 1952. 6. K. Noshiro, On the theory of schlicht functions, Journal of the Faculty of Science, Hokkaido Imperial University, Sapporo, (I), 2 (1934-1935), 129-155. 7. H.A. Schwarz, Ueber einige Abbildungsaufgaben, J. Reine und Angew. Math., 70 (1960), 105-120. 8. S. E. Warschawski, On the higher derivatives at the boundary in conformal mapping, Trans. Amer. Math. Soc., 38 (1935), 310-340. 9. J. Wolf, L'integral d'une fonction holomorphe et h partie r6elle positive dans un demiplan est univalent, C.R. Acad. ScL Paris, 198 (1934), 1209-1210. DEPARTMENTOF MATHEMATICS UNIVERSITYOF SOUTH FLORIDA TAMPA, FLORIDA,U.S.A. (Received April 24, 1971)