Journal of Computational and Applied Mathematics 105 (1999) 367–369
The Landau problem for bounded nonvanishing functions
Zdzis law Lewandowski, Jan Szynal∗M. Curie-Sklodowska University, 20-031 Lublin, Poland
Received 28 October 1997; revised 10 February 1998
Dedicated to Professor Haakon Waadeland on the occasion of his 70th birthday
Abstract
The problem of nding sup|a0+a1+· · ·+an|for holomorphic bounded and nonvanishing functionsf(z) =a0+a1z+· · · in the unit disk |z|¡1 is discussed. c1999 Published by Elsevier Science B.V. All rights reserved.
MSC:30C50
Keywords:Bounded nonvanishing function; The Landau problem; Krzy˙z conjecture
1. Let H(D) denote the set of holomorphic functions in the unit disk
D={z∈C:|z|¡1}:
We consider the following families of the functions:
B:={f∈H(D): f(z) =a0+a1z+· · ·; |f(z)|¡1; z∈D};
B0:={f∈B:f(z)6= 0; z∈D};
:={!∈B:!(z) =c1z+c2z2+· · ·; z∈D}: (1)
With no loss of generality we may assume that for f∈B
0 we have the normalization
a0= e
−t; t ¿0: (2)
The problem of determining maxf∈B0|an| is very hard to handle and only a partial solution to the so-called Krzy˙z conjecture is known (see e.g. [4]). The function
F(t;z) = exp
−t1 +z
1−z
= e−t+ ∞
X
n=1
An(t)zn (3)
∗Corresponding author.
368 Z. Lewandowski, J. Szynal / Journal of Computational and Applied Mathematics 105 (1999) 367–369
plays an important role in this problem. In [2] (see also [5]) Landau proved that
sup univalent (say f∈Bs) then Fejer [1], Szeg˝o [6] and Levin [3] proved subsequently, that for every
n∈N:
sup
f∈Bs
|a0+a1+· · ·+an|¡ K ≃1:6: : : : (5)
However this result is not sharp.
2. Considering the Landau problem for B0 we nd the following partial result:
Theorem 1. If f∈B
Both upper bounds (6) and (7) are sharp.
Proof. Using the representation formula for the function f∈B0 [4, p. 258]:
f(z) = exp
Applying the well-known inequalities:|c1|61; |c2|61−|c1|2 after elementary but tedious calculations
of several extrema we arrive at the following inequalities:
|a0+a1|6e
from which (6) and (7) follows.
In (6) the extremal function has the form f1(z) =F(12;−z): In (7) the extremal function is more
Z. Lewandowski, J. Szynal / Journal of Computational and Applied Mathematics 105 (1999) 367–369 369 Remark. We conjecture that for anyf∈B0 andn∈N there exists an absolute constant L ¿1 such
that
sup
f∈B0
|a0+a1+· · ·+an|6L:
Observe that the function (3) could not be extremal because of the fact that
A0(t) +A1(t) +· · ·+An(t) = n
X
k=0
e−tL(−1)
k (2t) = e −tL(0)
n (2t); A0(t) = e −t
and
e−t|L(0)
n (2t)|61; [7];
where L()
n denotes the Laguerre polynomial of order :
References
[1] L. Fejer, Uber die Koezientensumme einer beschrankten und schlichten Potenzreihe, Acta Math. 49 (1926) 183–190. [2] E. Landau, Abschatzung der Koezientensumme einer Potenzreihe, Arch. Math. Phys. 21 (1913) 42–50.
[3] V. Levin, Uber die Koezientensumme einiger Klassen von Potenzreihe, Math. Z. 38 (1934) 565–590.
[4] Z. Lewandowski, J. Szynal, On the Krzy˙z conjecture and related problems, in: Laine, Martio (Eds.), XVIth Rolf Nevanlinna Colloquim, Walter de Gruyter, Berlin, 1996, pp. 257–268.
[5] O. Szasz, Ungleichungen fur die Koezienten einer Potenzreihe, Math. Zeit. 1 (1918) 163–183. [6] G. Szeg˝o, Zur Theorie der schlichten Abbildungen, Math. Ann. 100 (1928) 188–211.
