Note on the growth of Area functions

c Journal of Technical University at Plovdiv Fundamental Sciences and Applications, Vol. 6, 1998 Series A-Pure and Applied Mathematics Bulgaria, ISSN 1310-8271
0 2π
mα
1 (1 − r )α
,0 < α < 1;
|f ′ (reiθ )| dθ ≤ c f
0
m1
1 1 1 1 − r log 1− r
,
where c denotes a universal constant which may change from line to line or even within a line. The inequalities A(r, f ) ≤ c f
References
[1] D. J. Hallenbeck, Growth of the Area Function for Multipliers of Analytic Fractional Cauchy Transforms. J. Tech. Univ. at Plovdiv, Fundam. Sci. Appl,Ser. A Pure Appl. Math., 5, 1997, 13-17.

= 2π
0 ∞ k =0
ˆ(k + 1)|2ρ2k (k + 1)2 |f
r
ρ dρ
= 2π
k =0 ∞
ˆ(k + 1)| (k + 1) |f
2
2 0
ρ2k+1 dρ
= π
k =0
ˆ(k + 1)|2 r 2(k+1) (k + 1)|f
Applying Parseval’s relations
arXiv:0807.3940v1 [math.CV] 24 Jul 2008
Note on the growth of Area functions
Peyo Stoilov, Roumyana Gesheva
Abstract Let A(r, f ) =
0 0 r 2π
|f ′ (ρeiθ )|2 ρ dρ dθ ,
H∞
f
mα
1 (1 − r )α
m1
,0 < α < 1;
A(r, f ) ≤ c f
H∞
f
1 1 1 1 − r log 1− r
follow immediately from the Main Lemma. The same inequalities were obtained by D. J. Hallenbeck [1] with the help of more complicated method.
2π n 2π
f (reiθ ) f ′ (reiθ )reiθ dθ =
0 k =0
ˆ(k + 1)r k+1 (k + 1)f
0 ∞
e−i(k+1)θ f (reiθ ) dθ
= 2π
k =0
ˆ(k + 1)|2 r 2(k+1) (k + 1)|f
Note on the growth of Area functions ... we obtain 1 A(r, f ) = 2 which implies 1 A(r, f ) ≤ sup |f (reiθ )| 2 θ
2
Main Lemma.
1 A(r, f ) ≤ f 2 Proof.
r 2π r 2π H∞ 0
Lemma 1. If f ∈ H ∞ , then |f ′(reiθ )| dθ
A(r, f ) =
0 0 r
|f ′ (ρeiθ )|2 ρ dρ dθ = 2π
0 ∞
θ )|2 dθ ρ dρ
where r = |z | < 1. A(r, f ) is the area of { z : |z | ≤ r } under f . In this note we prove that 1 A(r, f ) ≤ f 2
2π H∞ 0
|f ′ (reiθ )| dθ
and based on this result, a new proof of the inequalities of D. J. Hallenbeck for the Area functions of multipliers of fractional Cauchy transforms is given.
1
Introduction.
Let D denote the unit disk in the complex plane and T - the unit circle. Let M be the Banach space of all complex-valued Borel measures on T with the usual variation norm. For α > 0, let Fα denote the family of functions g for which there exists µ ∈ M such that g (z ) =
T
1 dµ(ξ ), (1 − ξz )α
z ∈ D.
(1)
We note that Fα is a Banach space with the natural norm g
0 0
Fα
= inf { µ : µ ∈ M such that (1) holds } .
1991 Mathematics Subject Classification:Primary 30E20, 30D50 Key words and phrases:: Analytic function, Area function, Cauchy transforms, multipliers.
r 2π mα
= sup { f g
Fα
: g
Fα
≤ 1} .
A(r, f ) =
0 0
|f ′(ρeiθ )|2 ρ dρ dθ
for r = |z | < 1. A(r, f ) is the area of { z : |z | ≤ r } under f . In [1] D. J. Hallenbeck determined the sharp growth of A(r, f ) for f ∈ mα . In this note we give new proof of these results.
40
Peyo Stoilov, Roumyana Gesheva
Department of Mathematics Technical University 25, Tsanko Dijstabanov, Plovdiv, Bulgaria e-mail: peyyyo@mail.bg
37
38
Peyo Stoilov, Roumyana Gesheva
Definition 1. Suppose that f is holomorphic in D. Then f is called a multiplier of Fα if g ∈ Fα ⇒ f g ∈ Fα . Let mα denote the set of multipliers of Fα and f For f analytic in D define
2π
39
2π
f (reiθ ) f ′ (reiθ )reiθ dθ ,
0
1 |f ′ (reiθ )| dθ ≤ f 2
2π H∞ 0
|f ′ (reiθ )| dθ .
0
3
Applications of the Main Lemma.
2π
It was proved by Hallenbeck and Samotij that |f ′(reiθ )| dθ ≤ c f