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24Trudy Petrozavodskogo gosudarstvennogo universiteta
Seria “Matematika” Vypusk 15, 2009
UDK 517
THE SHARP UPPER BOUND FOR K(A3 - AA2) IN U'a
I. Naraniecka
In this note we determine the exact value of max Re(A3 —
AA2), A e R, within the linearly invariant family U4 introduced by V. V. Starkov in [4]. For A = 0 the sharp estimate for |A3| follows. If a =1 the corresponding result is valid for convex univalent functions in the unit disk.
1. For given a > 1, we consider the class of holomorphic functions in the unit disk D = {z : |z| < 1} of the form
f (z) = z + A2Z2 + A3Z3 + ... (1)
which are defined by the formula
f '(z)=expj-2^ log(1
2n
---it y
— ze
d^(t))\ , (2)
where ^(t) is a complex function of bounded variation on [0, 2n] satisfying the conditions
f‘ 2n f‘ 2n
/ d^(t) = 1, / |d^(t)| < a. (3)
J 0 J 0
The class Ua has been introduced by Starkov in [4]. The idea of studing such a class is justified by at least two facts:
1) The class Ua appears to be a linearly invariant family in the sense of Pommerenke of order a, and can be used for studing the universal invariant family Ua [3].
2) The class U'a generalizes essentially the class Vk=2a of functions with bounded boundary variation (Paatero class ) and in the sequel convex univalent functions K = U[.
© I. Naraniecka, 2009
In [5] V. V. Starkov has found sharp bound for |A3| within the class U'a which disproved the Campbell-Cima-Pfaltzgraff conjecture about max | A31 in Ua.
In this note we determine
max Re (A3 — AA2), (4)
f eu'a
for real A, which as a corollary (A = 0) gives the above result of Starkov.
Justification of studing such a functional is highly motivated by corresponding result for the class of univalent functions S [2] (Bombieri Conjecture). As a method we are going to use is the variational method of Starkov for U'a [5].
2. Problem of finding (4) is equivalent to
max Re (C2 — ACi), A G R, (5)
f eua
where f'(z) = 1 + C1z + C2z2 + ..., f G Ua.
Because J(f) = Re (C2 — AC1) is a linear functional, then according to a result of Starkov [4] the extremal function f0(z) is of the form
f0(z) = (1 — ze-it1 )-2ai (1 — ze-it2)-2“2, (6)
where
ti, t2 G [0, 2n] (7)
and
ai + a2 = 1 and |ai | + |«21 = a. (8)
One cand find that the coefficients of f0 are given by
c2
ci = 2(aie itl + a2e it2), C2 = ^ + aie 2itl + a2e 2it2. (9)
Therefore the problem is reduced in finding the maximal value of
^(ai, a2; ti, t2) = R^2 [aie-itl + a2e-it2]2 + [aie-2itl + a2e-2it2]
—2A [aie-itl + a2e-it2]}
where ti, t2, ai, a2 are satysfying the conditions (7) and (8) Moreover same extra conditions follow from the extremality of f0 [4] (see below). We will start with simple technical lemma.
Lemma 1. If ai = |а1|ei^1 and a2 = |a2|e®^2 and
ai + a2 — 1
|«i | + |a21 = a > 1
then
|ai|
sin ^2
sin(^2 — A) ’
|a2 |
— sin ^i
sin(^2 — ft) '
(11)
(12)
Moreover, ft and ft satisfy the condition
ft + ft ft — ft + ft + ft a — 1
i-----------= a cos------------------•<=> tan — tan — =--------------------------
2 2 a +1
A. (13)
22 Proof. The system (11) can be written in the real form:
|ai | cos ft + |a21 cos ft = 1 |ai | sinft + |a21 sinft =0 |ai | + |a21 = a.
Solution of the first two equations by Cramer’s rule is unique and given by (12). (If sin(ft — ft) = 0 then the above system has no solution).
Substitution of (12) into the equation |ai| + |a2| = a gives (13) after slight calculations.
The following lemma plays important rule.
Lemma 2. The extremal function f0 for functional (5) has real coefficient ci .
Proof. If f0 g Ua is an extremal function, then for any e G (0,1), the following variation f£ of f belongs to Ua:
fe(z)
(f0(s))i-£ (f0(s))£ds
0
1 + ci(e)z + C2(e)z2 + ... G Ua.
(14)
But
c2(e) — Aci(e) = c2 — 2ieIm c2 + e(1 — e)(|ci|2 — Re ci) — A(ci — 2ieIm ci) = (c2 — Aci) — 2ieIm c2 + 2iAeIm ci + e(|ci|2 — Re ci) + o(e)
which implies
J(fe) = J(f0) + e(|ci|2 — Re c2) + Re o(e).
The extermality of f0: J(fe) < J(f0), when e ^ 0, gives the condition |ci|2 — Re c2 < 0 which implies Im ci = 0, due to the form of our functional (5).
. If Im [aie-itl + a2e-it2] = 0 then either: both e-itl and e-it2 are real, or e-it2 = eJtl = e-itl.
Denote:
0^-0 V/a2 — 1
cos p = —, sin p = -,
aa
3 — a2 . 3%/ a2 — 1
cos f = -------- , sin f =
(15)
ti/a2 + 3’ a%/ a2 + 3
t = p + f, x = t + p
We have:
Theorem 1. If f g U'a and f'(z) = 1 + ciz + c2z2 + ... then
max Re(c2 — Aci) = $(t0) feua / \ /
where
$(t) = a2 + (3 — a2) cos 2t + 3\Ja2 — 1 sin 2t — 2A ^cos t — Va2 — 1 sin t j
(16)
and
t 0 = t 0 (a, A) G (0, 2n) (17)
is the root of the equation: A sin x — %/a2 + 3 • sin (2x — 2t) = 0, for which f" (t0) < 0.
Proof. Let f g U/ and
f/(z) = ex^ —2J^ log (1 — ze-it) d^(t^ = exp{f(z)} (18)
= 1 + Ci z + C2z2 + ....
The functional J(f) = Re(C2 — ACi) is a linear and continuous on the compact family U/ and therefore it attains its sharp bounds on it. V. V. Starkov [3] has proved, that if F(f) = F(f)is Frechet differentiable and its differential functional on U/ with differential Lv(h), and max Re F(f) is attained for a jump function ^(t) with n jumps at points tj, j = 1,..., n and jumps 6j = arg d^n(tj) (we assume that at least two jumps 6j are different), then the following system of equations holds
[Re [e^ [|f(z,tj)]}
[Im [(ei0j — el9m) (L,
0 (19)
g(z,tj)] — [g(z,tm)]^ = 0.
In our case J(f) = Re (C2 — ACi) s Frechet differentiable and its differential is given by the formula:
Lv(h) = {hexpf}2 — A {hexpf}i,
where h = —2log(1 — ze-it) = g(z,t) and {F(z)}p denotes the p-th coefficient of F.
In our problem the extremal function has the form :
f/(z) = 1 + ciz + c2z2 + ... = (1 — ze-itl )-2al (1 — ze-it2)-2“2
where ti,t2 G [0, 2n], ai + a2 = 1, |ai| + |a2| = a.
Because the Freechet differential is equal to
L(log f0) [ —2log(1 — ze Jt)] = e 2it + 2e it(ci — A)
the conditions (19) take the form
"Im [e^1 (e-2itl + e-itl (ci — A))] = 0 Im [eift (e-2it2 + e-it2 (ci — A))] = 0
Im [(e^ — e^2) (e-2itl — e-2it2 + 2(ci — A) (e-itl — e-it2))] = 0.
(20)
The information that for the extremal function f0 the coefficient ci is real i.e.
Im ci =0 sinp2 sin(pi — ti) — sinpi sin(p2 — t2) = 0 (21)
implies eit2 = e-itl which gives t2 = —ti, or that e-itl and e-it2 are real. In the case when e-itl and e-it2 are real we obtain either contradiction or the result for Ui = K which is in the Corollary at the end of the papers.
In the case eit2 = e-itl i.e. t2 = —ti the first two equations of (20)
are
jsin(pi — 2ti) + (ci — A) sin(pi — ti) = 0 |sin(P2 — 2t2) + (ci — A) sin(P2 — t2) =0
which together with (21) for t2 = —ti implies that p2 = —pi. Substitution p2 = —Pi into (12) and (13) give
(22)
|ai| = |a21 = a; cos Pi = -1; cosP2 = -1;
2 a a
. Va2 — 1 . — V a2 — 1
sin pi =-------------; sin p2 =----------------.
(23)
Putting now ti = t G [0, 2n], t2 = —ti = —t, ai = 2e®^, a2 = 2e we
obtain
Re(c2 — Aci) := $(t) = a2 + (3 — a2) cos2t + 3^a2 — 1 sin2t
'______________________ (24)
— 2A(cos t — \J a2 — 1 sin t).
Using notations (15) we obtain:
$(t) = $(x) = a2 + a^a2 + 3 cos(2x — 2t ) — 2Aa cos x. (25)
The equation $'(x) = 0 is equivalent to
—A sin x + Va2 + 3 sin(2x — 2t ) = 0 (26)
or
4(a2 + 3) sin4 x — 4A Va2 + 3 sin 2t sin3 x + [A2 — 4(a2 + 3)] sin2 x
(27)
+ 2A\Ja2 + 3 sin2T sin x + (a2 + 3) sin2 2t = 0,
which ends the proof.
Corolary. If f g Ui = K then
max(A3 — AA2) = 1 + |A|, A G R.
z
The extremal functions have the form f0(z) =-----------.
w 1 ± z
Bibliography
[1] Godula J. Linear-invariant families / J. Godula V. V. Starkov // Tr. Petroz. Gosud. Univ. Seria ”Mathematica”. 5(1998). 3-96 (in Russian).
[2] Greiner R. On support points of univalent functions and disproof of a conjecture of Bombieri / R.Greiner, O.Roth // Proc. Amer. Math. Soc. 129(2001). 3657-3664.
[3] Pommerenke Ch. Linear-invariant Familien analytischer Funktionen / Ch. Pommerenke // Math. Ann. 155(1964). 108-154.
[4] Starkov V. V. The estimates of coefficients in locally-univalent family Ua / V. V. Starkov // Vestnik Lenin. Gosud. Univ. 13(1984). 48-54 (in Russian).
[5] Starkov V. V. Linear-invariant families of functions / Dissertation. 1989. 1-287. Ekaterinburg (in Russian).
Department of Applied Mathematics
Faculty of Economics, Maria Curie-Skiodowska University,
20-031 Lublin, Poland
E-mail: iwona.naraniecka@hektor.umcs.lublin.pl
