On functions that are bivalent in the unit circle

O N F U N C T I O N S T H A T A R E B I V A L E N T

I N T H E U N I T CIRCLE(1)

By MARTHA WATSON

in Bowliug Green, Kentucky, U.S.A.

1. I n t r o d u c t i o n . A func t ion f(z) is sa id to be p -va len t in the open unit

circle E i f it is regular and assumes no value more than(2) p t imes for z in

E . A well known conjec ture states tha t if

(1.1) w = f = ~ b,z" n = l

is univa lent in E, then for n > 1

(1.2) Ib.I =< . I b , l .

A. W. G o o d m a n [2] has made the ana logous conjecture tha t if f (z) is in the

class ~//'(p) o f funct ions (1.1) which are p-va len t and regular in E, then for

n > p

(1.3) lb.] < p 2k(n + p)! = ~= ( p + k ) ! ( p - k ) ! ( n - p - 1 ) ! ( n 2 - k 2) I b k l �9

In this work we restrict our a t ten t ion to the conjecture (1.3) in the case p = 2,

and we in t roduce the term bivalent funct ions for the funct ions in the class ~//~(2).

F o r this class the conjecture (1.3) simplifies to

0) This paper is a condensation of a dissertation at the University of Kentucky, written under direction of Professor A. W. Goodman, to whom the author expresses sincerest appre- ciation. This research was supported by the National Science Foundation.

(2) The standard definition requires also that f(z) assume some value exactly p times. For our purposes it is more convenient to consider a q-valent function as being also p-valent whenever q ~ p.

383

384 M A R T H A W A T S O N

(1.4) Ib.I < n(n234)1O, I + . ( . 2 _ = 6 1) Ib21'

and when n = 3 this becomes

(1.5) Ib31 51b, l+41b2 I

T h e first step in the proof of (1.5) was made by Goodman [3], who proved

that (1.5) is valid for the subclass of bivalent functions that are starlike and

have real coefficients. Later Goodman and Robertson [5] extended this to

prove the more general inequality (1.3) in the case of p-valently starlike func-

tions with real coefficients. M. S. Robertson [12] has subsequently shown

that the hypothesis of real coefficients may be removed if the added restriction

is made that bl = b 2 . . . . . bp_ 2 = 0. Further, W. K. Hayman [6; 7] has

shown that if bl=b 2 . . . . . bp_, = 0 , then 19~+,l<=2p16~l. Note that,

as a special case, this shows the validity of the inequality (1.5) when bl = 0.

However, despite these results, the inequalities (1.4), (1.5) are still unsettled

for arbitrary functions of ~ (2 ) .

Although we are not able to prove (1.4) or (1.5), we consider certain sub-

classes and show that any extremal function for which I b, ] is maximal must

have certain properties. Since the extremal function conjectured by Good-

man [2] has these properties, our work serves to strengthen his conjecture.

I f f ( z ) ~ r ( 2 ) then so also is emf(ei~ and hence by a double rotation

we can always arrange for bl > 0, and, for some preassigned j , we can also

arrange that bj < 0.(3) Since this double rotation leaves lb,] unchanged,

there is no loss of generality as far as the conjecture (1.4) is concerned.

Denote by V(p) the subclass of functions f(z)~//'(p) for which bl > 0 ;

and let V(2, m) be the subclass of V(2) consisting of functions which have at

least m distinct simple critical points in E .

In order to make this paper self-contained, we rederive the conjecture (1.4)

for the class ~e~(2) of bivalent functions in w The theory of normal families

is used in w to prove the existence of a solution to the problem of maximizing

lb, I, n > p, for fixed I bl I, l bz 1,'", ]be 1' in the class ~f(p). In w we summarize

(3) We make the normalization bj ~ 0 rather than bj ~ 0 in order to use the standard form of the conjectured extremal functions in our work.

ON FUNCTIONS THAT ARE BIVALENT IN THE UNIT CIRCLE 385

the variational formulas to be used. Application of these formulas is made

in w Theorem 5.1 states that if f (z) has a critical point and is extremal for

[b,[ , fixed n > 2, for fixed Iba [ , [b2t , then f ( z ) m u s t satisfy one of three

conditions; while Theorem 5.2 shows that the conjectured extremal function

does satisfy one of these conditions. In the case of maximizing [ b 3 [, for fixed

[ bl 1, ] b2l, Theorem 5.3 eliminates most functions with two critical points

from consideration as extremal functions, and Theorem 5.4 shows that no

function with three or more critical points is extremal. The last theorem,

Theorem 5.5, gives information as to the nature of the domain onto which

f ( z ) ~ V(2) maps E if f (z) is extremal.

2. The c o n j e c t u r e d e x t r e m a l f u n c t i o n s . The procedure used here to

obtain the conjecture (1.4) is carried out by Goodman [2] for p-valent functions,

p > 2, from which the general conjecture (1.3) is derived.

Consider the function

(2.1) w = P(u) = a i u + a 2 u z = ~ b . z" , n = l

where

Z ~ nzn. (2.2) u = u(z) (1 - z) 2 .= ,

A second degree polynomial is always bivalent in the entire complex plane;

and since u = z / ( l - z ) 2 is univalent in E, the composite function

w = P [ z / ( 1 - z) 2] is bivalent in E. Now

n(rl 2 1) (2.3) u z = u2(z) = ~ z". n=l 6

Consequently, using (2.2) and (2.3) in (2.1),

(2.4) b, = na 1 + n(n2 - 1)a 2 - - - o

6

For n = 1,2, the relation (2.4) yields

386 MARTHA WATSON

(2.5) b t --~ a 1 ,

b2 = 2al + a2.

We now suppose that b 1 and bz are given. We can then find the required at

and a 2 necessary to give this pair (bl, b2) in the expansion (2.l). Indeed (2.5)

yields a t = h i ,

(2.6) az = b 2 - 261.

But then (2.4) may be written as

(2.7) b, = n(n 2 "b-4, bl + n(n 2 - 1)b z 3 6 "

Note that for the special functions of the form (2.1) the maximal value for

I b, I' for fixed [bl l ' t b2 I' is obtained when b t , b z are real and have opposite

signs, in which case the equality sign holds in the conjecture (1.4). Thus if

we select b I > 0 and b 2 < 0, the function defined by (2.1) and (2.2) becomes

It is this two parameter family of bivalent functions F(z ) , depending on b t

and b2, that gives the conjectured extremal functions for the problem of

maximizing lb, I when ]bt I and ]b21 are fixed.

Equations (2.6) show that if bl > 0, b2 < 0, then at , a 2 have opposite

signs. In this case, F(z) has exactly one critical point in E. For P'(u) vanishes

at u = - a~/2a2; and since - a l / 2 a 2 > 0, this point is in the image domain

of E under u - u(z) . Further this is the only critical point of F(z), for u = u(z)

is univalent in E. Observe that the critical point C of F(z) is real and positive,

and satisfies the equation

2?2 ) (2.9) C 2 + \ b t - 3 C + l = 0 .

Also, the coefficients b, of F(z) in (2.8) are all real.


3. The ex i s t ence theorem. Since we wish to use results from the theory

of normal families, we have framed our definition of valence so that a q-valent

function is p-valent whenever p > q. For we want the limit of a sequence

of p-valent functions to be p-valent. As an example, consider the family of

functions formed by the sequence z + (n + 1)z2/2n, n = 1,2, ..-. Each mem-

ber is bivalent in E, but the limit function is univalent in E.

Denote by W(p;] b I 1, ]b z 1,..., ] b, 1) the subclass of functions f (z) ~ r

f (z) = ~ fl,z", for which ]/3jl = ] b~], j = 1,2, . . . ,p . Stated precisisely, we n = l

have

T h e o r e m 3.1. Let bl,bz,. . . ,b" be an arbitrary set of numbers not

all zero. For the subclass r(P;lb, l,lbel,...,Ib, I) an. each fixed g > p,

the,'e is a function F(z) e ~(p;I b, 1, [ha 1,-.., [b~ [), F(z) = ~ B,z", which n = l

has the property that if f (z) ~ r ], ]b2 ],'", ]bp 1), f (z) = ~ fl, z", then n = l

=< [BN[.

Proof . Let ~ (p ) denote the family consisting of functions

(3.1) f ( z ) = s ~,z"6 Y:(p) n = l

which satisfy the condition

(3.2) max {]eil , j 1,2,. . . ,p} = 1,

and let ~'(p;Ja, J,la2],...,Jap[ ) denote the subfamily of .~'(p) of functions

(3.1) for which [:ejl = [aj[ , j = 1,2, . . . ,p . For each f (z) ~ W(p) there exists

a constant c such that cf(z)~ ~(p) . Thus, to show the existence of an ex-

tremal function, for fixed N > p, in the class Y:(p; [ b x ], [b2 [, ' . ' , ] b,[) it

suffices to show the existence of such a function in the corresponding family

~-(p;]a,l , la2l, . . . ,]ap]), where c tay]=lb j l , j = l , 2 , . . . , p , e a positive

constant.

388 MARTHA WATSON

By [9, p. 67] the family ~-(P; I a, I, l a~ l, "", I a , I) is quasinormal of order

at most p, since each f ( z )~ :(p, lall,la2l,...,la, l) assumes the values 0

and 1 at most p times. Then by [-9, p. 70] the family is normal, since each

f (z ) in (p;la, I) has the property (3.2) and

[aj] <~ 1 j = 1 ,2 , . . . ,p .

Further note that because of the condition (3.2) the limit of a convergent

sequence of functions from ~ (P ; I a l I, t a2 I, "", lap I) is not a constant. But

then, by [10, p. 8], the limit of a convergent sequence of functions

from ~(p;lal l ,[a21, . . . ,[up[) is p-valent and is therefore also a function in

: ( p ; l a , l , l a z l , . . . , l a , I). Thus the family ~-(p;la, l, la2l , . . . , la , I) is normal and compact. Consequently, by [11, p. 144], the extremal problem [ bN [ = max.

has a solution within the family :(p;[al I, I), and this completes

the proof of the theorem.

4. C o l l e c t i o n of v a r i a t i o n a l f o r m u l a s , l f f ( z ) has a critical point in

E, then we can use the following formula due to Goodman [4].

oo

T h e o r e m 4.1. L e t f ( z ) = ~ an zn, a l > 0 , map E onto a Riemann n = l

suJface R with at least one simple branch point, say f (C) = B, where C ~ E,

C ~ O, and f"(C) # 0. Let R* be the Riemann surface formed from R by

moving the branch point B to B* = B + 2, where 2 is a sufficiently small

complex number, while holding the boundary and any other branch point

of R fixed. Let f*(z) be the uniquely determined function which maps E

onto R*, with f*(0) = f ( 0 ) , f* ' (0) > 0. Then

z f ' ( z ) {C + Zp2 1 + Cz ~7~ (4.1) f*(z) = f ( z ) - ~ C ~ z + 1 - - - ~ z ~d j + O()'2)'

where

(4.2) 1

P = - - Q = P , c~f, , (c) '

P , Q # O , ~ .

In particular i f f (z)~3e ' (2) , then so also is f*(z) .

ON FUNCTIONS THAT ARE BIVALENT IN TItE UNIT CIRCLE 389

The above theorem is easily generalized to the case wheref(z) has m simple

critical points in E. If C~, Cz,-" , C,, ~ E are distinct simple critical points of

f(z), Ck -r 0, k = 1,2,.. . , m, we set

1 (4.3) Pk = 2 , , Qk = Pk, Pk, Qk ~ O, oo.

Ck f (Ck)

We move each of the branch points f ( Q ) = Bk by amounts 2k,2 k = O(21)

k = 1,2,.-., m, while the image domain of E under f ( z ) is otherwise heId

fixed. Let f* ( z ) map E onto the domain thus obtained, with f * ( 0 ) = f ( 0 ) ,

f* ' (0) > 0. Then

(4.4) f*(z) = f ( z ) '~k k k + 1 Qk,~k "4- 0(22).

In particular i f f ( z ) e V(2m,), then so also is f*(z) .

The Marty variation ['8] is obtained by using a linear transformation of E

onto E.

Theorem 4.2. I f f ( z ) s V ( 2 , m) has the form (1.1), then for each

f~ E E, o~ v~ O, the function

- - 0 ~ ) f~(z) = f ~ - - f ( - -c 0 --f(z) + [ f ' (0) - - f ' (z)]~ + z2f ' (z)~ + O(~ 2)

(4.5)

= Y, l b . - (n + 1)b.+1 + (n - + 2) n = l

is also in the class V(2, m).

Exterior variational formulas have been used by Goluzin [1] and Schiffer

[13]. But for our purposes we prove

T h e o r e m 4.3. Let f ( z ) ~ V(p) and suppose there exists a 6 > 0 such

that w = f ( z ) assumes none of the values from a circle [w - w~ [ < 5, for z

in E. There exists an q > O such that i f IhJ[<t l , j = l , 2 , . . . , k , and if

w j , j = 1 ,2 , . . . ,k , are k fixed points in the circle [ w - wa [ < 6, then the

function

390

(4.6) f , ( z ) = f ( z ) +

is in the class V(p).

MARTHA WATSON

k f2(z ) hj

j =1 f ( z ) - - wj

Proof . Obviouslyf,(z) is regular in E. For tile proper valence it suffices

to show that for z l , z 2 e E , f , ( z O =f , (z2) if and only if f ( z l ) = f ( z 2 ) .

Clearly, if f ( z 0 =f(z2) , then f , ( z l ) = f , ( z 2 ) . Conversely, we wish to show

that if f ( z l ) #f(z2) , then f , ( z 0 # f , ( z 2 ) .

Thus suppose z t , z 2 e E such that f ( z l ) # f ( z 2 ) . If f , ( z l ) = f , ( z 2 ) , then

or

k ~C2(Z "~ f ( z , ) + 2~ h. - / ( z 2 ) + L nj . . . .

j = 1 ' f ( z l ) - wj j =1 f(z2) - - Wj

(4.7) ( f ( z l ) - - f ( z 2 ) ) 1 + )2 hj 1 - = O. j = 1 ( f ( z l ) - w j ) ( f ( z z ) - w j )

But then, by our hypothesis that f (z l ) # f(z2), equation (4.7) implies that

k ( Wi2 ) (4.8) 1 + Y~ hj 1 - = O.

j=~ ( f ( z l ) -- wj)(f(z2) -- wj)

Note that if I w j - w l ] =e j , j = 1 ,2 , . . . ,k , then O < e j < 6 and

(4.9) ] w - w j ] > 6 - e j - b j > O , j = 1 , 2 , . . . , l r

Hence (4.9) holds for all values w assumed byf (z ) , z in E. Further if ] wj] = r j,

then by inequality (4.9)

( ~ Z ~ W,/'2 /,.j2 --

W~.)(?(Z 2) W j) < - - - - = ~ j ' - - - - = •j2

and the modulus of the left hand side of (4.8) is greater than or equal to

k 1 - Y~ I h j ] ( 1 + ~j ) .

j = l

j = 1,2,...,/r

ON FUNCTIONS THAT ARE BIVALENT IN THE UN1T CIRCLE 391

Set r/j = 1/(k + 1)(I + ~j), j = 1 ,2 , - . - , k , and q = min {ql ,t/2 , " " , q k } �9 Then if

[hj] < q, j = 1 , 2 , . . . , k , we have

k k 1 - - ~ [ h j l ( l + r 1 . . . . > 0

j = l k - 1 "

Thus equation (4.6) cannot hold and we must have f . ( z l ) - ~ f , ( z 2 ) .

If, in the above t h e o r e m , f (z) is in V(2), then clearly I wj I > O, j = 1, 2,. . . , k.

So that in a ne ighborhood of the origin we can write

(4.10) f ,(z) =f(z) - ~Z hj ~2 f(z)"+l - ~ b,,z", j = l n = l Wjn n = l

and this the desired exterior variation formula.

(5.1)

5. A p p l i c a t i o n s . For brevity, we introduce the notat ions

~o = 3b 3 - - b~,

(5.2) ?k = (k + 1)bk+l 2bkb2 ( k _ l ) D k _ l , k = 3,4, .,.. bl

Theorem 5.1. with

Let k be a fixed integer, k > 2, and let f ( z ) ~ V ( 2 , 1 )

o~ (5.3) f ( z ) = ~2 b,z".

n = l

l f f ( z ) is extremal fo r Ib, I with fixed [b~ I,Ib2I, then at least one of three

conditions (a), (b), (c) listed below must be satisfied, f o r each critical point

C o f f ( z ) :

(a) b ~ + l = b ~ + ~ I;

3b3 2b22 (b) - --b~--[ = b l , i.e. ?o = O;

392

(c)

where

(5.4)

(5.5)

(5.6)

Proof.

M A R T H A W A T S O N

k - 1 k~,2 ~.~ f l jC 2 k - j - 2 31- f l jC j = O,

j = 0 j = O

flj -- (j + 1)bj+17o, 0 =<j =< k - 3,

2b~ ~_ /~,_~ = ( k - O a , - , ~ o - b,a~ - b , 3b3 - - K - , ) ~ , ,

1 , - b t ] + "

We prove that if none of the conditions (a), (b), (c) are satisfied

then f ( z ) is not extremal. Let C ~ E be any one of the critical points o f f ( z )

and let f*(z) = ~2 b,*z" be the function obtained when the branch point n = l

f (C) = B is moved a small amount 2. Thenf*(z) ~ V(2, 1) and by Theorem 4.1,

b,z = b , z " - P2 1+2 ~- +Q~ l + 2 ~ C " z " +O(22). n = l n = l n = n n = l / 1

With error O(22),

b* = b , [ l - Re(P2)],

(A) b~ = b 2 [ 1 - 2 R e ( P 2 ) ] - b, [ P 2 + QC2],

. ~

b~ = b k [ I -- kRe(P,~)] - I~ (k - j )bk_ j ~ + QCJI �9 j = l

Applying Marty's variational formula, equation (4.5), to f*(z ) , with c~ = 0(2), we obtain the function

f*(z) = ~2 [ b * - ( n + l ) b * t ~ + ( n - l ) b * _ l ~ ] z " + O ( ~ 2) -- ~2 a,z ~. n = l n = l

Then, except for an error term of O(22),

ON FUNCTIONS THAT ARE BIVALENT IN TIlE UNIT CIRCLE 393

al = b * - 2b~a,

(B) a 2 = b~' - a b l e + b*~

~

ak b * - ( k + l ) b k + l ~ t + ( k - 1)b*- 1~.

Note that b*ct - b i ~ = O ( . ~ 2 ) , i = 1,2, . . . ,and, thus, that

al = b * - 2b*c~ = bl[1 - R e ( P 2 ) - 2b2~/bx] + 0(22).

In order to normalize the first coefficient a~, we consider the bivalent function

defined by

(5.7) g(z) ~, , , [ 2bz=] = a.z = I + R e ( P 2 ) + b~ ] f * ( z ) " n = l

From (A) and (B) it then follows that, with error O(2Z),

a* = b l ,

, - ~ - j ~ ,

~ 1 7 6

( k - j ) b k _ j ] - - (k - l)bk k-, , =bk - (k )bk+ - - + ~. ( k - - j )bk_ jC j Q1 a k ~)- ] k'z + 2

j : 1 j = t

+ [ ( k + l ) b k + l - 2 b - - k b - - z e ] a - ( k - 1 ) b k _ t ~ } bl

We next consider conditions that will enable us to choose ).,c~ • 0 such that

a * = b2 and la *l > Ibkl- This is possible if

bx\ /bE b i t ) Q , ~ (363 2b22'

and

394 MARTHA WATSON

k~21 (k-j)bk_j P2 + (k 1)bk + )2 (k-j)bg_fl] s QI Re (k )bk + Ci j= l j=l

+ ( k + l ) b k + l - ~ - - - c~-(k- l )bk- l~ > O.

Replacing the terms in ,~ and ~ by their conjugates does not alter the real

part of this expression. Then, using the notation (5.2), this condition is equi-

valent to the condition

k~'(k_j ) b~_..j " + (lI~) Re{[(k-l)b,, + j=t ( b i + Dk_jCJ)]P2 ~kC~} > 0 .

Write condition (1~) as

(5.8) (3b 3 262 ~

We recall that P, Q # 0, Go, and we observe that the coefficients of 2 and ,~

cannot be simultaneously zero; for this would imply that bl(1 - I C 12)/C = O, which is impossible since C e E and bi # 0. Thus by taking (5.8) and its con-

jugate we obtain the pair of equations, linear in cz and ~.

(i)

This set has a non-zero solution ~, ~ for each 2 r 0 provided that we do not

have either

or

(b) ~3b 3 - bt = b 1.

ON FUNCTIONS TttAT ARE BIVALENT IN THE UNIT CIRCLE 395

Assuming now that neither (a) nor (b) hold, we can choose cr 0 such

that condition 01) is satisfied.

I f we solve the system (i) for ~, then, using the notation (5.1),

If we put the result (5.9) in the left side C~(lIl) of condition (IIl), and replace

the terms involving ~ by their conjugates, we obtain

c~(II,) -= Re{(L02},

where, collecting like powers of C,

L~ = P?o ( k - 1 ) b k T o - R e ?k b2 3 / ) 3 - - b ~ - ] + /~2bl

-57] ,q + c

- b , ' A - b ~ 3b3 ]~k] + ~ ( k - j ) +[h,--F ?o �9 j = 2 \ CJ "

Using the abbreviations given by (5.4), (5.5), (5.6), this becomes

(5.10) L~ =. P + "E fijC j . ~ o C k - I " ~ J - 0 " j = O ,'

Since the argument of 2 is arbitrary, 2 can always be chosen so that

~(1I~) = Re{(L~)2} > 0, provided that we do not have L t = 0 . But, by

(5.10), since P/,,o C~-1 #O,L~ = 0 if and only if condition (c) is satisfied.

Thus, if f (z) fails to satisfy all three conditions (a), (b), (c), then f ( z ) is not

extremal for [b,,I.

We next test F(z), the conjectured extremal function of w defined by

(2.8) for given b~ > 0, b 2 < 0, with these conditions. We recall from w that

396 MARHTA WA'ISON

F(z) has all coefficients real and that the single critical point C is positive

and satisfies the equation (2.9).

In this case condition (a) has the form

b 2 1 ( b2 ) 2b + ~ - = - + ~b-~ + C �9

The plus sign leads to C = + 1 and this is impossible. The negative sign leads to

(5.11) b~ "

Using (5.11) in conjunction with (2.9) gives bz/bl = 6, and this is impossible

since b~b2 < 0 for F(z). Hence F(z) never satisfies condition (a).

When n = 3, equation (2.7) gives b3 = - 5bl + 462. Using this in con-

dition (b) and the fact that all of the coefficients are real, we find that condition

(b) is equivalent to

(2b22 - IZb2bl + 15bZ) 2 = b~ o r

4(b~ - 6bzb, + 7b~)(b2 - 4b0(b2 - 2b~) = 0.

Again this is impossible because b~b2 < O. Hence F(z) never satisfies condi-

tion (b).

But F(z) always satisfies the condition (c); and since this condition is so

complicated, this circumstance greatly strengthens the conjecture that F(z)

is extremal. More precisely, we have

T h e o r e m 5.2. Let F(z)= ]E b,z", where the b,, are defined by equa- t l = 1

tion (2.7). For each fixed pair bl,b2, with bl > 0 , b 2 =<0, and for each

fixed k, k > 3, the function F(z) sati.~es the condition (c).

P r o o f . We will prove that for each k > 3 the expression

2{ bz - 3 )C + 1 C 2 -~- \ b 1 ,


occurs as a factor of the left hand side L:(c) of the condition (c); and since

from equation (2.9) this factor is zero for F(z) , it will follow that F(z) satisfies

the condition (c), for each k => 3.

Using (2.7) in (5.1) and (5.2), and simplifying, we obtain

(5.12) 7o = 4-4-4-4-~(b~ - 6b2bl + 7b2)(b2 z - 6b2bl + 8b~) U 1

1 (5.13) ?k = - - - k ( k 2 - 1)(b~ - 6b~b, + 8b~).

3b,

Using these equations and the fact that b3 = - 5ba + 4b2, we obtain

k(k 2 - 1) (5.14) 7k(3b3bl - 2b 2 + b 2) = 7obl 6

Putting the results (5.12), (5.14), and the appropriate expressions, given by

(2.7), for the bj, in equations (5.4), (5.5), (5.6), we obtain

(5.15) flj = ~obl(j.3 L 2)2[j(j.q_ b2 ] 6 2 ) ~ - ~ - 2 ( j + 3 ) ( j - l ) , O < j < - k - 3 ,

(5.16) /~_2 = ~' ' (k -1 ) (k - 2) k ( k - 1 ) ~ - ( k + O ( 2 k - 3 ) ,

~ '~ ( k + l ) N 2(k+2) (5.17) ilk-1 = 6 - "

Also note that flj = flj., j = 0, 1,..., k - 1.

We claim that

k - 1 k - 2

(5.18) ~(c) -- ~ /~ jc~k-J-2+ Z /~jc j j =o j =o

-~ CZ +I b,-6 C+I)( ~2 2 c9C 2'-j-'* + ~oO~JC ' , ] \ j = O J=

where the fl~ are defined by equations (5.15), (5.16), (5.17), and the ~j are

defined as follows:

398 MARTHA WATSON

yob1~. (5.19) aj - ~" tJ + 2)2(J + 1)(j + 3),

(5.20) = k(k2_ 1)(k- 2).

We define ,51, j = O, 1,..-, k - 2 by

5.21) C 2 + - 6 C + 1 j]~'~o ~ + k;3~ OCJ Cj]" "= j = 0 /

k - 1 k - 2 = ~ 8jC 2k-j-2 -F ~ 8jC j.

j = O j = O

Using (5.19), (5.20) in (5.21), we obtain

8o = ~o = ?obl = rio,

For 6 j , 2 < j < k - 3 , we have

[2bz ) 8y = ~j+ k b 1--6 o~y_l+c~.i_2

-- 7~ [(J + 1)2j(j + 2) ~b2- 2(j + 1)2(j + 3 ) ( j - 1)] - fly.

For 8k_ z and 6 k_l, we have

[2b~z ) 8k-2 = C~k-2 + k b l - 6 ~k-a+ek-4

-- Y~ [k(k-1)~~21-(k + l ) ( 2 k - 3)

[2b2 ) 8k_ 1 = 20~k_ 3 -{-" \ bt - 6 ~ k - 2

' ~ ( k+ l ) b2 ] -- 6 ~ [ - 2(k + 2) = fig-l,

O < j = < k - 3 ,

= ~ - 2 ;

O N F U N C T I O N S T H A T A R E B I V A L E N T I N T H E U N I T C I R C L E 399

Tiffs proves the identity (5.18) and completes the proof of the theorem.

We can give a second proof(4) that depends more Ola concepts, and less

on computation than the first proof. However it turns out that a careful pre-

sentation of this second proof actually requires more space than the first

proof. Hence we merely outline the steps. Details may be found in the author 's

dissertation in the University of Kentucky Library.

Let J / b e the class of bivalent functions that map E onto a Riemann surface

consisting of one full sheet and one sheet minus a single linear slit extending

to infinity, with the sheets tied together at a single simple branch point. We

observe (1) that F(z), obtained in w belongs to this class ~ ; (2) that the

variation of a branch point leads from one function of this class to another

in this class; (3) that the Marty variation also preserves membership in the

class ~ ; and finally (4) that in the class ~ the conjecture (1.4) is valid. Con-

sequently if F(z) does not satisfy condition (c), we would have a contradiction

to item (4). Hence F(z) satisfies condition (c), for each integer k > 3.

T h e o r e m 5.3. Let f ( z ) e V(2,2), antl let

f ( z ) = ~ b,z", b 3 < O,

have critical points CI,C2EE. I f the product C1C 2 is not real, then f ( z )

is not extremal for ]b31, with fixed ]b, I'l b21.

= l/C k f (Ck)#O, o%k= l ,2 . Proof . Note that C 1 # C 2 , C k # O , Pk 2 ,,

If f * ( z ) = ~, b'z" is the function obtained by applying formula (4.4), for n : l

m = 2, to f ( z ) , where the branch points f (Ck)= B k are moved by amounts

2k, k = 1,2, (with 22 = 0(21)), then

b ' z " = ~, b,z" - - -~ - - 1 + 2 n = l n = l n = l = n = l

+ Z Qklk 1 + 2 C~z" +O(,t~). k = l n = l

(4) This second proof is due to A. W. Goodman.

400

Thus, with error 0(12),

MARTHA WATSON

( 2 ) b* = b l 1 - R e E Pk2k ,

k=l

( D 2 ) b* = b 2 1 - 2Re 1 ek2k - bl E 2k + QkCgIg = k = 1 \ ~ " k

- bl E I~ + Q~C~Ik �9 k=l

Note that if we can choose 11 r 0 such that

) (I2) E 2 k+ QkCkl k = O, k=l

and arg22 such that, for such 11,

z Re E Pk).k = 0

k=l (II2)

and

(III2) Re ~ 2 k+ QkCklk > O, k=l

then, by the relations (D2), we would have b* = bl , b2* = b2, [b~'[ > [b3 l-

But we can always solve for 21 so that condition (I2) is satisfied. Consider

the system

(ii)

P1 P2

Wl w2

Q2 P1Cl l l -q - Q~llll = -P2C2~2 - C-~72,

where the second equation is obtained by taking the conjugate of the first.

Since the "determinant of coefficients" of each side of the system (ii) is

IPk =12 <1-- Ickl') 0, /

k = 1,2,


we obtain a non-zero solution for 21 for each non-zero 22. If we solve for

2x in the system (ii), we obtain

(5.22) I C l 12 {F1)~2 + G2)~2 } , = le112(1_ ic , i,)

where

-2 2 P2Q1 ( _ 1 q- C1C2) , (5.23) F 2 -- 01C2

(5.24) G2 _ Q2Q~ (C~- C~). C1 C2

We now wish to write the left hand side 5e(II2) of condition (II2) in the form 2

.LP(II2) =- Re ~] Pk2k ----Re{(L2)22}. To do this, we put the solution for 21, k=l

given by (5.22), (5.23), (5.24), in ~(112) and take the conjugate of the ,~2

term. A computation then gives

P2(C2 -- Ca)(1 -- C1C2) (5.25) L2 = C2(1 + I C ' ]2) , L 2 r 0

Consequently, we can write the condition (II2) as

} (IIz*) ~(II2) = Re [[ C2(1 + I C , [2) j)l 2 = 0.

Similarly, we wish to obtain the left hand side ~(III2) of condition (III2)

in the form

S(III2) = Re ~2 2 k + QkC21k = Re{(M2)22}. k=l

Note that we can always choose are "~2 SO condition (III2) is satisfied, provided

that M2 -~ 0. But, replacing QkCg22k by its conjugate, we can write

402 MARTHA WATSON

2

5~ = Re }2 (1 + Ck 4) k = l

P k 2,. Ck 2

Again, using equations (5.22), (5.23), (5.24), we find

P2 { (7tC2(1 + C~)(I -- C2C22) (5.26) M2 - C~ I C, ] s (l - I C, I*)

+ c , c ~ o + c t ) ( c ~ - c~) (1- lc, l')

+ [Cxi2(1 + C4)}

P2 - - 8 7 c~lc,?

where S is defined by this equation. We next write S as a polynomial in C2

and simplify it by finding two of the roots. Setting

4

s = s ( c 9 ~ E ~ jc{ , j=O

we see that ao = a4 = t C, 12 and az = O. Since I C, I z # O. we have S(Cz) ~ O.

Further note that

[C 112(1 "~ C4)(1 -- [C~ [4) + I c ' 120 + C~) = O, S ( C , ) = - (~ -- ]C , ]~)

and that

s = c ~ ( l - [ c , I ~) + - - ~ , - ' ~ ' + ~) = o .

Hence S ( C 2 ) has a factorization of the form

(c ~, + c b s(c2) = - (c2 - c o o - c , c 9 c , c ~ + 0 + I c l l ~) C2 + r

o r


(5.27) S = S(C:) = (C2 - c O O - C, c2)

(1 + lc , l ~) [ - c , o + l c , I ~ +c, c2)

- c , ( c ~ + c ] [ c , 1~ + c ,c , ) ] .

But then equat ion (5.26) becomes

P~ P 2 ( c ~ - c o d - C, c2)

M 2 -- C11Cl l2 S = C21C112(1 q - I C I I 2) [ - c , o + [ c, 12 + c, c o

- c , (c l + click[ 2 + c, c2)],

and, using L2, given by (5.25), we can write M 2 = L2N2, where

- c , (1 + [ c , 12 + c , c ~ ) - c , (c~ + c ~ l c , I ~ + c , c~) (5.28) N 2 = c~lc, p

But L 2 # 0. Thus if the expression Ne # 0, then M 2 5~ 0. But N 2 = 0 implies

tha t

I , + l c , l~+c,c~l = [c2[]c2(1+]c,[~)+c, I,

which is impossible since 1C21 < 1 and it is easy to prove that

11 +lc, l'+<c~l > Ic~<1 + Ic, b + c,I

Hence N2 --/: 0, and, therefore, M 2 -~ 0. Also note that condit ion (III=) can

be writ ten

( I I I2") s = Re {(M2)22} = Re{(L2N2)22} > 0.

I f we let a rgLa = ~b, a rgN2 = 0, a rg22 - - -7 , then condit ions (IIa*) and

(III2*) are equivalent to

cos ( r + ~,) = 0 ,

cos (4~ + 0 + ~) > 0 .

Since the complex number 22 is arbi t rary, we can select ~ = a rg2 z so tha t

404 MARTHA WATSON

these conditions are satisfied, unless 0 = 0 or 0 = n. Thus with [bll and

[b21 fixed we can increase l ba ] if N2 is not real. For N2 is real if and only

if N2 - N2 --- 0; or, using equation (5.28), if and only if

C 2 [ - C,(1 + C~C2 + [C~ I ~) - CiC2(C, + Cz + cz [C~ 12)]

+ C2[C,(1 + C,C2 + l c , [2) + c,c (c, + + I t , = 0.

But simplifying the expression on the left hand side, we obtain the condition

(C~C2 - Cat2)(1 + [C112)(1 - 1C212) = 0,

which is true if and only if C1C 2 = C~C2. Thus N2 is real if and only if the

product C1C2 is real. Hence if CIC 2 is not real, we can always increase [ ba[

while holding I b, [ and [b2 [ fixed.

The difficulty which arose in the preceding theorem does not arise if there

are three critical points.

T h e o r e m 5.4. I f f ( z ) ~ V(2,3),

oo

f (z) = Z b,z", b a < O, n = l

then f (z) is not extremal for [bal, with fixed [b l [ , Ib2[ .

P roof . If f ( z ) has distinct critical points C1, C2, C a G E , then

C k ~fi O, Pk = 1 / C k 2 f " ( C k ) :fi 0 , k = 1, 2, 3, and we can apply formula (4.4),

for m -- 3, to f ( z ) . If f *(z) = ~ b.*z" is the function thus obtained, then n=l

the equation set (D3) that gives b.* in terms of b s is identical with the se t (D2) ,

except that now each sum on k runs from 1 to 3. Just as before we wish to

select 21 so that

(Ia) ~ 2k + QkCk,~k = 0; k = l

22, for such 2x, so that

([h)

ON FUNCTIONS THAT ARE BIVALENT IN THE UN1T CIRCLE 4 0 5

3

R e ~2 pkJ.k = O; k = l

and arg 23, for such 2~, 22, so that

3( ) ( i i i 3 ) Rek__~ 1 Pk = U~& + GC~L >o .

Then we would have bl*=b,, b2*- -b2 , Ib3*[ > l b 3 [ .

To solve for 21, consider the system

(iii)

pl 3(p, )

g l C 1 ) ~ 1 q_ Q1 g k C k ~ k ..[_ .

Since the "determinant of coefficients" of the left hand side of the system

(iii) is

Iv'll~ 0 - l c l l ' ) ~ 0, [c,

we can always solve for 21 . If we do so, we obtain

(5.29) 3

x, = iP, l ~ = l c , i, ) =

where, for k = 2, 3,

(5.30) P k Q 1 - 2 " Fk = ~ ( - 1 + C, G ) ,

(5.31) ~ - q~Q_'-(c?- c : ) . CkC~

If the left hand side of condition (II3) is denoted by ~(II3) , then we wish

to determine Hk such that

3 3

s = Re ]E Pk2k = Re E Hk2 k. k = l k = 2

406 MARTHA WATSON

Using equations (5.29), (5.30), (5.31), we find

H k = Pk(Ck -- C1)(1 -- C1Ck), k = 2,3. Ck(1 + I C112)

But then we can write condition (IIa) as

a 3 Pk(C k - C 1 ) ( 1 _ C1CR ) 013") s = Re ,=2 ~ Hk2* = Re k=2 ~ C,(I-1C~12 ) 2 k = 0.

For each non-zero 23, we select 2 2 SO that

(5.32) P3C2(C3 - Cl) ( l - - CiC3) 2 22 = - P2C3(C2 C1)(1 -C~2) 3 5/= 0,

and condition (II3*) is satisfied.

We must show that when 22 ~ 0 and 23 :~ 0 are related by (5.32), then 21,

determined by equation (5.29), is not zero. But a computation yields

P 3 C I ( I - C 1 C 3 ) ( C 3 - - C 2 ) / ~ Q 3 C 1 ( C 3 - - C l ) ( C 3 - -

(5.33) 21-~C3(C2_Cl)(1_1C, 12) 3 - ~ - 1 _ C l C 2 ) ( 1 _ ] ~ I ] 2 ~ 3 .

and 21 = 0 for 2 3 -5 ~ 0 only if

o r

1 - C1C3 C1 - C3

1 - C 2 C 1 C 1 - - C 3

cS-- = 1- F1"

But this is impossible, since the left side is greater than 1, and the right side

is less than 1, for C1,C2, C3~E. Consequently, we do obtain a non-zero

solution for 21 for each 23 ~ 0.

Finally, we wish to obtain the left hand side ~(I I I3) of condition (1II3) in the form

) ~cf(III3); = Re ]~ 2k + QkCk2~k = Re{(L3)23) . k = l


Putting the results (5.32) and (5.33) in s we obtain

L 3 - P3(1 - (~C3)(C3 - Cz)(1 + C~)

c , G ( c ~ - c , ) o - l c~ I ~)

P3(C3 -- C , ) ( C 3 - C2) ( I + C 4)

CIC3(1 - e t C 2 ) ( l - ] C 112 )

(5.34) P3(C3 - C l ) ( l -- C 1C3)(]. + 6 2 4)

CzC3(C2 - C1)(1 - G C D e3 (i + G )

+ C~33

P3 = j Cl 12C2C2 T,

where

(5.35) T--- T(C3) C, c 2 G ( 1 - G c 3 ) ( c ~ - c 2 ) ( l + c'~)

(c~ - c , ) o - ]c~ I ~)

C1C2C3(C3 - C1)(C3 - C2)(1 q- 1~I)

(~ - G c : ) ( 1 - Ic~ 1 ~)

I c ' [ ~ c ~ ( c 3 - c ' ) ( l - G c ~ ) o +c~) ]~cdt+c~) . (c~ - c O O - G c 9 , ] c ,

Since we can now write condition (III3) as

(III3") ~ ( m 9 = Re{(L3)X~} > 0 ,

and since arg23 is arbitrary, the condition (III3) may always be satisfied pro-

vided that L3 :~ 0. But we see by equation (5.34) that L3 r 0 if and only if

T r Clearly T is a fourth degree polynomial in C 3, and if we write 4

T(C3)----- ]~ z, jC3, then z o = r ~ = l C , 12Cz. Note that since ]C,]ZCzr j=O

T(C3)~O. Further, direct substitution shows that Cz, C, , and I / ~ are

roots of the polynomial. But since ro = ~4, the product of the four roots of

the polynomial equation T(C3)-----0 is one. Consequently, the fourth root

is C1/C1C z. But then T(C3) = 0 if and only if C 3 = C, ,Cz ,1 /C, ,Cj /C,C2.

Note that I/C1, C,/C1C2 (~ E. Thus all four solutions are excluded by the con-

ditions of our problem that C3 :~ C~, C2 and C 3 e E. But then T = T(C3) r 0

and therefore L 3 ~ 0. This completes the proof of the theorem.

408 MARTHA WATSON

Theorem 5.5. Let k be a fixed integer, k > 3. I f f (z) ~ V(2), given by

(1.1), omits an open set lw - wl I < 6,for z ~ E , t hen f (z) is not extremal for

Ibk[, with fixed [b,l,lb21.

Proof . If we apply the formula (4.10), with k = 2, to f ( z ) , we obtain

2 co f ( z ) ,+ l ~o (5.36) f , (z) =f (z ) - j~lhJS'= =1 --wj" - ,=i ~ b, ,z" ,

where hl,h 2 are complex constants and w2 is in the omitted set [w - w~] < 6.

We know that if [h~ [, I h2 [ are sufficiently small, thenf(z) s V(2).

The relation (5.36) yields

(5.37) ~., b , , f = ~ b,z" - ~ hj ~2 k=l n

n = l n = l j = l n = l Wj

and hence

We can choose

b l , = b I ,

b2, = b2 - ~1 + ~ b12"

(5.38) h i - Wl h2, w2

since w 2 -r 0. Also, since w~ r 0, we have hi r 0 for h 2 -~ 0. But then, for

the choice (5.38) of h i , b~, = bl, b2, = bE- If we compute bk,, replacing

hi by - wlh2/w 2 in equation (5.37), then

h2 (5.39) bk, = bk + -~-zs-G(w2),

where G(w2) is a polynomial of degree ( k - 1) in wz. Note that the constant

term is - b ~ . Consequently, G(w2) ~ 0. But, then there are only ( k - l )

values of w2, for fixed wl, for which G(w2)= 0, and thus we can always

choose w 2 such that G(w2)~ O. Finally, by relation (5.39), we can choose

argh2 so that I b k , ] > l b k l . Thus f ( z ) i s not extremal for [bk[.


REFERENCES

1. G. M. Goluzin, Method of variations in the theory of conformal representation 1, Mat. Sbornik 19 (61): 2 (1946), 203-236.

2. A. W. Goodman, On some determinants related to p-valent functions, Trans. Amer. Math. Soc. 63 (1948), 175-192.

3. A. W. Goodman, On the Schwarz-Christoffel transformation and p-valent functions, Trans. Amer. Math. Soc. 68 (1950). 204-223.

4. A. W. Goodman, Variation of the branch point for an analytic function, Trans. Amer. Math. Soe. 89(1958), 277-284.

5. A W. Goodman and M. S. Robertson, A class of multivalent functions, Trans. Amer. Math. Soc. 70 (1951), 127-136.

6. W. K. Hayman,Someapplicationsofthetransfinitediameter to the theory of functions, Journal d'Analyse Mathdmatique (Jerusalem) 1 (1951), 155-179.

7. W. K. Hayman, Symmetrization in the theory of functions, Tech. Rep. l l, Navy Contract N6-ori-106 Task Order 5, Stanford Univ., Calif., 1950.

8. F. Marty, Sur le module des coefficients de MacLaurin d'une fonction univalent, Comptes Rendus (Paris) 198 (1934), 1569-1571.

9. P. Montel, Le~;ons sur les familles normales de fonctions analytiques et leur appli- cations, Gauthier-Villars, Paris, 1927.

10. P. Montel, Leqons sur les fonctions univalentes ou multivalentes, Gauthier-Villars, Paris, 1933.

11. Z. Nehari, Conformal mapping, McGraw-Hill, New York, 1952. 12. M. S. Robertson, Multivalently star-like functions Duke Math. J. 20 (1953), 539-549. 13. M. Schiffer, Variation of the Green function and theory of the p-valued functions,

Amer. J. Math. 65 (1943), 341-360.

DEPARTMENT OF MATHEMATICS

WESTERN KENTUCKY STATE COLLEGE,

BOWLING GREEN, KENTUCKY, U.S.A.

(Received January 7, 1966)

On functions that are bivalent in the unit circle

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Transcript of On functions that are bivalent in the unit circle