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26Труды Петрозаводского государственного университета
Серия “Математика” Выпуск 3, 1996
YAK 517
THE MAXIMUM OF SOME FUNCTIONAL FOR HOLOMORPHIC AND UNIVALENT FUNCTIONS WITH REAL COEFFICIENTS
WlESLAW MAJCHRZAK AND ANDRZEJ SZWANKOWSKI
In the paper the maximum of the functional a2a™(a3 — aa%) in the class Sr of functions f(z) = z + anZn, an =
holomorphic and univalent in the unit disc is obtained for a real and kj m positive integers.
0. We consider a functional
H(f) = 0*2a™ (o3 - aat) (1)
for k,m = 1,2,..., a E IR, defined on the class Sr of functions
oo
/(z) — Z -h ^ ^ anZ , CLn — Qjn, (2)
n=2
holomorphic and univalent in the unit disc A.
In papers [2], [3], [6], the extremal values of functional (1) were determined in the cases:
1° m = 0, k = 0,1, 2,...,
2° k = 0, m = 1, 2, —
In the case m = 0 and k = 0,1,2,3,..., the maximum of the functional \H(f)\ for functions / belonging to the well-known class S was also determined [1], [5], [8].
The aim of the present paper is to obtain the maximum of functional (1) in the class Sr.
© Wieslaw Majchrzak and Andrzej Szwankowski, 1996
1. The functional H(f) is continuous, whereas the class Sr is compact in the topology of locally uniform convergence in the unit disc, therefore there exist functions /* E Sr, called further extremal ones, for which
H{f*) = max J?(/).
It is known [2] that each extremal function from the class Sr is a solution of a differential-functional equation which, in the case of the functional #(/), has the following form:
№ where
Bi
B2
B0
Bif(z) + B2 B2z^ + B\z3 + Bqz2 + B\z + B2
P(z)
at1* 3
4a™-1
,-1
z £ A,
Z~
(3)
\(ka3 + 2mal)(a3 - aal) + 2(1 - a)ala3\ , (4)
(5)
(6)
[m(a3 - aaj) + a3] , = (2 + 2m + k)a2a™{a3 — aa^),
with that the right-hand side of the equation (3) is non-negative on the circle |z| = 1 and has at least one double zero on this circle; besides, the coefficient B0 is positive.
It is easy to notice that no function (2), for which <22 = 0 or a3 = 0, is an extremal function. Consequently, in our further considerations we assume the coefficients 0,2 and a3 of extremal functions to be different from zero.
At present, we shall succesively consider all the admissible cases of the factorization of the numerator of the right-hand side of equation (3), also taking account of the vanishing or the non-vanishing of the coefficients Bi, B2.
2. At first, we consider the case when equation (3) has the form
*/'(*)
№ J
B1f(z) + B2 f2(z)
(z - £l)2{z - e2r)(z - e2r X)
z E A, • B2 0,
(a)
where £i = ±1, £2 = ±1, r E (0,1).
Comparing the coefficients in the numerators of the right-hand sides of equations (3) and (a), we obtain, among other things,
2+&H)=t <7»
where £ = £\£2 = ±1.
Whereas, integrating equation (a) and comparing the constant terms of the expansions of both sides in Laurent series, we get
2+(r+^)e r + r~1+2e 1-r 2(a2 + 2ei)ei
(r-.-r)£ 2 + (r + r-L)f S T+7 ^ 2 + r + r-' ' (8)
whence
a2 = -2ei, ei = ±1, (9)
which, in consequence, yields
a3 = 34 = 3. (10)
Taking account of (9) and (10) in (7), from the forms of Б0, B2 we obtain
3(2 + 2m + k) (3 — 4ск) 1 Зш + 3 /ii\
3 +3m —4am ' = * + ^
The study of the dependence of a on r described by formula (11) in
the cases £ = ±1 as well as the examination of the values of the functional H{f) for a2 and аз of forms (9) and (10), where £\ = ±1, imply
JIemma 1. If the extremal function satisfies equation (a), then
H(f) = 2fe3m(3 - 4a) for а<аи k,m = 1,2,..., (12)
and
H(f) = 2h3m(4a — 3) forae U +“) ,
A: = 1,3,..., m = l,2,..., (13)
7 3k 3(8m+3fc+8)
where ai — 4fc+8, а2 — 4(8т+3&+6) •
Values (12) and (13) are taken by the functional H(f) for the Koebe function only. For a E (ai, a2) and k,m = 1,2,... as well as for a > a2 and k = 2,4,..., m = 1,2,..., the extremal function does not satisfy equation (a).
3. Let us now consider equation (3) under the assumption that Bi / 0 and B2 — 0. After simple calculations we get Bq = whence equation
(3) takes the form
№
№
z2 + a,2Z + 1 / x
----------------------, z e A. (b)
It follows from the general properties of equation (3) that
a22=4
and, in consequence,
4 a m
0*3
m + 1
Summing up, we obtain JIemma 2. If the extremal function satisfies equation (b), then
ra+l
/ \ m-\-i
H(f) = 22rn+k+2mrn I J (14)
for a = 3^3 and m — 1,2,3,..., k = 1,3, 5,___________
For the remaining values of a, the extremal function fails to satisfy equation (b).
Value (14) is taken by the functional H(f) for the Koebe function only.
4. The successive form of equation (3) to be considered is
zf'iz)
№ J
Bif(z) + B2 {z - z0)2(z - z0)2 ,
P(Z) = B2---------------?----------' 2 e A' W
where Zq = e*^, ‘i/j G IR, under the condition ^1^2 / 0.
Comparing the coefficients in equations (3) and (c), we get
^ = -4 cos ip, (15)
£>2
TD
—— = 2 + 4 cos2 ф. (16)
^>2
After integrating equation (c) and making use of the fact ([4]), that there exists хеШ such that f(exx) = — as well as of (15) and (16), we obtain
0,2 — 2 cos ф(—1 H- log cos ф), (17)
аз = 1 + 2 cos2 ф[ 1 + 2(—1 + log cos ф) log cos ф\, (18)
_ 1 Mi 1 + 2 cos2 ф[1 + 2(—1 + log cos?/’) log cos?/’]
а 4 Mi + 2(1 + 2 cos2 ф) cos2 ф(—1 + log cos ф)2
(19)
where Mi = Л:(14- 2 cos2 ф) + 4(2 + 2m + &) cos2 ф(—l + logcos^) log cos ф,
and ф G (0, 7t/2), k,m = 1, 2, 3,__________
Consequently, we have proved
JIemma 3. If the extremal function satisfies equation (c), then the value of the functional H{f) is given by 0,2 and аз defined by formulae (17) and (18) where ф is the function inverse to increasing function (19).
5. Now, consider the case when equation (3) is of the form
zf'iz)
№
P(z)
Bi ■ B2 ф 0,
with that
I=t4’
Bo
Bo
= 6.
(d)
(20)
(21)
Taking account of (20) in equation (d), after integrating we obtain
T4 f(z)
\/lT4 f(z) №
± 2 log
(yi =F 4/(z) + l)
= - ± 2 log z — z + С (22)
where C is a constant, y/l = 1, log(—1) = 7ri.
Comparing the constant terms, we get
C = ±2 + a2 T 21og(=Fl). (23)
On the other hand, it is well known that there exists a point z = elx,
x E IR, for which f(elx) = ±|. This and (22) imply
Re (7 = 0. (24)
Finally, from (23) and (24) we have
a2 = T2,
and so,
a3 = 3.
Putting the above values of the coefficients a2 and a3 in (20) or (21), we obtain
3k
a~ 4& + 8 ’
To sum up, we have shown JIemma 4. If the extremal function satisfies an equation of form (d), then
H(f) = 2fe3m(3 - 4a)
for k, m = 1,2,3,..a = 3fe
4fc+8 '
________ h —
4fc+8 ’
For a / 4^8’ ^ = 1,2,3,..., the extremal function fails to satisfy equation (d).
6. To finish with, let us consider the case when B\ — 0 and B2 / 0 in equation (3). Then this equation is of the form
zf'(z)]2 1 _ (z — zp)2(z — Z]_)2 ^
№
P{z)
where |^o| = |^i| = 1 and z\ / zq.
Comparing the coefficients of equations (25) and (3), we get z\ — —z0 and Zq = 1 or Zq = — 1. However, it turns out that, for zq = —1, the solution of equation (25) is not holomorphic in the disc A.
Finally, equation (25) takes the form
*/'(*)
№ J
(z-l)2(z + l)2
P(z)
(e)
Integrating equation (e) and, next, comparing the constant terms, we obtain
/1
as — a2 = — 1.
Hence and from the condition B\ — 0 we have
9 2k 2m 2 — (k-\- 2) a H- y/~D
a\ =
(26)
or
0*3 =
a% =
a 3
2(1- a)(k + 2m + 2) Vd
(k + 4m + 2 )a — 2 (m + 1) +
2(1- - a)(k 2m 2)
k + 2m + I [k + 2)(1 — a)-y/D
2(1- a)(k + 2m + 2) Vd
(k + 4m + 2) a — 2 (m + 1)-
2(1- - a)(k 2m 2)
(k + 2)a]2 + Skma, £,*, m = = 1,2.
(27)
(28)
It follows from the conditions a\ < 4 and — 1 < as < 3 that system (27) has a solution only for a E (0,0:3) where
as —
9 k2 + 48m2 + 42 km + 42 k + 96m + 48 12 k2 + 64m2 + 56 km + 4 8k + 112m + 48 ’
k, m = 1, 2,3,.
(29)
while system (28) has a solution for a > 0, with that, for a = 1, a| = and as =
— 2m k-\-2m '
Remark 1 If as) is a solution of system (27) or (28), then, for any positive integers k, m and for a > 0
as — a a\ <0.
(30)
Remark 2 Ifk^m — 1,2,3,... and a E (0,1), then it follows from system
(27) that as > 0, whereas for system (28) — that as < 0.
The above considerations imply
JIemma 5. If к = 1,3,..., m = 1, 2,... and the extremal function satisfies equation (e), then the value of the functional H(f) is defined by system
(27) for а E (0, с^з) or by system (28) — for а E (0, +oo);
If к = 2,4,..., m = 1,3,... and the extremal function satisfies equation (e), then the value of the functional H{f) is defined by system
(28) for а E (0, +oo);
If к = 2,4,..., m = 2,4,..., then there exists no extremal function satisfying equation (e).
7. Let us introduce the following notations:
M2(fc, m, 'ip(a)) = a2<2™(a3 — aa\)
where a2,a3 are defined by formulae (17), (18), and 'ijj(a) is the function inverse to the increasing function a = if; E (0,7r/2), given by the
formula (19);
M3(k, m, a) = a2a^(a3 — aal) where a2 ans as are defined by equations (27);
M±(k, m, a) = a2<2™(a3 — aa^)
where a2 ans as are defined by equations (28).
Lemmas 1-5 and the continuity of the functional H(/) in the compact class Sr, by proceeding similarly as in papers [2], [6], imply
Theorem 1. For any function f E Sr:
1° if k = 2,4,..., m = 2,4,..., then
H(f)<
2k3rn(3-4a) for а < ai,
М2(к,т,ф(а)) for a > 2° if к = 1, 3,..., m = 1, 2,..., then
' 2k3rn{3 — 4a) for а < аь
М2(к,т,ф(а)) for a\ < a < «5,
Ms(k,m,a) for «5 < a < «2,
2fc3m(4a — 3) for a > a2;
3° if k = 2,4,..., m = 1,3,..., then
( 2fc3m(3-4a) for a<au
H(f) < \ M2(fc,m,a) for ai < a < a4,
[ M4(fc,m,a) for a > «4,
where a\ = 4^8 , «2 = 4(8m+3fc+6) > W^e a4 > a5 are TOOtS of the
equations M2(k,m,tip(a)) = M4(fc,ra,a) and
M2(fc,ra,7/>(a)) = M3(kJmJa)J respectively.
All the estimates given above are exact.
Remark 3 Theorem 1 (3°) implies the well-known estimate of the functional a™ (<23 — aal) for m = 1, 3, 5,..., a E IR [6].
For m — 2,4,..., from Theorem 1 (1°) we obtain the estimate of the functional a^(a3 — aa^) for a < m + 1 only. For a > ra + 1, it is known [6] that the only function extremal with respect to this functional is the function with the coefficients a,2 = 0 and = 1. This function, however is not extremal with respect to the functional H(f) investigated in our paper.
From Theorem 1 we do not obtain directly any estimate of the functional a%(as — cm|), either (see [2]), on account of the necessary assumption m / 0 used in the proof of Lemma 1.
References
[1] Jakubowski Z.J., Majchrzak W., Szwankowski A., On some extrem al problem in the class S of functions holomorphic and univalent in the unit disc, Ann. UMCS 40 no. 7 (1986), 63-78.
[2] Jakubowski Z.J., Szwankowski A., On some extremal problem in the class of holomorphic symmetric univalent functions, Comment. Math. 29
(1990), 195-207.
[3 ]----, On the minimum of the functional a™(a3 — aa|) in the class of
holomorphic and univalent functions with the real coefficient s, Bull. Soc. Sci. Lettres de Lodz 41 no. 11 (1991), 69-78.
[4] Kirwan W.E., A note on extremal problems for certain classes of analytic functions, Proc. Amer. Math. Soc. 17 (1966), 1028-1030.
[5] Majchrzak W., Szwankowski A., On the maximum of the Jakubowsk i functional in the class of holomorphic and univalent functions, Mathematica
(Cluj), 36 (59) no. 2 (1995), 169-177.
[6] Majchrzak W., Szwankowski A., The bounds of some functional f or holomorphic and univalent functions with real coefficients, Bull. Soc. Sci.
Lettres de Lodz 45 no. 20 (1995), 27-37.
[7] Schaeffer A.C., Spencer D.C., Coefficient regions for schlich t functions, Amer. Math. Soc., Colloq. Publ. 35 (1950).
[8] Szwankowski A., Estimation of the functional \a3 — a.a\\ in the class S of holomorphic and univalent functions for a complex , Acta Univ. Lodz. 1 (1984), 151-177. Abstracts Conference on Analytic Functions, Kozubnik 1979, 54.
Institute of Mathematics University of Lodz ul. Banacha 22, 90-238 Lodz, Poland
