On an estimate of Robertson’s functional in the class of odd bounded univalent functions

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PR ACE MATEMATYCZNE XXIX (1990)

Krystyna Zyskowska (Lôdz)

On an estimate of Robertson’s functional in the class of odd bounded univalent functions

Introduction. Let us denote by S the family of functions of the form F (z) = z + A 2 z2 + A 3 z3 + ... + An z" + ...,

univalent and holomorphic in the disc E = {z: \z\ < 1}. Let S{2) stand for the class of odd univalent functions having in E the expansion

(1) H(z) = z + C3z3 + C5z5 + ... + C 2n + 1z2" + 1+ ...

It is well known that H e S(2) if and only if there exists a function F e S such that

In 1936 M. S. Robertson ([6]) posed the conjecture that n — 1

1+ z ^\C 2^{k + i \ 2} n = 2, 3 ,...

fc=i

For n = 2, it follows immediately from the well-known result of K. Lôwner ([4]); for n — 3, it was solved by M. S. Robertson ([6]), and for n = 4 — by S.

Friedland ([2]). In the general case, the proof of Robertson’s conjecture was obtained by L. de Branges in 1984 in his paper [1] concerning the solution of the famous conjecture of Bieberbach. This result additionally emphasizes the importance of Robertson’s problem.

Let S(M) be a subclass of S of functions satisfying in the disc E the condition |F(z)| < M, M > 1. Denote by S(2)(y/M) the class of odd univalent functions of form (1), bounded by v/M , that is, \H(z)\ < у / м , z e E .

. Evidently, for each function F e S (M), the function H (z) = yJF{z2) belongs to Si2)(y/M) and vice versa.

Making use of this relationship, we obtain .

(2) С3 = Ы 2, C5 = H A , - i A 22).

15 — Roczniki PTM — Prace Matematyczne XXIX

342 K . Z y s k o w s k a

From the well-known estimate of the modulus of the coefficient A 2 in the class S{M) one knows that ([5])

(3) |C3Kl-77> M>1,

the Pick function w = P(z,M ),

z e E , |e| = 1.

One also knows the estimate of the functional \A3 — аАЦ, for any real a, in the class S(M) ([3]); in the case a = 1/4, the maximum of this functional is not attained for the Pick function.

In view of the above facts, it seems purposeful to determine the maximum of the functional

with equality in (3) holding only for P(0, M) = 0, given by the equation

(4)

)2 (l+£w)² ’

(5) #-(Я) = |С3|2-НС5|2

in the classes S{2){y/M), M > 1.

In the paper we shall be occupied with this problem in the case when M ^ 3; the studies on its solution for the remaining M, i.e. M e( 1, 3), are being continued.

1. Let us observe that relations (2) and the properties of the classes S(M) imply that the determination of the maximum of functional (5) is equivalent to the determination of the maximum of the functional

(6) G(F) = { \A212 + [Re i (A 3 — i Л2)]2, FeS(M ).

Of course, for this purpose, it suffices to determine the upper bound of G {F) in the subclass S* (M ) of S (M) of functions of the form (cf. the Lôwner theorem ([4]))

F (z) = lime1 J{z, t), m = logM,

t-1-ш

where f(z , t) is a holomorphic function of the variable z in the disc E, I f( z , t)| < 1 for z e E, /(0 , t) = 0 and / / ( 0, t) > 0, and f(z , t) is, for 0 ^ t < m, a solution of the equation

f = A + k f dt J 1 - k f ’

satisfying the initial condition/(z, 0) = z. The function к — k(t), |/c(f)| = 1, is any function continuous in the interval <0, m> except a finite number of points of discontinuity of the first kind.

Since the coefficients A 2 and А ъ of functions of the class S*(M) are expressed by the formulae (cf. [4], [3])

m

A2 — — 2 J e~xk(x)dx, о

m m

A 3 = — 2 J e~2xk2(x)dx + 4(j e~x k(x)dx)2, m = logM,

о о

therefore it follows from (6) that we have to determine the maximum of the expression

m m

(7) G(F) = (f e~x cos в (т) dx)2 + ( j e~T sin в (т) dx)2

о о

m m

+ 4{3(J e~x cos 0 (x) dx)2 — 3(J e~xsin9(x) dx)2

о 0

m

— 4 J e~2xcos2 0(x) dx+1 —e~2m}2, 0

where 0(x) = argfc(i), 0 (t)g<O, 2ti), over all possible functions k(x) satisfying the assumptions of the Lowner theorem cited above.

In the further part of the paper, we shall make use of the following two lemmas ([3]):

Lemma A. I f : 1° A(t) is any real function of a real variable x, defined and continuous in the interval <0, m> except a finite number of points of discontinuity of the first kind; 2° |A(t)| ^ e~x for x e (0 , m); and 3°

m

(A. 1) J ^{X2 (t})dx ^ m e ~ 2m,

0 then

m

(A.2) (J X(t)dx)2 ^ (me~2m — ve~2v)m 0

where v, 0 ^ v ^ m, is the root of the equation

m

(A3) J X2(x)dx = me~2m — ve~2v.

0

For each v e (0 , m>, there exists a constant function X(x) = c such that in (A.2) equality holds. Then the relation me2 = me~2m — ve~2v must take place.

LemmaB. I f a function a(t) satisfies assumptions 1° and 2° of Lemma A and the condition

m

(B.l) \X 2(x)dx> m e~2m,

0

344 K. Z y s k o w s k a

then we have

(B.2) JJ Я(т)dx\ < (v + \ ) e - v- e ~ m о

where v, 0 ^ v ^ m, is the root of the equation

m

(B.3) j X2(x)dT = (v + %)e~2v — ²^~2m.

о

Estimate (B.2) is sharp for every v and the equality sign occurs only if А(т) = ± ^ (t) where

. . fe~v for 0 ^ t < v, Ж = < _ T , . .

[e lor v ^ t ^ m.

Put

m

x = J e-Tcos Q(t)cIt, о

m

(8) у — J e_ tsin0(i)dT,

о

2(t) = e~Tcos Q(t).

From the properties of the function к(т) and from the definitions of the functions 0 (t) and Л(т) it follows that Л(т) satisfies assumptions l°-2° of Lemma A and, moreover, either (A.l) or (B.l).

If condition (A.l) holds, then, in virtue of (A.3), (A.2) and (8), we have

(9) 0 ^ x2 ^ X a(v)

where

X A(v) = m(me 2m — ve 2v).

The function X A(v) is decreasing, where v e ( 0 , m ) when me(0, j ) and v e ( 0 , v*) when m > j, v* is the root of the equation OA(u) = 0 where

Qa(v) = me~2m — ve~2v. The function X A(v) takes its values from the interval

<0, m2 e~2m}.

Since from (8) and (A.3) we have m

J e~ 2z sin2 в (z) dv = ^(1 —

m

e~2m)— § e~2zcos2 в (т)dx = ^(1 — e~2m]

0 0

it follows from Lemma A that

(10) 0 ^ У2 ^ Ya (и)

where

YA{u) = m(me 2m — ue 2u), u = QA 1[ \ { \ —e 2m) — QA(vj].

It is not difficult to verify that YA(u) is an increasing function, taking values from the interval (0, m2e~2m}.

If condition (B.l) holds, then in virtue of (B.3), (B.2) and (8) we have

(11) 0 < x 2 ^ 2 f B(t;)

where X B(v) = [(y + \)e~ v — e~m]2. The function X B(v) is a decreasing function of the variable v, 0 ^ v < m. Besides, X B(v) takes its values from the interval

<m2 e~2m, (1— e~m)2)>.

Denoting QB(v) = (y + ²)e~ 2v—je ~ 2m, analogously as above we obtain

(12) 0 < у 2 < У в(н)

where

YB(u) = [(m+ 1)е-“- е ~ т]2> и = Од 1 Й(1 —e“ 2m)—Ob(ü)].

The function YB(u) is increasing and takes values from the interval

<m2e~2m, (1— e~m)2}.

The conclusions above, resulting from Lemmas A and В and concerning the properties of x2, y 2 defined by equalities (8), will be helpful in our further considerations.

When seeking for the maximum of the functional G (F) defined by formula (7), we shall base ourselves on Lemmas A and B, taking account of the inequality x2 + y2 ^ (1— e~m)2> m > 0, following from (3). We shall also use some elements of the method applied by M. S. Robertson ([6]).

2. Suppose first that condition (A.l) holds. So, consider some subset of functions 0 (t), thus some subset of functions k(t), and in consequence, some subclass of the family S(M).

In virtue of (A.3) and (8), expression (7) takes the form (13) G(F) = jtf(x 2, y 2, v)

= x2 + y2+ i[ 3 x 2 — 3y2 — 4(me~2m — ve~2v) + l — e~2m~\2, m = log M.

It follows from notations (8) we have adopted as well as from inequality (3) that

x2 ^ (1 — e~m)2— y 2.

On the other hand, by (A.2) and (9), we have x2 ^ X A (v).

By using the above relations and the properties of x2 and y2 following from Section 1, seeking for the greatest value of the expression s f ( x 2, y2, v) defined by equality (13) will be reduced to the determination of the maxima of some functions of the variable v for admissible values of v.

Since s i (x2, y2, v) is a quadratic function of the variable x2, therefore, for fixed admissible y2 and v, we have

(14) s f(x 2, y2, v) ^ s f (min {X A{v); (1 - e ~ m)2- y 2}, y2, v)

346 K. Z y s k o w s k a

or

(15) s /{ x 2, у2, v) ^ sf(0, y2, v).

Note that if m ^ log 3, then

(16) (1 — e~m)2 — X A(v) > m2e~2m, ue< 0, v*};

consequently, it follows from (₁₀) that then the inequality 0< ( l - e - " )2- y 2 ^ * »

does not hold.

If

(17) x A( v ) ^ ( l - e ~ m)2- y 2,

then, taking account of the form of X A(v), from (14) and (13) we have (18) sé{x2, y 2, v) ^ X A(v), y 2, v)

= m(me~2m — ve~2v)-\-y2+ ^{(3m — 4)(me~2m — ve~2v) — 3y2 + 1 — e~2m}2.

The function on the right-hand side of (18), being a quadratic trinomial with respect to y 2, attains its maximum at the end-points of y2. From (17) we have

0 ^ y2 ^ (1 —e~m)2 — X A(v),

and since, for m ^ log3, inequality (16) holds, therefore from the properties of the function YA(u) it follows that

0 ^ y2 ^ m2 e~2m.

In consequence, from (18) we have

stf (x2, y2, y)< j/ (X A(v), m2 e~2m, v) = A t (v) or

^ ( x 2, y2, V) < X A(v), 0, v) = A 2(v), whereas from (15) we have

s # (x2, y2, v) ^ stf (0, m2 e~2m, v) = A 3 (v), or

л /

(x2, y2,

<

(0, 0,

) =

where, in virtue of (13),

A x (v) = m(me~2m — ve~2v) + m2 e~2m

+ i [(3m — 4)(me~2m — ve~2v) — 3m2e~2m + 1 — e~2m]2,

A 2 (v) = m(me~2m — ve~2v) + l[{3m—4)(me~2rn — ve~2v)+ l — e~2m]2, A3 (v) = m2e-2m + ^ [ — 3m2 e~2m — 4{me~2m — ve~2v) + 1 — e~2m]2,

Aa{v) = 4[ — 4(me~2m — ve~2v) + l —e~2m]2,

where m ^ log3, ue<0, v*) and v* is the root of the equation me~2m — ve~2v = 0 (cf. Section 1).

Fixing m ^ log3 and examining the functions Ak{v), к = 1, 2, 3, 4, for v e{0 , v*), we successively obtain

(19) A 1(v) ^ ^i(O) = 2m2 e~2m+ ^[4me~2m — (1— e~2m)]2 = A(1)(m), (20) A 2(v) ^ A2(0) = m2 e~2m + i[{3m — 4)me_2m+ l — e~2m]2 s A(2)(m), (21) A 3(v) ^ A 3(v*) = m2 e_2m + 4 [3m2 e~2m—(1 — e~2m)]2 = A(3)(m), (22) A 4 (v) ^ A 4 (v*) = i ( l - e ~ 2m)2 = A(4)(m).

Consequently, if we make use of Lemma A, then, for a fixed m ^ log 3, the maximum of the functional G(F) does not exceed maxk=1 2 3 4Л(к)(т).

3. Proceeding similarly as in Section 2, we shall now use Lemma B. So, suppose that condition (B.l) holds; then in virtue of (B.3) and (8), expression (7) will take the form

(23) G{F) = @(x2, x 2, v)

= x 2 + y2 + {{3x2 — 3y2 — 4[(v + $)e~2v—| e " 2m] + 1— e~2m}2.

As before, it is known that x2 < (1 — e~m)2 — y2, whereas by (B.2) and (11), x2 < X B(v). Treating J?(x2, y 2, v) as a quadratic function of the variable x2, for fixed admissible y2 and v, we have

(24) (fft (x2, y2, v) ^ ^(m in{JfB(t;); (1 - e ~ m)2- y 2}, y2, v) or

(25) @(x2, y2, D) a ( 0 , y2, v).

If

(26) 0 ^ (1 — e~m)2 —y2 ^ X B(v),

then from (24) and (23) we have

(27) ^ ( x 2, y2, u) ^ ^ ( ( l - e ~ m)2- y 2, y2, u)

= (1 ^— 6 ~ m ) 2 + 4⁽3 ( 1 ^— £?~m)2 ^— 6y2 ^— 4 ^{£ (}u- \ -2)e ~ 2v ^— 2& ~ 2m ~\ -f- 1 ^— e ~~2mJ 2 .

Since, by (26),

(1 - e - m)2- X B(v) ^ y2 < (1 - e ~ m)2,

therefore from the character of the function on the right-hand side of (27) we obtain that

m (x2, y2, v) ^ a ( ( l - e ~ m)2- y 2, ( l - e ~ m)2~ X B(v), v) = B,(v) or

(28) ^ ( x 2, y2, V) < Щ(1 — е m)2 y2, (1 - e m)2, v) = B2{v),

348 R . Z y s k o w s k a

where

(29) В, (!>) = ( 1 - e - ”)2

+ [(3v2 + 4v + 2 )e -2v- 6 ( v + l ) e ' ve - m + 4 e -2m- ( l - e ~ m) ( l - 2 e - m)]2, (30) B2(v) = ( l - e ~ m)2

+ [(2v + l ) e - 2v- e - 2m + ( l - e - m) ( l - 2 e ~ mïï2, 0 ^ v < m.

If

(31) XB( » K ( l - e - " )2- y 2,

then, taking account of the form of X B{v), from (24) and (23) we have (32) & (x2, y 2, u) ^ @(XB{v), y2, u) = [(v + 1)e~v — e~mY + y2

+ 4 (3 [(i> + l)e~ v — e~m]2 — 3y2 — 4 [(v + 2)e ~2v- ' i e ~ 2m] + 1 — e~2m}2.

Since, by (31),

0^ у2^ (1- е - '" )2- Х » ,

therefore from the character of the function on the right-hand side of (32) we get that

@ (x \ y2, ») < & (X B(v), (1 ») 3 B, (») or

(33) @(x2, y 2, (XB (v), 0, v) = B3 (u), where

(34) B3(v) = [(v + l)e~ v- e ~ m]2

+ i[(3v2 + 2 v + l ) e - 2v- 6 ( v + l ) e - ve ' m + 4e~2m + l] 2 ' v e ( 0 , m), B 1(v) is defined by equality (29). >

From (12) and the properties of the function YB(u) it follows that 0 ^ y2 ^ (1—e“ m)2. Hence and from condition (25) we get

(35) 0Hx2, y 2, v) < &(0, (l — e~m)2, v) = B2(v) or

^ ( x 2, y2, v) ^ ^ (0 , 0, y) = B4(v), where

(36) B4 (») « J [2 (2u +1) e - 2y - e - 2m - 1] 2, v e ( 0 , m), B2(v) is defined by equality (30).

Let us fix m ^ log 3 and occupy ourselves with the determination of the maxima of the functions Bk{v), к = 1, 2, 3, 4, for v e ( 0 , m>.

Consider first the function B 1 (u) defined by formula (29). We have B'x(v) = -4 v e ~ vg1(v)-hl (v)

where

g1 (v) = (3v+ \)e v — 3e m,

hi(v) — (3v2 + 4v + 2)e~2v—6(v + i)e~ ve~m

- H 4 e “ 2 m - ( 1 - e ~ m) ( 1 - 2 e ~ m) , O ^ v ^ m .

Note that if m ^ lo g 3 , then g1{v)'^0 for v e ( 0 ,m ) , and h i(v )= -2 v e ~ vg1(v),

with that, for m ^ log 3,

hx (0) > 0, hx (m) < 0.

Consequently, in the interval <0, m) there is a point v such that h1 (i;) = 0.

From the form of Bi(v) it follows that the function Bl (v) has then a local minimum at the point v. In consequence, the greatest value of B l (v) in the interval <0, m> is attained at the end-point of this interval. It is not hard to check that if m ^ lo g 3 , then B 1(0)~B 1(m) > 0, thus

(37) 5 , {v) < B t (0) = (1 - e ~ m)2 + [(1 - е ~ т){1 - 2 e “ M)]2

= 5 (i)(m), m ^ log 3.

If we consider the function B2 (v) defined by formula (30), we shall obtain (38) B2(v) ^ B2(0) = (1 — e-m)2 + [(1 — e~m)(2 —e~m)]2 = B(2)(m).

Let us next occupy ourselves with the function B3 (v) defined by formula (34). The derivative of this function is of the form

B3(v) = —ve~vÊ(v) where

Ê{v) = 2 [{ v + i)e -v~ e ~m]

+ [ ( 3 v - l) e " 0- 3 e ”w] [(3v2 + 2v + l)e~ 2v — 6 (v+ l)e~ v e~m + 4e~2m + 1], v e (0 , m).

Since ê {0) = -~2e~m(l — e~m)(l — 6e~m), therefore 5(0) ^ 0 when log3

^ m < log 6 and 5(0) < 0 when m > log 6. It can be shown that Ê(m)> 0 when m > log 3. Examining the function В (у), after arduous considerations one obtains, in consequence, that if log 3 ^ m < log 6, then В (u) ^ 0, whereas for m > log 6, the function Ê(v) has exactly one zero i’oe(0, m). Hence it appears that, for log 3 ^ m ^ log 6, the function B3 (u) is decreasing; if m > log 6, then

350 K . Z y s k o w s k a

B3(v) has a local maximum at the point v0. Summing up, we get B3(v) <£ B3(0) = ( l - e-" ) 2 + [ ( l—г - ”)( 1 -2 е-" )]2

= B(1)(m) when log3 ^ m ^ log 6, (39) B3(v) ^ B3(v0) = B(3)(m) when m > log6,

where üoe(0, m) is the root of the equation Ê(v) = 0, that is,

(40) 2[(v0 + l)e~Vo — e~m'] + [(3v0 — l)e~Vo — 3e~m'] [(3i>o + 2t>0 + l)e -2,;o

— 6(v0 + l)e~voe~m + 4e~2m+lJl = 0.

In the case of the function B4(v) defined by formula (36), in an easy way one obtains that

(41) B4{v) ^ B4(0) = i ( l - e ~ 2m)2 = J3(4)(m).

4. In Sections 2 and 3 we were using Lemma A or Lemma B, respectively, with the well-known estimate of the modulus of the coefficient A 2 in the classes S (M) being taken into account. The method applied allows us to infer that, for a fixed m ^ log 3, the sought-for maximum of the functional G (F) does not exceed the maxima of the numbers A ik)(m), B(k)(m), к = 1, 2, 3, 4, given by formulae (19)-(22), (37)-(39) and (41), respectively.

It is not difficult to verify that since m ^ log 3, therefore the following inequalities are satisfied:

Aw (m )< B {1)(m) for к = 1 , 2 , 3 , A(4)(m) = B(4)(m) < B(1)(m).

From the examination of the function B3(v) in Section 3 it also follows that if m > log 6, then

B(1)(m) < B3(v0) = Bi3){m).

Hence we obtain that, for log 3 ^ m ^ log 6, G(F) < max {B(1)(m), B{2)(m)}, whereas, for m > log 6,

G(F) ^ max {Bi3)(m), B{2)(m)}.

It can be checked that, for log3 ^ m ^ logé, we have B(2)(m) > Ba)(m) and, for m > l o g6, we have B(2) (m) > B{3) (m) and, of course, limm-oo = 5 > 2.

Note that the equality G(F) = B{2)(m) would hold only if the equality in Lemma В held in those cases where we had obtained the function B{2) (m). So, it follows from (38) that the equalities in (28) and (35) would have to hold for v = 0. Consequently, by the use of Lemma B, the equality G (F) = B{2) (m)

would be satisfied only if v = 0, i.e. if x 2 = ( l —e~m)2, which is impossible in both the cases because y2 = ( 1 — e~m)2.

Thus, from the considerations above we get that

(42) G (F )^ <

(1 —e~'”)2 + [(l — e~m)(l — 2e~m)]2

[(u0 +

e-w]2 +

2i;0

21)0

— 6 ( v 0 + l ) e ~ Voe ~ m + 4 e ~ 2 m+ l ] 2

when log 3 ^ m ^ log 6,

when m > log 6, where v0 is the root of equation (40), 0 < v0 < m.

It still remains to prove that estimate (42) is sharp. Indeed, when log 3 < m ^ log 6, equality holds in (42) for the Pick, function defined by equality (4) for e = +1, m — log M.

* In order to show that, for m > log 6, equality holds in (42), it suffices to prove, in view of (33) and on account of Lemma B, that there exists a function 0* (t), 0 ^ t ^ m, for which y2 = 0, that is,

m

(43) j e~x sin 9^{x) dx = 0

о and IА (т)| = Ц (t).

Let v0, voe(0, m), be a solution of equation (40), and 0* (r) the function defined by the formulae

Then

cos 0# (t) ex Vo for 0 < t < v0, 1 for v0 ^ t ^ m.

sin 0* (t) ± y j 1 —e2{x~Vo) for 0 ^ t ^ v0, 0 for v0 ^ t < m,

whence one can easily obtain the formulae for the function /с+ (t) = ew*(x).

Obviously, Я# (т) = e ~xcos 0^(1) = ц{т). By choosing different signs in portions of the interval <0, m>, one can make condition (43) satisfied. Indeed, consider, for instance, the function

X _____________ Vo _____________

<p(x) = — e2(x~Vo)dx — J e~xy j \ — e2(x~vo)dz, x e < 0, v0).

о x

It is continuous in the interval <0, v0), <p(0) < 0, <p(v0) > 0, thus there exists a point xoe(0, r 0) suc^ tbat <p(x0) = 0. Putting then

sin 0* (t)

J \ - e 2(x- V0\ 0 ^ t ^ x 0, - y J \ - e 2(x- V0\ *0 < T < vo>

o, v0 ^ t ^ m,

we finally obtain condition (43).

352 K. Z y s k o w s k a

We have thus shown that there exist functions of the classes S (M) realizing the equality of estimate (42), with m = log M. Thereby, we have proved that the following theorem holds.

Th e o r e m. I f H is any function of the form (1 )from the class S{2)(y/M), then the following sharp estimates

\c3\2+\c5\2 ^ <

1 1Y l — +

M (v0 + l)e'

i - l V i - i

M l M

1

when 3 M ^ 6,

M

+ - (3uo + 2i;0 + l)e~ 2t7°

- M ^ + l ) e ~n + W + l hold, where i’oe(0, log M) is the root of the equation

1

when M > 6

(v+ l)e~v- M + { 3 v - l) e ~ v

M (3v2 + 2v + l)e 2v—— (v + l)e v + ^-pi+\

M M ⁰.

References

[1] L. de B r a n g e s, A proof o f the Bieberbach conjecture, Acta Math. 154 (1985), 137-152.

[2] S. F r ie d la n d , On a conjecture o f Robertson, Arch. Rat. Mech. Anal. 37 (1970), 255-261.

[3] Z. J. J a k u b o w s k i, Sur les coefficients des fonctions univalentes dans le cercle unité, Ann. Pol.

Math. 19 (1967), 207-233.

£4] K. L o w n er, Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, Math.

Ann. 89 (1923), 103-121.

[5] G. P ic k , Über die konforme Abbildung eines Kreises auf eim schlichtes und zugleich beschrânktes Gebiet, Sitzungsber. Akad. Wiss. Wien Abt. II a (1917), 247-263.

[6] M. S. R o b e r t s o n , A remark on the odd schlicht functions, Bull. Amer. Math. Soc. 42 (1936), 366-370.