New York Journal of Mathematics New York J. Math.

New York Journal of Mathematics

New York J. Math. 17a(2011) 165–175.

Counterexamples on non-α-normal functions with good integrability

Rauno Aulaskari, Shamil Makhmutov and Jouni R¨ atty¨ a

Abstract. Blaschke products are used to construct concrete examples of analytic functions with good integrability and bad behavior of spheri- cal derivative. These examples are used to show that none of the classes Mp^#, 0< p <∞, is contained in theα-normal classN^αwhen 0< α <2.

This implies thatM_p^#is in a sense a much larger class thanQ^#_p.

Contents

1. Introduction and results 165

2. Proof of Theorem 1 168

3. Proof of Theorem 2 171

4. Proof of Theorem 3 171

5. Proof of Theorem 4 172

References 174

1. Introduction and results

LetM(D) denote the class of all meromorphic functions in the unit disc D={z:|z|<1}. A function f ∈ M(D) is called normal if

kfk_N = sup

z∈D

f^#(z)(1− |z|²)<∞,

wheref^#(z) =|f⁰(z)|/(1 +|f(z)|²) is the spherical derivative off atz. The class of normal functions is denoted by N. For a given sequence {z_n}^∞_n=1 of points in D for which P∞

n=1(1− |z_n|²) converges (with the convention

Received August 2, 2010.

2000Mathematics Subject Classification. Primary 30D50; Secondary 30D35, 30D45.

Key words and phrases. Dirichlet space, normal function, Blaschke product.

This research was supported in part by the Academy of Finland #121281;

IG/SCI//DOMS/10/04; MTM2007-30904-E, MTM2008-05891, MTM2008-02829-E (MICINN, Spain); FQM-210 (Junta de Andaluca, Spain); and the European Science Foun- dation RNP HCAA.

ISSN 1076-9803/2011

165

RAUNO AULASKARI, SHAMIL MAKHMUTOV AND JOUNI R ¨ATTY ¨A

z_n/|z_n|= 1 for z_n = 0), the Blaschke product associated with the sequence {z_n}^∞_n=1 is defined as

B(z) =

∞

Y

n=1

|z_n| z_n

z_n−z 1−z_nz. Allen and Belna [1] showed that the analytic function

f_s(z) =B(z)/(1−z)^s,

where B(z) is the Blaschke product associated with{1−e⁻ⁿ}^∞_n=1, is not a normal function if 0< s < ¹₂, but satisfies the integrability condition

Z

D

|f_s⁰(z)|dA(z)<∞.

It is well-known that if f is analytic in Dand satisfies Z

D

|f⁰(z)|²dA(z)<∞,

that is, f belongs to the Dirichlet space (analytic functions in D with bounded area of image counting multiplicities), then f ∈ N. Concerning the normality, the question arose if

(1)

Z

D

|f⁰(z)|^pdA(z)<∞, 1< p <2,

implies f ∈ N. Yamashita [9] showed that this is not the case since the function

(2) f(z) =B(z) log 1

1−z,

where B is a Blaschke product associated with an exponential sequence {z_n}^∞_n=1 whose limit is 1, is not normal but satisfies (1) for all 1 < p < 2.

Recall that a sequence{z_n}^∞_n=1 is exponential if (3) 1− |z_n+1| ≤β(1− |z_n|), n∈N,

for some β ∈ (0,1). It is well known that every such sequence {z_n}^∞_n=1 satisfies

(4) Y

k6=n

z_n−z_k 1−z_nz_k

≥δ, n∈N,

for someδ=δ(β)>0, and is therefore an interpolating sequence (uniformly separated sequence).

The basic idea in this note is to find a function f that satisfies (1) (or another integrability condition) but the behavior of f^# is worse than the behavior of the spherical derivative of a nonnormal function necessarily is.

To make this precise, for 0 < α < ∞, a function f ∈ M(D) is called α- normal if

sup

z∈D

f^#(z)(1− |z|²)^α<∞.

The class of allα-normal functions is denoted byN^α.

Theorem 1. Let B be the Blaschke product associated with an exponential sequence{z_n}^∞_n=1 whose limit is 1. Let 1≤α <∞, 0< p <2 and

(5) fs(z) = B(z)

(1−z)^s, 0< s <∞.

Then f_s6∈ N^α for alls > α−1, but Z

D

|f_s⁰(z)|^pdA(z)<∞ for all s∈(0,2/p−1).

It is easy to see that fα−1 ∈ N^α. Moreover, the following result proves the sharpness of Theorem 1.

Theorem 2. Let B be the Blaschke product associated with an exponential sequence {z_n}^∞_n=1 such that |z_n− ¹₂|= ¹₂, =z_n >0 and limn→∞zn = 1. Let 0< p <1. Then

f²

p−1(z) = B(z) (1−z)^2p−1 satisfies

Z

D

|f⁰2

p−1(z)|^pdA(z) =∞.

The following result is of the same nature as Theorem 1.

Theorem 3. Let B be the Blaschke product associated with an exponential sequence{z_n}^∞_n=1 whose limit is 1. Let 1≤α <∞ and

f(z) = log 1 1−z

B(z) (1−z)^α−1. Then f 6∈ N^α, but

(6)

Z

D

|f⁰(z)|²(1− |z|²)^2α−2+εdA(z)<∞ for all ε >0.

Wulan [8] showed that the function f, defined in (2), satisfies

(7) f 6∈ [

0<p<∞

Q^#_p but f ∈ \

0<p<∞

M_p^#,

where

Q^#_p =

f ∈ M(D) : sup

a∈D

Z

D

(f^#(z))²g^p(z, a)dA(z)<∞

and M_p^#=

f ∈ M(D) :kfk²

Mp^# = sup

a∈D

Z

D

(f^#(z))²(1− |ϕ_a(z)|²)^pdA(z)<∞

.

RAUNO AULASKARI, SHAMIL MAKHMUTOV AND JOUNI R ¨ATTY ¨A

Here ϕ_a(z) = (a−z)/(1−az) is a M¨obius transformation and g(z, a) =

−log|ϕ_a(z)|is a Green’s function ofD.

Keeping (7) in mind, we will show that the function f_s, defined in (5), belongs to Mp^# for certain values ofs.

Theorem 4. Let B be the Blaschke product associated with an exponential sequence {z_n}^∞_n=1 whose limit is 1. Then the function f_s, defined in (5), satisfies

f_s∈ \

0<p<∞

M_p^#

for all 0< s≤1.

Theorems 1and 4 have the following immediate consequence.

Corollary 5.

\

0<p<∞

M_p^#6⊂ [

0<α<2

N^α.

Using [6, Theorem 3.3.3], with α = 2−2/q, we see that if f ∈ M(D) satisfies

sup

a∈D

Z

D

(f^#(z))^q(1− |z|²)^2q−4(1− |ϕ_a(z)|²)^pdA(z)<∞,

for some 2< q <∞ and 0≤p <∞, thenf ∈ N². In view of this fact and Corollary5 it is natural to ask the following questions.

Question 6. For which values of p the classM_p^# is contained inN²? Question 7. Is the class

B^#= (

f ∈ M(D) : sup

a∈D

Z

D(a,r)

(f^#(z))²dA(z)<∞ )

contained inN²?

Recall that Mp^#=B^# for all 1< p <∞, see [7,8].

Before embarking on the proofs of the results presented in this section, a word about the notation. We will writeA.Bif there is a positive constant C such that A ≤CB. The notation A &B is understood in an analogous manner. In particular, ifA.B .A, we will write A'B.

2. Proof of Theorem 1

The factfs6∈ N^α for alls > α−1 follows at once by the following lemma, see also [7, Theorem 4.4.2].

Lemma 8. Let 1≤α < ∞ and let B be an interpolating Blaschke product associated with the sequence {z_n}^∞_n=1. If g is analytic inD and satisfies

|g(z_nk)|(1− |z_nk|)^α−1 → ∞, k→ ∞, for some subsequence{z_nk}^∞_k=1, then Bg6∈ N^α.

Proof. Since B is interpolating, there existsδ >0 such that (Bg)^#(znk)(1− |z_nk|²)^α =|B⁰(znk)||g(z_nk)|(1− |z_nk|²)^α

≥δ|g(z_nk)|(1− |z_nk|)^α−1 → ∞, k→ ∞,

and henceBg6∈ N^α.

To prove the assertion on the integrability, let first 0< p≤1 ands < ²_p−1.

Then Z

D

|f_s⁰(z)|^pdA(z)≤ Z

D

|B⁰(z)|^p

|1−z|^psdA(z) + Z

D

s^p|B(z)|^p

|1−z|^p(s+1)dA(z) =I₁+I₂, whereI2<∞ sinces < ²_p−1. To estimateI1, write

B⁰(z) B(z) =

∞

X

n=1

|z_n|²−1 (1−z_nz)(z_n−z) so that

|B⁰(z)|=

∞

X

n=1

1− |z_n|² (1−znz)(zn−z)

∞

Y

k=1

|z_k| zk

z_k−z 1−zkz

(8)

≤

∞

X

n=1

1− |z_n|²

|z_n−z||1−znz|

|z_n−z|

|1−znz||B_n(z)|

≤

∞

X

n=1

|ϕ⁰_z

n(z)|, whereB_n(z) =Q

k6=n

|z_k| zk

z_k−z

1−zkz.Therefore

(9) |B⁰(z)|^p≤

∞

X

n=1

|ϕ⁰_zn(z)|^p, and the H¨older inequality yields

I₁ ≤ Z

D

∞

X

n=1

|ϕ⁰_z

n(z)|^p

! dA(z)

|1−z|^ps

=

∞

X

n=1

(1− |z_n|²)^p Z

D

dA(z)

|1−z_nz|^2p|1−z|^ps

≤

∞

X

n=1

(1− |z_n|²)^p Z

D

dA(z)

|1−z_nz|^2pµ _µ¹ Z

D

dA(z)

|1−z|^{psλ 1}_λ

,

RAUNO AULASKARI, SHAMIL MAKHMUTOV AND JOUNI R ¨ATTY ¨A

where µ ∈ (_2−sp² ,²_p) and λ∈ (_2−p² ,_sp²) such that λ⁻¹+µ⁻¹ = 1. The last integral above is finite because psλ < 2. Moreover, we may choose µ such thatpµ >1. Then

Z

D

dA(z)

|1−z_nz|^{2pµ 1}

µ

. 1

(1− |z_n|²)^2p−2/µ, and it follows that

I1 .

∞

X

n=1

(1− |z_n|²)^µ2−p ≤(1− |z₁|²)^2µ−p

∞

X

n=1

β^{(n−1)(µ2−p)}<∞

since _µ² −p >0.

Let now 1< p <2 and s < ²_p −1. Then Z

D

|f_s⁰(z)|^pdA(z)

≤2^p−1 Z

D

|B⁰(z)|^p

|1−z|^ps dA(z) + 2^p−1 Z

D

s^p|B(z)|^p

|1−z|^p(s+1)dA(z) =I1+I2, where I₂ <∞ sinces < ²_p −1. By using the inequality (9) with p= 1 and the Schwarz–Pick lemma, we obtain

I1≤2^p−1 Z

D

|B⁰(z)|

(1− |z|)^p−1|1−z|^ps dA(z)

≤2^p−1

∞

X

n=1

(1− |z_n|²) Z 1

0

1 (1−r)^ps+p−1

Z 2π 0

dθ

|1−znre^iθ|²dr

'

∞

X

n=1

(1− |z_n|²) Z 1

0

dr

(1−r)^ps+p−1(1− |z_n|r).

Choose the conjugate indexes µ and λsuch that µ <1/(ps+p−1). Then the H¨older inequality implies

I₁.

∞

X

n=1

(1− |z_n|) Z 1

0

dr (1−r)^µ(ps+p−1)

¹_µZ 1 0

dr (1− |z_n|r)^λ

_λ¹

.

∞

X

n=1

(1− |z_n|) 1

(1− |z_n|)^λ−1λ =

∞

X

n=1

(1− |z_n|)^1λ.

Since {z_n}^∞_n=1 is exponential, it follows that I1<∞, and we are done.

3. Proof of Theorem 2

Denote D⁻ = {z ∈ D :=z ≤ 0}. By [3, Theorem 1] there exists δ > 0 such that|B(z)| ≥δ for all z∈D⁻. Since 0< p <1, we have

Z

D

|f⁰2

p−1(z)|^pdA(z) +|f2

p−1(0)|^p ' Z

D

|f2

p−1(z)|^p(1− |z|²)^−pdA(z) (10)

≥ Z

D⁻

|B(z)|^p(1− |z|²)^−p

|1−z|^2−p dA(z)

&δ^p Z 1

0

ds

1−s =∞, and the assertion follows.

Note that an application of the asymptotic equality in (10) gives an al- ternative way to prove the case 0< p <1 in Theorem 1.

4. Proof of Theorem 3 Denote

gα(z) = (1−z)^1−αlog 1

1−z, 1≤α <∞.

Then Lemma8 implies thatf =Bgα 6∈ N^α.

To show that (6) is satisfied for all ε > 0, observe first that |f⁰|² ≤ 4(|g⁰_α|²+|g_α|²|B⁰|²). Now

I₁ = Z

D

|g_α⁰(z)|²(1− |z|²)^2α−2+εdA(z) .

Z 1 0

log e

1−r 2

(1−r)^2α−2+ε Z 2π

0

dθ

|1−re^iθ|^2α

dr

. Z 1

0

log e

1−r 3

(1−r)^ε−1dr <∞ for all ε >0, so it suffices to show that

I₂ = Z

D

|g_α(z)B⁰(z)|²(1− |z|²)^2α−2+εdA(z)<∞.

By using the inequality (9) with p = 1 and the Schwarz–Pick lemma, we obtain

I2 ≤ Z

D

|g_α(z)|²|B⁰(z)|(1− |z|²)^2α−3+εdA(z) .

∞

X

n=1

(1− |z_n|²) Z 1

0

log e

1−r 2

(1−r)^ε−1 Z 2π

0

dθ

|1−z_nre^iθ|²dr

'

∞

X

n=1

(1− |z_n|²) Z 1

0

log e

1−r 2

(1−r)^ε−1(1− |z_n|r)⁻¹dr.

RAUNO AULASKARI, SHAMIL MAKHMUTOV AND JOUNI R ¨ATTY ¨A

Choose the conjugate indexespandqsuch thatp(1−ε)<1. Here we assume, without loss of generality, thatε∈(0,1). Then the H¨older inequality implies I2.

∞

X

n=1

(1− |z_n|) Z 1

0

log e

1−r 2p

(1−r)^p(ε−1)dr

!¹_p Z 1

0

dr (1− |z_n|r)^q

¹_q

.

∞

X

n=1

(1− |z_n|) 1 (1− |z_n|)^q−1q

=

∞

X

n=1

(1− |z_n|)^1q.

Since{z_n}^∞_n=1 is exponential, there existsβ ∈(0,1) such that (1− |z_n+1|)≤ β(1− |z_n|) for all n∈ N. Therefore I₂ .(1− |z₁|)^1qP∞

n=1β^n−1q <∞, and we are done.

5. Proof of Theorem 4

This proof uses ideas from [2]. Let 0 < p < ∞ and 0 < s ≤ 1. Let δ^∗ =δ/4, and consider the pseudohyperbolic discs

D_n=D(z_n, δ^∗) ={z:|ϕ_zn(z)|< δ^∗}.

Then Dn∩D_k =∅ ifn6=k. Denote gs(z) = (1−z)^−s, so that

|f_s⁰(z)| ≤ |B⁰(z)||g_s(z)|+|B(z)||g⁰_s(z)|.

Further, denote E₁=∪^∞_n=1D_n and E₂=D\E₁. Then kf_sk²

Mp^# = Z

D

|f_s^#(z)|²(1− |ϕ_a(z)|²)^pdA(z)

≤2

sup

a∈D

I₁(E₁) + sup

a∈D

I₁(E₂) + sup

a∈D

I₂(E₁) + sup

a∈D

I₂(E₂)

, where

I1(F) = Z

F

|B⁰(z)g_s(z)|²

(1 +|B(z)g_s(z)|²)²(1− |ϕ_a(z)|²)^pdA(z) and

I₂(F) = Z

F

|B(z)g⁰_s(z)|²

(1 +|B(z)g_s(z)|²)²(1− |ϕ_a(z)|²)^pdA(z) forF ⊂D.

Recall that {z_n}^∞_n=1 satisfies (4) for some δ = δ(β) > 0, and therefore

|B(z)| ≥γ =γ(δ)>0 for all z∈E2 by [3, Theorem 1]. It follows that I₁(E₂) =

Z

E2

|B⁰(z)g_s(z)|²

(1 +|B(z)gs(z)|²)²(1− |ϕ_a(z)|²)^pdA(z)

≤ 1 γ²

Z

E2

|B⁰(z)|² γ²|g_s(z)|²

(1 +γ²|g_s(z)|²)²(1− |ϕ_a(z)|²)^pdA(z)

≤ 1 4γ²

Z

D

|B⁰(z)|²(1− |ϕ_a(z)|²)^pdA(z)≤M₁ <∞

for all a∈D by [4, p. 208 (2)]. Moreover, I₂(E₂) =

Z

E2

|B(z)g⁰_s(z)|²

(1 +|B(z)g_s(z)|²)²(1− |ϕ_a(z)|²)^pdA(z)

≤ 1 γ⁴

Z

E2

|g_s⁰(z)|²

|g_s(z)|⁴(1− |ϕ_a(z)|²)^pdA(z)

≤ s² γ⁴

Z

D

1

|1−z|^1−3s(1− |ϕ_a(z)|²)^pdA(z)≤M₂ <∞ for all p >0 and 0< s≤1.

To estimate the integrals over E1, note first that

|ϕ_a(z)|=|ϕ_ϕa(w)(ϕw(z))|, z, w, a∈D, and hence

|B(z)|=

∞

Y

n=1

ϕzn(z)

=|ϕ_zk(z)|

Y

n6=k

ϕzn(z)

=|ϕ_zk(z)|

Y

n6=k

ϕ_ϕzn(zk)(ϕz_k(z)) .

This yields

(11) 3δ

4 |ϕ_zk(z)| ≤ |B(z)|<|ϕ_zk(z)|, z∈D(z_k, δ^∗).

Therefore Z

Dk

|B(z)|²|g_s⁰(z)|²

(1 +|B(z)g_s(z)|²)²(1− |ϕ_a(z)|²)^pdA(z)

≤ Z

Dk

|ϕ_zk(z)|²|g_s⁰(z)|²(1− |ϕ_a(z)|²)^pdA(z)

= Z

D(0,δ^∗)

|w|²|g⁰_s(ϕ_zk(w))|²(1− |ϕ_a(ϕ_zk(w))|²)^p|ϕ⁰_z

k(w)|²dA(w) .(1− |ϕ_a(zk)|²)^p

Z

D(0,δ^∗)

|g⁰_s(ϕzk(w))|²|ϕ⁰_z

k(w)|²dA(w), where

Z

D(0,δ^∗)

|g_s⁰(ϕzk(w))|²|ϕ⁰_z

k(w)|²dA(w)

≤ Z

D(0,δ^∗)

|ϕ⁰_z

k(w)|²

|1−ϕz_k(w))|^2(s+1)dA(w) .(1− |z_k|)^4−2(s+1)≤1.

RAUNO AULASKARI, SHAMIL MAKHMUTOV AND JOUNI R ¨ATTY ¨A

Hence

I2(E1) = Z

E1

|B(z)g⁰_s(z)|²

(1 +|B(z)g_s(z)|²)²(1− |ϕ_a(z)|²)^pdA(z) .sup

a∈D

∞

X

k=1

(1− |ϕ_a(z_k)|²)^p <∞ by [5].

A similar reasoning as above together with the Schwarz–Pick lemma yields Z

Dk

|B⁰(z)|²|g_s(z)|²

(1 +|B(z)g_s(z)|²)²(1− |ϕ_a(z)|²)^pdA(z) .(1− |ϕ_a(z_k)|²)^p

Z

D(0,δ^∗)

|g_s(ϕ_zk(w))|² (1− |ϕ_zk(w)|²)²|ϕ⁰_z

k(w)|²dA(w), where

Z

D(0,δ^∗)

|g_s(ϕ_zk(w))|² (1− |ϕ_zk(w)|²)²|ϕ⁰_z

k(w)|²dA(w)

≤4 Z

D(0,δ^∗)

|ϕ⁰_z

k(w)|²

(1− |ϕ_zk(w)|²)^2+2sdA(w).(1− |z_k|)^2−2s≤1.

It follows that I1(E1) =

Z

E1

|B⁰(z)|²|g_s(z)|²

(1 +|B(z)g_s(z)|²)²(1− |ϕ_a(z)|²)^pdA(z) .sup

a∈D

∞

X

k=1

(1− |ϕ_a(zk)|²)^p <∞.

Putting everything together, we obtain f ∈Mp^#. References

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University of Eastern Finland, Department of Physics and Mathematics, Cam- pus of Joensuu, P. O. Box 111, 80101 Joensuu, Finland

[email protected]

Department of Mathematics and Statistics, Sultan Qaboos University, P. O.

Box 36, Al Khodh 123, Oman, UFA State Aviation Technical University, UFA, Russia

[email protected]

University of Eastern Finland, Department of Physics and Mathematics, Cam- pus of Joensuu, P. O. Box 111, 80101 Joensuu, Finland

[email protected]

This paper is available via http://nyjm.albany.edu/j/2011/17a-10.html.