We discuss the de Branges weight systems and give a counterexample uncovering pitfalls

Grinshpan, Anatolii

Grinshpan, Arcadii Z.

English

Caltech Authors - Main

PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 133, Number 9, Pages 2605–2608
S 0002-9939(05)07934-7
Article electronically published on March 31, 2005
A NOTE ON DE BRANGES’ WEIGHTS
ARCADII Z. GRINSHPAN AND ANATOLII GRINSHPAN
(Communicated by Juha M. Heinonen)
Abstract. We discuss the de Branges weight systems and give a counterex-
ample uncovering pitfalls.
1. Preliminaries
For a positive integer n and t ≥ 0, a de Branges weight system is a system of
nonincreasing functions σm(t) satisfying the diﬀerential recurrence
(1) σm − σm+1 = − σ˙m
m
− σ˙m+1
m + 1
, m = 1, . . . , n, σn+1 ≡ 0.
The weight system deﬁned by the initial conditions σm(0) = n+1−m plays a key
part in de Branges’ famous proof of the inequality
n∑
m=1
(n + 1−m)(m|cm|2 − 4/m) ≤ 0
for the logarithmic coeﬃcients cm of normalized univalent functions, conjectured by
Milin and implying the Bieberbach conjecture [1] (results, history and bibliography
can be found in [2, 4]).
In fact, de Branges’ theorem contains the inequality
(2)
n∑
m=1
xm(m|cm|2 − 4/m) ≤ 0
for any xm that are the initial values σm(0) of nonincreasing weights σm. An explicit
or asymptotic knowledge of numbers x1, . . . , xn such that (2) holds potentially
opens the door to other interesting estimates. Not all of these tuples arise as initial
values of the nonincreasing solutions of (1), that is via de Branges’ method, and
their description is unknown. However, a complete description of all nonincreasing
solutions of (1) is given in [3] in the framework of a more general setting.
Because of certain implications, our attention has recently been drawn to [6]
which claims to obtain (2), by de Branges’ method, for any choice of the initial
Received by the editors January 6, 2004.
2000 Mathematics Subject Classiﬁcation. Primary 30C50, 30C55, 30C75.
Key words and phrases. Univalent functions, de Branges’ weight systems, associated Legendre
functions.
c©2005 American Mathematical Society
2605
2606 ARCADII Z. GRINSHPAN AND ANATOLII GRINSHPAN
conditions satisfying
(3) −xm + 2
n∑
k=m
(−1)k−mxk ≥ 0, m = 1, . . . , n.
The purpose of this note is to explain why neither the method nor the result of [6]
is correct.
Let D = {z : |z| < 1} be the open unit disk. We will denote by {f}m the
coeﬃcient of zm in a power series f . The class S is formed by all functions f
that are analytic and univalent in D and normalized by f(0) = 0, f ′(0) = 1. The
logarithmic coeﬃcients of f ∈ S are the coeﬃcients of log (f(z)/z). The Koebe
function
K(z) =
z
(1− z)2
is in S. It maps D onto the plane slit along (−∞,−14 ]. For t ≥ 0, the Pick function
P (z, t) is deﬁned implicitly by
K(P (z, t)) = e−tK(z).
It is evident that P (z, 0) = z. For t > 0,
P (z, t) = e−tz + 2e−t(1− e−t)z2 + e−t(3− 8e−t + 5e−2t)z3 + . . .
is a univalent map of D onto D cut along the real axis from −1 to the point
1+2
(√
e2t − et− et). For each m ≥ 1, the coeﬃcient {P (z, t)}m is a polynomial in
e−t of degree m. A relevant property of the Pick function is given by the diﬀerential
equation
(4) P˙ = −P 1− P
1 + P
.
2. The weight system
Every set of solutions of (1) is uniquely determined by the initial conditions and
can be written in terms of the coeﬃcients of powers of the Pick function. Indeed,
by (4) Pm(z, t) satisﬁes the m-th equation of (1), 1 ≤ m ≤ n. Therefore
(5) σm =
n∑
k=m
xk
{
Pm
}
k
, m = 1, . . . , n,
is the general solution of (1). One can also write the general solution in terms of the
special functions. The most general nonincreasing solutions of (1) are determined
by the initial data
(6) σm(0) =
n∑
k=m
akβmk, m = 1, . . . , n,
where βmk are deﬁned by means of the associated Legendre functions Pmk,
βmk = 2m
(k −m)!
(k + m)!
∫ 1
0
P 2mk(x)
xdx
1− x2 ,
and ak are arbitrary nonnegative numbers [3]. In particular, the last three betas
for each m are as follows:
(7) βmm =
1
4m
(
2m
m
)
, βm−1 m =
2
4m
(
2m
m
)
, βm−2 m =
1
4m
5m− 2
2m− 1
(
2m
m
)
.
A NOTE ON DE BRANGES’ WEIGHTS 2607
Equations (1) can be written in the form
− σ˙m
m
= −σm + 2
n∑
k=m
(−1)k−mσk
=
n∑
k=m
(−1)k−m(σk − σk+1), m = 1, . . . , n.
Hence a necessary and suﬃcient condition for the weights σm to be nonincreasing
is given by
(8)
n∑
k=m
(−1)k−m(σk − σk+1) ≥ 0, m = 1, . . . , n.
By adding these inequalities inductively we obtain a simple necessary condition
σ1 ≥ · · · ≥ σn−1 ≥ 2σn ≥ 0.
Obviously, (8) reduces to (3) for t = 0. However, (3) by itself does not guarantee
the nonincreasing behavior of weights. To see this, let x1 = 2, x2 = 2, x3 = 1, and
let the remaining xk be zero (n ≥ 3) in (5). Then
σ1 = e−t(9− 12e−t + 5e−2t)
is not monotone. Worse, one can check using (7) that for every n ≥ 3 the initial
data of the form (. . . , 2, 2, 1) forces an−2 in (6) to be negative, which contradicts
monotonicity. So even if (2) is true in this case, de Branges’ method cannot be
applied to prove it.
3. A counterexample
We now show that merely under assumption (3) the inequality (2) can fail on S.
For each t, (8) describes a half-cone of vectors (σ1, . . . , σn) in Rn. It is generated
by the linear combinations of vectors
(1, 0, 0, . . . , 0), (2, 1, 0, . . . , 0), . . . , (2, 2, . . . , 2, 1)
with nonnegative coeﬃcients. Hence the conclusion of [6] is correct if and only if
(2) holds for the initial data of the form
(2, . . . , 2, 1, 0, . . . , 0).
To show that (2) actually fails for vectors (2, 2, 1, 0, . . . , 0), we use a construction
of the Schaeﬀer-Spencer type [5]; see also [2, 4] for details and references.
For −1 ≤ ε ≤ 1, the function
g(z) = z(1− z)−1−ε(1 + z)−1+ε, z ∈ D,
is starlike and belongs to S. Given 0 < t ≤ 1, deﬁne w(z, t) by the equation
g(w(z, t)) = tg(z).
Note that w(z, t) is a univalent map of D into itself. Furthermore, let
F (z, t) =
w(z, t)
t(1− w(z, t))2 .
Clearly, F (z, t) ∈ S for every admissible pair of parameters ε and t. Write
log
F (z, t)
z
= c1z + c2z2 + c3z3 + . . . .
2608 ARCADII Z. GRINSHPAN AND ANATOLII GRINSHPAN
A routine calculation gives
c1 = 2(1− (1− ε)(1− t)),
c2 = 1 + 4εt(1− ε)(1− t),
c3 =
2
3
+ 2(1− ε)(1− t)(2ε2t(1− 3t) + 1
3
(2t(1 + t)− 1)).
It is now easy to see that the functional
J(F ) = 2(|c1|2 − 4) + 2(2|c2|2 − 4/2) + (3|c3|2 − 4/3)
= 2|c1|2 + 4|c2|2 + 3|c3|2 − 40/3
can assume positive values. In fact, the pairs (ε, t) close to (.65, .85) result in the
values of J which are slightly greater than .002. Thus the result of [6] is false for
every n ≥ 3.
References
[1] L. de Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), no. 1-2, 137–152.
MR0772434 (86h:30026)
[2] P. Duren, Univalent functions, Springer-Verlag, New York, 1983. MR0708494 (85j:30034)
[3] A. Grinshpan, Monotone functions of certain motions, preprint, Cal Tech, 2003.
[4] A. Z. Grinshpan, Logarithmic geometry, exponentiation, and coeﬃcient bounds in the theory
of univalent functions and nonoverlapping domains, Handbook of Complex Analysis: Geomet-
ric Function Theory, V. 1 (Ed. by R. Ku¨hnau), 273–332, North-Holland, Amsterdam, 2002.
MR1966197 (2003m:30041)
[5] A.C. Schaeﬀer and D.C. Spencer, The coeﬃcients of schlicht functions, Duke Math. J. 10
(1943), 611–635. MR0009631 (5:175i)
[6] Xie Ming-Qin, A generalization of the de Branges theorem, Proc. of the AMS 125 (1997), no.
12, 3605–3611. MR1415355 (98b:30018)
Department of Mathematics, University of South Florida, Tampa, Florida 33620
E-mail address: azg@math.usf.edu
Mathematics 253-37, California Institute of Technology, Pasadena, California 91125
Current address: Department of Mathematics, Oklahoma State University, Stillwater, Okla-
homa 74078-1058
E-mail address: agrinsh@math.okstate.edu


A note on de Branges' weights

A generalization of the de Branges theorem,

Branges, A proof of the Bieberbach conjecture,

Logarithmic geometry, exponentiation, and coeﬃcient bounds in the theory of univalent functions and nonoverlapping domains,

Monotone functions of certain motions,

The coeﬃcients of schlicht functions,

Univalent functions,

https://authors.library.caltech.edu/24760/1/GRIpams05.pdf

A note on de Branges' weights

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