ANNALES ACADEMIAE §CTENTIARUM FENNICAE

Series A

I. Å,1ÅT-§{E&äATICA

/å90

OI{ ESTIMATING OF

A FOURTH ORDER FUNCTIOI\AL FOR BOUNDBD

UNIVALBNT FUNCTIOI\S

BY

JTILIAN tr A\YR.YNOI\'trC2. and OILI TAet^{I

IIELSINKI L971SU O MAL AINE l{ TIE D }tr AKATtr MIr\

Communicated 13 November 1970 by Launr MlrneERG

I(ESKLTSKI RJAPAIN OFIELSINI{I 1971

Preface

The authors are indebted to Mr K. Kuvaja for tabulating on a computerIBM f 620 the function å* appearing in this pa,per.

This work was supported in part by a Scholarship of the X'innish lllin-istry of Education at the University of Helsinki.

1. Introiluction and statement of results

The authors are concerned with the class ,S(är) of urrivalent functions

f(") : brz I brzz + "', o < å1 < 1,

which map the closed unit disc into itself. This class has often been studiedin the equivalerrt from of functions

r : (rlbl)f ,

whose coefficients we shall denote by o*. Both classes have tleen developedquite extensively since 1950 (cf. e.g. ll4l, l4), and [9]). In this paper theauthors investigate the problem of finding the sharp estimate for the func-tional of the four-th order

B : l% - pazes + qall , p, qreal,

in d.ependence ort p, q, and år, w-here / ranges over §(är).Given teal p,q, and. ör, 0 < ö, ( I, let

M(r): -3(1 -år) [1 -5p-p2 lI2q -(11 -ep-p'tt2q)br]- åts -f p -l 2p, - t2q * (2 - 3p - ,p, { t2q)brlr

+ +(7 - 3p + 3p' - r2q)rz I !r(a - ") ({+(2 - p)'

+ åt5 - ep I t2q - (8 - rlp 4 r2q)b, - $1s - ep I t2q)fl2j8

- *fs - ep I r2q - (8 - lrp I t2q)b, - *ts - sp I tzq)xl),

where r is supposed, to be real. Further, let D d.enote the set' of pairs(p,q) for which there is a b(p, q) such that if 0 ( år {b(p, g), then thecorresponding M satisfiesthecondition M(x) <0 for 01r<2(I-br).

Ann. Acad. Sci. Fennicre A. r. 490

Table ITable of values of the functions b* estimated from below

i\6oiI u' t..l

m-4 -3 -2 -r

l

i

!

I

I

t2l110

-- 11

- t0-9__8

-l--6

-5--4

-3__ €)

-10

t2

3

4

o

6

7

II

IO

t112

.000.018.r08

.000 .009 .103

.000 .000 .096

.000 .000 .085

.000 .000 .071

.000 .000 .051

.000 .000 .025

.000 .000 .000

.000.000.000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.172 .189 .205

. r 66 .185 .202

.r60.179.197

.152 .r73 .r93

.t44 .167 .188

.134 .159 .lg2

.123 .I50 .t7 5

. 109 . 140 .t67

.086 .128 . r58

.048 .1r3 .t4i

.000 .096 .r34

.000 .074 .r19

.000 .039 .099

.000 .000 .07 4

.000.000.041

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.000 .000 .000

.220 .234 .248 .260

.2L7 .232 .246 .259

.2r4 .230 .245 .259

.zlt .227 .243 .258

.207 .225 .21L .257

.202 .22L .239 .255

. 197 .2I8 .237 .254

.t92 .214 .234 .253

. r 85 .209 .231 .25t

.177 .204 .228 .249

.168 .r97 .221 .217

.157 . r 90 .2L9 .245

. t 43 . I 80 .2t3 .212

.125 .r69 .206 .238

.103 .154 .tg7 .233

.073 .135 . r 86 .228

.032 .109 .r71 .220

.000 .073 . r 50 .210

.000 .0r6 .rr9 .r96

.000.000.07r .t74

.000 .000 .000 .138

.000 .000 .000 .067

.000 .000 .000 .000

.000 .000 .000 .000

.000 .000 .000 .000

.272.283 .294

.27 2 .283 .295

.27 2 .281 .295

.27 t .294 .296

.27 t .285 .297

.271 .295 .298

.270.285.300

.270 .286.30I

.27 0 .287 .302

.269 "287 .304

.269 .288 .306

.268.289.308

.267 .290 .31 l

.266.291 .31.{

.265 .293 .318

.264 .29ö .322

.262.297.328

.259.300.335

.256 .304 .344

.250 .309 .357

.24L .31 7 .37 5

.224 .329 .4A4

.183 .346 .450

.051 .351 .502

.000 .250 .500

.304.3r4.323

.305 .315 .325

.306 .3L7 .327

.308.3r9.329

.309 .32L .331

.3rr .323 .334

.3 r 3 .325 .33 7

.315.328.340

.317.331 .344

.320.334.348

.323 .338 .352

.326 .312 .357

.330 .317.363

.334 .353 .370

.3.10 .360 .378

.346 .368 .388

.354 .378 .399

.365.391 .4t4

.378.407.432

.396.429.457

.42t .459 .491

.46L .506 .542

.523 .57 7 .619

.594.656.701

.625 .700 .750

.332i

.ss+lI

.3361

.33e1

.sa'.zlI

.345i

.s+al

.tszlI

.356il

.3601l

.sool

.szri

.3781

.3861I

.3e51

.+oolI

.4rei

:21'.4oo

i

.481 i

.518i

.57 21

.652i

.zesi

.zs;i

In all cases listod above, where ö*(p, q) ^,

.000, rve have b*(p, q) : 0. For

-12 < 6p { 12, -12 < 6q! -4 we also have b*(pt, il : 0.

Given a (p,q) in D, let, b*(p, g) denote the least upper bound of b(p,q),while let b*(p, q) : 0 otherwise. Hence D is the set of (p, g) such thatb*(p,q)> 0. The table enclosed gives the values of b* for various pand q. For more clarity this is also given in the form of a malr in the(p, q)-plane with level-lines corresponding to fixed values of ö*.

The results obtained in this paper may be formulated. as follori-s.Theorem l. If (p, q) and, f bel,ong to D and, S(br), resgtecticely. where

0 ( ä, !b*(p,q), then the correspond,i,ng B d,oes not erceeil

B* : å(r - ail + 2(r -år),1å - Bp * 4q- (s* - 5p * aq)b,).

The esti,mate ,is sharp for euery p, q, anil, br. All the ertremal fu,ncti,ons aregi,aen by the formula

Jur,raN tr a.urn'rNo\\rycz and Ollr Tarvrur, On estimating of a fourth ord.er

§§{

*

§r

sr

-§

.!e

"O

-P

+i0)

€o

P

ho

6oP-{ac)

Sr

aa)

I

a

F]

r-

b0

l={

1l

iiilIl

.4. .4.-ir Lvvl

A§i Hr

\r'-)e ,t-§ .O

§

ii

ll

§Yr

"§

Ann. Acacl. Sci. Fennictp. A. r. 49{J

f!(") : e-bP-'1brP(et"z)), P(z) : zl$ - ")', l"l ! r, -n < c ! n .

Theorem 2. If p 3 E and f belongs fo §(ör), where

(p-2)'*p2-p-aq-l*ffiff+ < b, < t rot q < *(P' - P + Z),

while

(p-2)'-3(p'-p-4q+1\, i-. *:, =u'

< I for qZ*(P'- P * *) '

then the corresltond,ing B d,oes not erceecl,

B** : 3(1 - öi) .

?he esti,mate is sharp for euery p, q, anil, br. All the ertremal functions aregi,uen by the formul,a

f!*(") : "-*i'-t16ri1"'""71, i1"7 : zll - z3)Ps,lzl ! r, -n<c{n .

The paper is conclud.ed by few applications and remarks. In particular,the authors obtain the sharp estimates of the third coefficients of

s(z):f*,hk\:L* l"P): f@,il\z):'*'T,(4, z, 1r

Actually, these classical functionals (cf. [1], [12], and [13]) gave reason tointroduce the fourth order functional B of the present paper. The func-tional B is also an analogue of the functional lo, - poZl, p real, con-sidered in the case u'here I < p < I in [14], formula (37), and in thegeneral case in [3].

2. Proof of Theorem 1

The proof is analogous to that given in [7] for 3 - t,a4l.

We start with the estimate

(1) lan- Zarar* 13r*l* r?ur* *3@r- a,1) + zrr(ar- f;al)i

= å(t - Öi) - tbrl*ri' f 2izri'z(l - är)

| @zl2(r - bi) - 2brRe(xrdr)

whichholdsfora,yt1r n1tr, (not,necessarilyreal). Irrequalit5'(I) isasimpleconsequence of the Grunsky-Nehari inequalities (cf. [7]. inequality (+)).Let us choose fr1, fi2 to be real. Then we get

Julrerr tr A§.RrrNolilrcz and Or,lr Telrmr, On estimating of a fourth order 7

ian - 2ara, + t}"l { 2rr(a, - Z"Zl I r?a, * rZ@, - al)l

= å(t - Öi) - *b,lo,l' + 2nl(t - b1\

^ 2rrbrRe a, f xilt - t\1 .

Here the left-hand side is not less than

Re[an - 2ararl *Zo?r+ 2rr(ar- 2"3) + xlas* rllar- a!11

: Re[aa - pezas + qal]

-l- Re[(p - 2)arar- (q - l|lol+ 2rr)' | *iar* r\Q- lo|li1,

where

).:az-2"3.Therefore

(2) Re(an - para, a r1a?r)

5 å(t - äl) + (2 - 1t)Re(t,i,) + (å - Ze +q)Ir'eal-*brl*,|'

- 2rr(Re ), * btRe nr) + 2rc1\ - bt - | Re ar)

-f r30 - bi - Re 2 1- { ne a!) .

Now rve notice that la, - a,il < | - b?,, (cf. [15], formula (19)), whence

"30-öl -Re1++Reol) >0.Therefore u.e cltoose

(3) it2:0.

On the other hand, thele is tro loss of getreralitv if x-e asslrme

(4) ut - p{t.2e.. - ,ln| > (l . Re e, > t) ,

since this normalization carl alu-avs be achier-ed b1' a properly chosen

rotation. Consequentl5,, from (2), (3), and (4) s'e infer

(5) B:a+-gazas*q*}

=3(t-ti) +(2 -p)Il,e(ar)') -(1å -'*p=q)Rea?,

- äbrlo,rl'- 2r,(Iie I -l b,,Re nr) 1 2.i';(t - br - rvRe ar) .

The estimate (5) implies, in particular.

(6) r-br-åR"a2>o.

We shall consider, separatel;r, two cases: I - b, - åR" &z:0 and1-br-åR"02)0.

Suppose first that | -b, - åR" &z:0. It is u'ell known (cf. [7])that this implies f : fff, whence

Ann. Acacl. Sci. l'enniciB A. I. 490

brz I brarzz -l brarz| * bradza.l____ _ brz

(-1 + brz ! bra,rzz I brarzs + . . .)' - (L - z1z

and' consequentl/' az: 2(r - br) ,

as:3-8Ö1+5bi,an: 2(2 - 10å1 + l5öi - ?bl) .

Hence

B:10 - ai) + 2(L - b,),[*(r - 4b,) -p(s - bbr) + 4q(r - ö,)]D*

-D

Suppose next that l_ br-åR" az)O. Then we can minimizethe right-hand side of (S) b5, choosing

Re ,1 f brRe a,

'', :

2(t - br) --R"- CI,

and find the most favourable estimate

(7) B<å(t -ail+ Q-dRe(a,2)

* (r'z - ze + q),,eal- *brlo,l,-

=u _ nffi*

The right-hand side of (7) may be rewritten in the form

ått - bi) - e - dbrB"r,za2 I (rå - f;p I q)P"e o,l

- (2 - p)Im arlm ,1 - tbrknza,(Re ,1 * b, P"e ar)zr- (2 - p)Re u, (Re L - ärRe rr) - ,1, _ 4f_ n" *

and it attains its maximum with respect to Re 2 { brP"e o,, treated as

the only variable for

Re ,i * brF,a ar: (2 - p)Re nr(i *- br - *Re or; .

Hence

B < å(t - öi) - (f, -p)örRero, + (rt" - *p +q) Reaj - $brlmza,

- (2 - lo) Im arhn ). + +(2 - p)zRez«r(l - b, - $Re ar)

or, after a rearrangement,

B < å(t - ai) + *ftz - p)2 - (s - p)zbr)P"e2a,

- (?, - ie + Lp' - q)R'e3a, - (å - -asp i- 3q)It'e arlmza,

- (2 - p)Im arlm i - lrbrlmza, .

Jur,rex Lewnnrowrcz and Or,r,r Tertut, On estimating of a fourth order I

Thus rve have removed the paremeter B,e,X. In ord.er to remove Im /'we introduce the quantities

R'e o, : 2(L - br) - *'0 I x < 2(l - b)'Im ar: A ,Tm )': T - brlmar.

Under these notation we have

B = å(r - ai) + *ftz - p), - (3 - p),brll2(L - br) - r)'

- ({, - ie + Lp, - il12(r - b,) - *lu

- (å - |p * sq)12(t - b,) - r)y' - (2 - p)yq * (Z - ilb'y': co * crr { crrz * cr*'- åts - 9p t l2q

- (8 - rrp I t2q)br- t@ - ep { L2q)mlyz - (2 - p)yq ,

where cs, c1, c2, c, do not depend on tr. Direct calculation gives

co: 3(r - ai) + 2le - p), - (B - p,)br7 (r - ö,)'

- s(å - *e + *p' -() (I - br)'

: å(r - ai) + 2(r - br)rtå - Bp * 4q - (2ro - 5p t aq)br)

:B*,cr: - 2lQ - p)z - (3 - p)'brl (1 - är)

+nGaz-Le+*P'-ilG-br)z: - (1 - år) [I - 5p - p' * t2q- (11 - 9p - p2 | t2q)brf ,

c, : *lQ - p)z - (3 - p)'b,.) - 6(& - äe + *p' - q) (I - ÖJ

: - å[g * p * 2p, - r2q + (2 - 3 - 2p, I t2q)br),

cs:tzz-ie+*P'-q:htt-Bp*3p2-L2q).

Therefore

(s) B - B* ( - (1-ör)U-5p-p'{t2q-(rr-e'p-p'1,t2q)br)r

- åte -l p * 2p'- r2q + (2 - 3p - ,'p'{ t2q)brlrz

+ å(7 - 3p * 3p' - L2q)uB

- Lfr - ep I L2q - (8 - 1110 ! t2q)b,

- *(s - sp * L2q)r)yz - (2 - p)aq .

I{orv we apply the id.entity

t t-tp Y- (2 - p)yr : lt - *pl@y, + *.n\- 11 - +el"\y + a _ir1 n)

Ann. Acad. Sci. Fennicre A. r. 490

wjth a free parameter a ) 0. Clearly,

, , --LO \,ir --lp',"\,u + ;;Or, ( o,

whence

(z - p)art < 11 - tpl@y, * n-rrt\ .

We utilize further the estimate (cf. [T], inequality (27,))

,f{*r-*(", lyz).Therefore

(e) - (2 - p)yr < lr - *pllt$" - rz -,az)o.-L I arol.Besides, it is easily seen that

(10) *(+* - 12 - yz)6r-t * Azot > 2lyl L*$" - rz - yz11|

whenever 4x - rz - y2 > 0, but this inequality must hord since the right-hand side in (8) is bounded from below for any a ) 0, and for the term- (2 - p)yq in (8) we have derived the estimate (9). If 4r - rz - y, > 0and y t' 0, in order to obtain equality in (10) u,e have to choose

a: ll@x - trz - A\y-rlä ,

whence (9) becomes

(tr) - (2 - p)yq { 12 - pl lyll}1$* - rz - sz))b .

If 4r - rz - Az : 0, (f f ) is an immediate consequence of (9). Finallv,if y: g, (11) is trivial. Combining (tt) with (8) we obtain

(12) B- Bx < - (l -ör)[l -5p-p2-lt2q- (il -9p-pzit2q)br]råts + p i 2p2 - t2q + (2 - Bp - 2p, -; t2c1)br)t:2

+ å(7 - 3p i 3p, - r2q)fi

- äfs - ep * t2q _ (8 _ lrp { L2q)b,

- *ts - ep * r2q)rlyz + lz - pl lyll+$c: - rz - yz))b .

rn order to find for fixed * the maximum of the right-hand side in(12) with respect to g we consider the function

(rs) Q@): - Uar+ 12-plyy -fu)*,-ey)+{a3(zV)+,where

U:*15 - ep * t2q- (8 - trla I t2q)br- *tr _sp * t2q)rl,V:Lar(4-r).

t0

Ju:-rax tr-awnyNorl,rcz and Or,r,r Tervnrr, On estimating of a fourth order ll

Since @ is even in y, we may, without loss of generality, assume thaty 2 0. Straightforvrard differentiation lead,s to the following conditionfor the value of y at internal extrema:

(14) it - LelU - *a\l(Y - ta\L : LTy .

Solving this equation for y, we obtain two possibilities:

yi: gv(r - {u,lL+Q - il, + url}L) ,

yZ: tv(r + {arfi+(2 - p)z + u,)}b) .

If U > 0, equation (1.4) requires 7 - ?y, > 0 since we look for positivevalues of y. Tn this case only yz will be permissible. If U < 0, we haveto demand V - *y, < 0, which leads to the only possibility yl. Con-sequently the internal extremal point, for Q satisfies the equation

(15)

Since

y2 : *T/{L U IL+Q - il'+ Url*}

Q'Q) : o,Q'(Y)--> - oo as Y ---> $V)*,

relation (15) leads to a maximum of Q, and. we find from (13) and (la)

max Q(y) : *V{l+e - r))z + 71218 - U} .

Thus from (12) 'r,r-e obtain

B- B* < - (l -ör)[1 - 5p-pzlt%q+ (2-3p-2p'{L2q)br]r

- åts * p * 2p, - t2q * (2 - zp - 2p, 1 L2q)br)rz

+ å(7 - z,p * zp, - rzq)r3 + ZV{l+e - d, + u\+ - U}

: - (1 - ör)[1 - 5p - pz * t2q- (ll - 9p - p2 { r2q)br)r

- åtS * p * 2p, - tzq. + (2 - 3p - rIt, 1 t2q)brlrz

+ hU - 3p + 3p' - L2q)r3 * **(+ - .u) ({}(z - p;z

+ å15 - ep * L2q - (8 - IIp ! L2q)br- åt; - sp i r2q)xlz';b

- *lr-sp * L2q- (8 - Lrp * r2q)b, - åt; - ep ! rzq)x))

: lrM(r) .

If (p, q) belongs to D and 0 ( är !b*(p, q), then, by the definitionof il[, we have M(*)< 0 for 01r<2(L-b1). Since r has beenrestricted to this interval, B - B* < 0, as desired.

X'inally we notice that, by (5), where we have assumed (4), in ord,er todemonstrate that B : B* can only hold for f : ff it is sufficient toshow that equality in (6) can only hold for f : /f , but this is a well known

t2 Ann. Acad. Sci. Fennicm A. r. 4ga

result (cf. [7]). On the other hand, for any f!, -n ( c ( z, we haveB : B*, as it u'as verified before by direct calculation. Thus the proofis completed.

3. Prool of Theorem 2

The proof is analogous to that given in [7] for B : lonl.We again start with the estimate (1), where we put 11 : årar, rf,: irar,

and choose ir, i, to be real. Then we get

lan I Qi, * il, - 2)a,a, { 6? - Z;, - ä, * !*l"lt<å(r - ail + lä,1(r - b?)1",1+lzöie - ö,) - 2;Lq-Lb,.7lo,l,.

Here the left-hand side is not less than

Refan f Qi, * å, - 21ara, + 6i - *å, - ä, + !*)ol1.We choose ä, so that zilr, * å, - 2 : - ?, i.e.

är:z-P-2L.Therefore

Relao - pazua * tii + *i, -f p - #1"'rl

= å(r - ai) + lz - p - 2ä,1 (t - b?)la,l

+ l2;?Q - år) - zåLbL - *br)lo,l, ,

whence

(16) Re(nn - para, { qasr)

< å(r - ail + lz - p - 2r,ie - bi)',o,'

+ l2;i$ - b,) - 2;&L - *b'),,,,'',

- tii + *å, + p - q- ]]yRe aj,

I{ow we notice that there is no loss of genera}ity if rre assume

(17) ct4- pttzcrs* qaur> 0, a,r: tt I i,u ,u ! 0 ,

since this normalization can always be achieved bv a properly chosenrotation. Consequently, from (16) and (17) we infer

B:at-?azas*q"ZS B** + 12 - p - zir1lr - b1)@' + o\l + l2nl1 - br)

- 2;,Lq - lzb,.f(u, * a,) - (ii + ti, + p - q - ++)(u, - Buaz),

Ju1us tr-awn:rNowrcz and Or,r,r Telrur, On estimating of a fourth order 13

where B** : åtr - ail. Since (1 - Öi) fuz { a\t 2 0, we choose ä,

sothat 2-p-2är:0, i.e.

Hence

or. after a, rearrangement,

(18) B - B** S*"rL(p - 2), - (p - 3)'br- *(p'- p - +q + *)ul

+ *rrl(p - z), - (p - z)rb, + *(p, - p - 4q + *)"1.

\Ve shall consider, separately, two cases:

fn both cases, by (6) and (17), we have

2(L br){u<0.

(1e)

and

(20)

(21)

Suppose first (19). By (21), in ord.er to obtain

(22) äurl@ - 2), - (p - 3),br- *(p'- p - +q * *)"1-t *o,l@ - 2), - (p - 3),b, + Z@' - p - 4q + *)%1 < 0

we hal.e to assume

(23) (p - 2)r- (p- 3),Ö, - L@, - p - 4q + ål (- 2)(r - br) ( o

and

(24) (p -2)'- (p - 3)'br*3@'-p - 4q +*) '0 < 0.

Irrequalities (23) and (24) are equivalent to

respectivelv, wltere, by (21), s'o should assume

(25)

ancl

(26)

(27)

t4 Ann. Aead. Sci. Fennica: A. I. 490

and

(28) (p - 2)'l (e - 3), < 1 .

On the other hand, as it can easily be verified, corrditions (19) ancl (28)imply

(p - 2)' . (p - 2)' * p' - p - aq I !@-3Y > @-3)r+er-e-4q+*

and (27). Consequently we have to assume (25) and. (28), or, what is thesame, (25) and p < $. Under these conditions we can asser-t (22).

Suppose next (20). By (2I), in order to obtain (22) we have to assume

(2s) (p - 2), - (p - 3),br-t@, - p - 4{t + å) .0 < 0

and.

(30) (p-2)r- (p-3),b,t*@r-p- 4{t +å) (- 2)(r - ör) ( o.

Inequalities (29) and (30) are equivalent to (26) and

(3r) nr=: (p - 3)' - 3(p' - p - 4q + +)'respectively, 'where, by (2I), w-e should assume (28) and

(P-2)2-3(P2-P-4qt1\132) \t L !' </ 1\"-t (p_ B)r_B1pr_p_Eq+Z): -'

On the other hand, as it can easily be verified, conditions (20) and (28)imply

(p - 2)' .\!_)'- s(e'- e - aq +rr)(p - 3)' : (p - :))z - 3(p2 - p - +q + *)

and (32). Consequently rl,e har-e to assume (31)and (28). or, rrhat is the same,(31) and e =*.

Under these corrditiorrs 'w'e can assert (22).Inequalities (18) and (22) 5'ield B - B** < 0, as desired.Finally we notice that, since (17) gives rro loss of generality, in order

to demonstrate that B : Bxx can onlv hold for f : ff* it is sufficientto shorv t}i.at B: .B*{< with additional corrditiorr (I7) can only hold forf : ,f§*. To this end we observe first that if

(p-2)zlp'-p-4q**t - rr +ffi( b' ( I for q si.tp' - p * t)'

rvhile

(p-2)r-3(pr-p-4q+7\ffi-nq;g <å, --< t ror qz*(p'-p*E),

Jur,rant trÄwnyNo$acz and Ou,r Taurur, On estimating of a fourth order 15

therr the expressions in the squa.re brackets in (22) are negative. Conse-quently (I8) yields that B - B** can only vanish for a, : u * i,a : 0.

Hence we must seek the extremum functions in the subclass of §(är)with or:9. Here, by (17), we assume that au-pazas+qoZ>o, i.e.aq,> O. For such functions we apply the estimate (1) with rr:ird,s,rl : irdr, and choose ilr, i, to be nonnegative. Then we get

an{ frrlarlz + zfrLlclnlz

< 3(r - ai) + 2fri$ - br)la,l, I fr,(r - b?)larl ,

u'hence

B - B** ! frr(r - bi)larl - 21fr, - i?Q - br) - tir)lorlr .

Therefore, if we put frr: t and frr: 0, the last, inequality becomes

B-B**{-2brlarz,where equality is possible only for functions in §(ör) rvith o, : e,r: g

and an) 0. Now u,e apply a known result (cf. [I5], pp. 13-14) that if./ belongs to §(är) and satisfies the conditions o, : as:O and an) 0,

then /:/f*. On the other hand, for any f!*, -, 1c{n, we haveB : B*tr, as it can easily be verified by direct calculation. Thus the proofis completed.

4. Conclusions

We begin with the applications announced in Section 1. Let

Flence, lry the definition of g,

1 + Zarz f Sarzz { 4anz3 + +I1 * azz * asz' * &+23 +

ancl, colrsequentl;,,

Bo: L, Br: az, Bz: 2a, - c['2, Bs: Scto - 3ct,ra, I a',

Therefore, by Theorem 2,

Next, let

Ilence, by the definition of h,

16 Ann. Acad. Sci. Fennicre A. I. 490

Zarz { 6arzz { L\anz\ +- Oo + C& + Czz'* Crr'+1+

1 + Zarz { Sarzz { 4anz3 +aud, consequently,

Ca: L, Cr: 2a,2, Cz: 6o, - aotr, Cs: L2an- L&arar! 8ol

Therefore, by Theorem 2,

lcrl < 8(r - u1),LS ä, ( r.tr'urther we remark that in Theorems I and 2 t'he intervals for Ö, can

be improved if we do not restrict the parameters r, and r, in (1) to realnumbers. However in this case calculations are much more complicated.Furthermore, one may try to generalize Theorems t and 2 by consideringcomplex p and. q.

Finally we remark that the same method. can be applied to analogous

functionals of higher ord,ers. A counterpart of Theorem 2 for the functionalof the fifth order

lau - paran - q"? ! raia, - sct'ni , p, Q, r, s real

is established. in our forthcoming paper [5]. ilore generally, rre introducean analogous ra-th order functional as follou's.

Let f be a function in §(år). Define the coefficients ,4-* 1oy therela,tion

- f(z) - l@o) . ,.l"e"-;; : ^,ioA^*"32^,1"1<

l, izol < l.

Clearly, A^r": Ar* for any (m, k). It' is rvell known that the coefficientsA-o pla.r an important, role in the Grunsky-Nehari inequalities (cf. [2],16l, [8], and l9l). In particular (cf. [8], p. 4), we have

Aoo: logÖ1 , Aoz:or-$af;, Aoz:&4-(l2ctl*å'3 ,

A$: a2, Au: ct"- ai, Arr- 0+- 2«'r«r-,_ a!'

A pollrromial of n-l variables e2 1...,ctn, n22. is said to be

related, Lo A*1", m { h: n, if(i) its coefficient at a, is 1,

(ii) each other of its coefficients is 0 if the corresponding coefficierrt

of A*1r, m. I k: n, is 0.

It carr easily be shown that if a pol;,reomial is related to A^y m I lc : n,then it is related. to each A*,t",, ffi' I k' : n. The modulus of a pol5'-

nomial *hich is related to some A^k, ffi + k : rz, calculated at the point(ar, . . . , ctn) determine d by some furrction / of S(Ör), is said to be an

n-th order Grunsky functional. The coefficients of this pol;rromial are calledthe coeffi,cients of the corresponding Grunskv functional.

(33)

Jur-rerv tr-awr,r-Nowrcz and Orrr Terrntr, On estimating of a fourth order 17

In view of the results obtained in [10], [I1], and [8] it seems to theauthors natural to pose the following:

Conjecture. Suppose that n,n)-Z, 'is an arb'i,trary 'i,nteger and,

B(arr...raoiPrgr...)

is an n-th ord,er Gruruky functi,onal wi,th coffici,ents gt, Q, . . . 7n the Euclid,ean(p,q.,...)-space there'i,s a neighbourhood, D. of (0,0,...) such that i,f the

point (gt, q,, . .) and, a furwti,on f correspond,ing to (ut, . . ., a^) belong to

D. and, S(br), respectiaely, then (33) d,oes not erceeil,

2(34) B^: n_t(I -å?-')for b, in some interuul

b"(p, q) < ä1 < 1, 0 < b"(p, q) < | .

The estimate is sharp for eaery ?, !1, . . . , and, bl. All the ertremal functionsare g'iuen by the formula

2

1","(") : e-i"P;rlbrP^("*")), P^(z) : zlG - z"-')^ ,

lzlll,-it<c{n.The authors believe that the estimate of (33) by (3a) can be established

with help of the method applied in [8], while the totality of the extremalfunctions can be determined. v'ith help of the method applied in [I5].These methods, however, ma5, fail in view of techtrical difficulties.

Institute of Mathematicsof the Polish Academl. of Scierrces,

Laboratory in 1,6d2,tr-.6di,, Poland

University of Helsinki,Helsinki, Finland

References

[1] GolusrN, G. M. (lonyann, l. M.): O Hootp(puqueurax o,quo,'rrrcrrrnx {yunquit. -

Mat. Sb. N. S" 12 (1943), 40*47.[2] GnuNsxv, IL: I(oeffizientenbedingungen fiir schlicht abbildende meromorphe

Funktionen. - Math. Z. 45 (1935),29-61.[3] Jerusows:<r, Z, J.: Sur les coefficients des fonctions univalentes dans le cercle

unit6. - Ann. Polon. Math. 19 (1967), 207-233.[4] JaNowsrr, W.: Le maximum des coefficients A, et ,4, des fonctions univalentes

born6es. - Ibid. 2 (1955), 145-160.[5] tr,twn,vrvowtcz, J, and T-rurrrn O.: On estimating of a fifth order functional for

trounded univalent functions. - Colloq. Math. lto appear].[6] Nnuenr, Z.: Some inequalities in the theory of functions. - Trans. Amer, Math,

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Sci. Fenn. Ser. A I 435 (1968), 26 pp.[9] --»- On the coefficient problem for bounded univalent functions. - Trans.

Amer. Math. Soc. 140 (1969), 461-474.[0] Srnvrrtnsrt, L.: The local solution of the coefficient problem for bounded schlicht

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[1I] -» Sharp estimation of the coefficients of bounded univalent functions nearthe identity. - (a) Bull. Acad. Polon. Sci. S6r. Sci. Math. Astronom.Phys. 16 (1968), 575-576. (b) Dissertationes llath. Rozprawy Mat.fto appear].

[2] Telrlrr, O.: On the maximaliza,tiort of the coefficients of schlicht and relatedfunctions. - Ann. Acad. Sci. Fenn. Ser. A I 114 (1952), 51 pp.

[3] -»- On certain combinations of the coefficients of schlicht functions. - Ibid.r40 (t952), t3 pp.

[f 4] - » - On the maximalization of tho coefficient a, of bounded schlicht functions.- Ibid. I49 (1953), t4 pp.

[5] -»- Grunsky type of inequalities, and dotermination of the totality of theextremal functions. - Ibid. 443 (1969), 20 pp.

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