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 classNαwhen 0< α <2. This implies thatMp#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
kfkN = sup
z∈D
f#(z)(1− |z|2)<∞,
wheref#(z) =|f′(z)|/(1 +|f(z)|2) is the spherical derivative off atz. The
class of normal functions is denoted by N. For a given sequence {zn}∞
n=1
of points in D for which P∞
n=1(1− |zn|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.
zn/|zn|= 1 for zn = 0), the Blaschke product associated with the sequence {zn}∞
n=1 is defined as
B(z) =
∞ Y
n=1
|zn|
zn
zn−z
1−znz
.
Allen and Belna [1] showed that the analytic function
fs(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
|fs′(z)|dA(z)<∞.
It is well-known that if f is analytic in Dand satisfies
Z
D
|f′(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
|f′(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
{zn}∞n=1 whose limit is 1, is not normal but satisfies (1) for all 1 < p < 2.
Recall that a sequence{zn}∞n=1 is exponential if (3) 1− |zn+1| ≤β(1− |zn|), n∈N,
for some β ∈ (0,1). It is well known that every such sequence {zn}∞n=1
satisfies
(4) Y
k6=n
zn−zk
1−znzk
≥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
The class of allα-normal functions is denoted byNα.
Theorem 1. Let B be the Blaschke product associated with an exponential sequence{zn}∞n=1 whose limit is 1. Let 1≤α <∞, 0< p <2 and
(5) fs(z) =
B(z)
(1−z)s, 0< s <∞. Then fs6∈ Nα for alls > α−1, but
Z
D
|fs′(z)|pdA(z)<∞
for all s∈(0,2/p−1).
It is easy to see that fα−1 ∈ Nα. Moreover, the following result proves
the sharpness of Theorem 1.
Theorem 2. Let B be the Blaschke product associated with an exponential sequence {zn}∞n=1 such that |zn− 12|= 12, ℑzn >0 and limn→∞zn = 1. Let
0< p <1. Then
f2
p−1(z) =
B(z)
(1−z)2p−1
satisfies
Z
D
|f′2 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 {zn}∞
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∈ Nα, but
(6)
Z
D
|f′(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<∞
Mp#,
where
Q#p =
f ∈ M(D) : sup
a∈D
Z
D
(f#(z))2gp(z, a)dA(z)<∞
and
Mp#=
f ∈ M(D) :kfk2
Mp# = supa∈
D
Z
D
(f#(z))2(1− |ϕa(z)|2)pdA(z)<∞
Here ϕa(z) = (a−z)/(1−az) is a M¨obius transformation and g(z, a) = −log|ϕa(z)|is a Green’s function ofD.
Keeping (7) in mind, we will show that the function fs, defined in (5),
belongs to Mp# for certain values ofs.
Theorem 4. Let B be the Blaschke product associated with an exponential sequence {zn}∞n=1 whose limit is 1. Then the function fs, defined in (5), satisfies
fs∈
\
0<p<∞
Mp#
for all 0< s≤1.
Theorems 1 and 4 have the following immediate consequence.
Corollary 5.
\
0<p<∞
Mp#6⊂ [
0<α<2
Nα.
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− |ϕa(z)|2)pdA(z)<∞,
for some 2< q <∞ and 0≤p <∞, thenf ∈ N2. In view of this fact and
Corollary 5 it is natural to ask the following questions.
Question 6. For which values of p the classMp# 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∈ Nα for alls > α−1 follows at once by the following lemma,
Lemma 8. Let 1≤α < ∞ and let B be an interpolating Blaschke product associated with the sequence {zn}∞
n=1. If g is analytic inD and satisfies
Proof. Since B is interpolating, there existsδ >0 such that
(Bg)#(znk)(1− |znk|
and the H¨older inequality yields
where µ ∈ (2−2sp,2p) and λ∈ (2−2p,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
and it follows that
I1 .
the Schwarz–Pick lemma, we obtain
I1≤2p−1 the H¨older inequality implies
3. Proof of Theorem 2
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
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.
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
for all a∈D by [4, p. 208 (2)]. Moreover,
To estimate the integrals over E1, note first that
Hence
I2(E1) =
Z
E1
|B(z)g′s(z)|2
(1 +|B(z)gs(z)|2)2
(1− |ϕa(z)|2)pdA(z)
.sup
a∈D
∞ X
k=1
(1− |ϕa(zk)|2)p <∞
by [5].
A similar reasoning as above together with the Schwarz–Pick lemma yields
Z
Dk
|B′(z)|2|g
s(z)|2
(1 +|B(z)gs(z)|2)2
(1− |ϕa(z)|2)pdA(z)
.(1− |ϕa(zk)|2)p
Z
D(0,δ∗)
|gs(ϕzk(w))| 2
(1− |ϕzk(w)|2)2
|ϕ′z k(w)|
2dA(w),
where
Z
D(0,δ∗)
|gs(ϕzk(w))| 2
(1− |ϕzk(w)|2)2
|ϕ′z k(w)|
2dA(w)
≤4
Z
D(0,δ∗)
|ϕ′zk(w)|2
(1− |ϕzk(w)|2)2+2s
dA(w).(1− |zk|)2−2s≤1.
It follows that
I1(E1) =
Z
E1
|B′(z)|2|g
s(z)|2
(1 +|B(z)gs(z)|2)2
(1− |ϕa(z)|2)pdA(z)
.sup
a∈D
∞ X
k=1
(1− |ϕa(zk)|2)p <∞.
Putting everything together, we obtain f ∈Mp#.
References
[1] Allen, H.; Belna, C.Non-normal functionsf(z) with∫ ∫|z|<1|f′(z)|dxdy < ∞. J.
Math. Soc. Japan24(1972) 128–131. MR0294649 (45 #3717) Zbl 0239.30038.
[2] Aulaskari, R.; Wulan, H.; Zhao. R.Carleson measures and some classes of mero-morphic functions.Proc. Amer. Math. Soc.128(2000), no. 8, 2329–2335. MR1657750
(2000k:30054) Zbl 0945.30027.
[3] Cima, J.; Colwell, P.Blaschke quotients and normality.Proc. Amer. Math. Soc.19
(1968) 796–798. MR0227423 (37 #3007) Zbl 0159.36704.
[4] Danikas, N.; Mouratides, Chr.Blaschke products inQpspaces.Complex Variables
Theory Appl.43(2000), no. 2, 199–209. MR1812465 (2001m:30040) Zbl 1021.30033.
[5] Ess´en, M.; Xiao, J.Some results on Qp spaces, 0< p <1.J. Reine Angew. Math.
485(1997) 173–195. MR1442193 (98d:46024) Zbl 0866.30027.
[6] R¨atty¨a, J. On some complex function spaces and classes. Ann. Acad. Sci. Fenn. Math. Diss.No. 124 (2001) 73 pp. MR1866201 (2002j:30052) Zbl 0984.30019. [7] Wulan, H.On some classes of meromorphic functions. Ann. Acad. Sci. Fenn. Math.
[8] Wulan, H.A non-normal function relatedQpspaces and its applications.Progress in
analysis, Vol. I, II(Berlin, 2001), 229–234,World Sci. Publ., River Edge, NJ, 2003. MR2032689.
[9] Yamashita, S.A nonnormal function whose derivative has finite area integral of order 0< p <2.Ann. Acad. Sci. Fenn. Ser. A I Math.4(1979), no. 2, 293–298. MR0565879
(81k:30042) Zbl 0433.30026.
University of Eastern Finland, Department of Physics and Mathematics, Cam-pus of Joensuu, P. O. Box 111, 80101 Joensuu, Finland
Department of Mathematics and Statistics, Sultan Qaboos University, P. O. Box 36, Al Khodh 123, Oman, UFA State Aviation Technical University, UFA, Russia
University of Eastern Finland, Department of Physics and Mathematics, Cam-pus of Joensuu, P. O. Box 111, 80101 Joensuu, Finland
