INTEGRAL MEANS OF THE DERIVATIVES OF BLASCHKE PRODUCTS

Description

INTEGRAL MEANS OF THE DERIVATIVES OF BLASCHKE PRODUCTS EMMANUEL FRICAIN, JAVAD MASHREGHI Abstract. We study the rate of growth of some integral means of the derivatives of a Blaschke product and we generalize several classical results. Moreover, we obtain the rate of growth of integral means of the derivative of functions in the model subspace KB generated by the Blaschke product B. 1. Introduction Let (zn)n≥1 be a sequence in the unit disc satisfying the Blaschke condition (1.1) ∞ ∑ n=1 (1? |zn|) < ∞. Then, the product B(z) = ∞ ∏ n=1 |zn| zn zn ? z 1? z¯n z is a bounded analytic function on the unit disc D with zeros only at the points zn, n ≥ 1, [5, page 20]. Since the product converges uniformly on compact subsets of D, the logarithmic derivative of B is given by B?(z) B(z) = ∞ ∑ n=1 1? |zn|2 (1? z¯n z)(z ? zn) , (z ? D). 2000 Mathematics Subject Classification. Primary: 30D50, Secondary: 32A70. Key words and phrases. Blaschke products, model space. This work was supported by NSERC (Canada) and FQRNT (Quebec).

• blaschke sequence


• zn


• positive continuous


• condition ∑∞


• points zn


• classical results


• function satisfying


• bergman space

Sujets

• McGill University
• Blaschke product
• Bergman space

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 31
• Langue: English

Extrait

INTEGRAL MEANS OF THE DERIVATIVES OF BLASCHKE
PRODUCTS

EMMANUEL FRICAIN, JAVAD MASHREGHI

Abstract.We study the rate of growth of some integral means of the derivatives
of a Blaschke product and we generalize several classical results.Moreover, we
obtain the rate of growth of integral means of the derivative of functions in the
model subspaceKBgenerated by the Blaschke productB.

1.Introduction

Let (zn)n≥1be a sequence in the unit disc satisfying the Blaschke condition

(1.1)

∞
X
(1− |zn|)<∞.
n=1

Then, the product
∞
Y
|zn|zn−z
B(z) =
zn1−z¯nz
n=1
is a bounded analytic function on the unit discDwith zeros only at the pointszn,
n≥Since the product converges uniformly on compact subsets of1, [5, page 20].
D, the logarithmic derivative ofBis given by

∞
X
′2
B(z) 1− |zn|
=,
B(z) (1−z¯nz)(z−zn)
n=1

(z∈D).

2000Mathematics Subject Classification.Primary: 30D50,Secondary: 32A70.
Key words and phrases.Blaschke products, model space.
This work was supported by NSERC (Canada) and FQRNT (Qu´bec).A part of this work was
done while the first author was visiting McGill University.He would like to thank this institution
for its warm hospitality.
1

2

Therefore,

EMMANUEL FRICAIN, JAVAD MASHREGHI

∞
X
2
1− |zn|
′iθ iθ
(1.2)|B(re)| ≤,(re∈D).
iθ2
|1−z¯nre|
n=1
If (1.1) is the only restriction we put on the zeros ofB, we can only say that
Z∞Z
X
2π2π
dθ
′iθ2
|B(re)|dθ≤(1− |zn|)
iθ2
0 0|1−z¯nre|
n=1
∞
X
2π
2
= (1− |zn|)
2 2
(1− |zn|r)
n=1
P
∞
4π(1− |zn|)
n=1
≤,
(1−r)
which implies
Z
2π
o(1)
′iθ
(1.3)|B(re)|dθ=,(r→1).
01−r
However, assuming stronger restrictions on the rate of increase of the zeros ofB
′
give us more precise estimates about the rate of increase of integral means ofBas
r
r→1. Themost common restriction is
∞
X
α
(1.4) (1− |zn|)<∞
n=1
for someα∈(0,1). Protas[15] took the first step in this direction by proving the
following results.
p
Let us mention thatH, 0< p <∞, stands for the classical Hardy space equipped
with the norm
Z
1
2π
p
dθ
iθ p
kfkp= lim|f(re)|,
2π
r→1
0
p
and its cousinA, 0< p <∞andγ >−1, stands for the (weighted) Bergman space
γ
equipped with the norm
ZZ
1
1 2π
2γp
r(1−r)dr dθ
iθ p
kfkp,γ=|f(re)|.
0 0π/(1 +γ)