Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 283–293

QUOTIENT DECOMPOSITION OF Q_{p}^{#} FUNCTIONS

Hasi Wulan

Shantou University, Department of Mathematics Shantou, Guangdong 515063, P. R. China; wulan@stu.edu.cn

Abstract. For 0 < p < ∞, Q_{p}^{#} is the class of meromorphic functions f defined in the unit disk Δ satisfying that sup_{a∈Δ} ∫∫_{Δ}(f^{#}(z))^{2}g^{p}(z, a) dA(z) < ∞, where g(z, a) is the Green function of Δ. A sufficient and necessary condition for the quotient f = f_{1}/f_{2} of two bounded analytic functions f_{1} and f_{2} to belong to Q_{p}^{#} is given. Also, we prove that there exists a class X of meromorphic functions on Δ such that Q_{1}^{#} ⫋ X ⫋ N, where N is the class of normal functions. This observation gives an affirmative answer to a question in the literature.

1. Throughout this paper, Δ, ∂Δ and dA(z) are the unit disk on the complex plane, the boundary of the unit disk and the Euclidean area element on Δ, respectively. For a meromorphic function f on Δ, the Ahlfors–Shimizu characteristic function is defined as

T (r, f) = \frac{1}{π} ∫_{0}^{r} t^{−1} ∫∫_{|z|<t}(f^{#}(z))^{2} dA(z) dt, 0 < r < 1.

A function f meromorphic on Δ is said to belong to the Nevanlinna class 𝒩 if

T (1, f) = lim_{r→1} T (r, f) < ∞.

The well-known R. Nevanlinna quotient theorem (cf. [Ne, p. 188]) says that every function in 𝒩 is the quotient of two functions in

H^{∞} = {f : f analytic in Δ and ∥f∥_{∞} = sup_{z∈Δ} |f(z)| < ∞}.

If a meromorphic function f belongs to 𝒩 , then f = IO/J, where I, J are inner functions whose greatest common divisor is 1 and O is an outer function in 𝒩 . Conversely, such a function f = IO/J belongs to 𝒩 . Up to some unimodular constants, the functions I, J, O are uniquely determined in this case. An inner function is a function I analytic on Δ, having the properties |I(z)| ≤ 1 for all z ∈ Δ and |I(e^{iθ})| = 1 a.e. on ∂Δ. An outer function is a function of the form

O_{ψ}(z) = exp(∫_{∂Δ} \frac{ζ + z}{ζ − z} log ψ(ζ) \frac{|dζ|}{2π}),

2000 Mathematics Subject Classification: Primary 30D35, 30D45.

where ψ ≥ 0 a.e. on ∂Δ and log ψ ∈ L^{1}(∂Δ).

In his thesis, Carleson [Ca] considered the classes T_{α}, 0 ≤ α < 1, of meromorphic functions f on Δ satisfying

(1.1)  ∥f∥_{α} = ∫_{0}^{1}(1 − r)^{−α} ∫∫_{|z|<r}(f^{#}(z))^{2} dA(z)dr < ∞,

and the class T_{1} of meromorphic functions f on Δ with

(1.2)  ∥f∥_{1} = sup_{0<r<1} ∫∫_{|z|<r}(f^{#}(z))^{2} dA(z) < ∞.

Obviously, we have T_{1} ⊂ T_{α} ⊂ T_{β} ⊂ T_{0} for all α, β ∈ (0, 1) with α > β. The class T_{0} coincides with the Nevanlinna class 𝒩 .

Carleson [Ca] found a partial quotient theorem for T_{α} and further Aleman [Al] gave a complete result to the quotient decomposition for T_{α}, showing that each function in T_{α}, 0 < α ≤ 1, is the quotient of two functions in H^{∞}∩T_{α}. Using a result of Cima and Colwell [CC] for the class of normal functions, Yamashita [Ya1] gave criteria for a Blaschke quotient B_{1}/B_{2} to be of UBC, the class of all meromorphic functions of uniformly bounded characteristic on Δ (cf. [Ya2]). Xiao [Xi] considered the subclass BIT_{α} of T_{α} and proved that each function in BIT_{α}, 0 < α < 1, is the quotient of two functions in H^{∞} ∩ BIT_{α}. Recently, we studied [AW] the same problem for Q_{p}^{#} classes which have attracted considerable attention and proved that each function in Q_{p}^{#} , 0 < p < ∞, is the quotient of two functions in H^{∞} ∩ Q_{p}. We first show in Section 2 that the converse is not true, that is, there exist functions f_{1}, f_{2} ∈ H^{∞} ∩ Q_{p} but f = f_{1}/f_{2} ∉ Q_{p}^{#} for any p, 0 < p < ∞. The aim of Section 2 is to give a sufficient and necessary condition for f = f_{1}/f_{2} ∈ T_{0} to belong to Q_{p}^{#} (Q_{p,0}^{#} ) for all p ∈ (0, ∞). In Section 3 we prove that there exists a class X of meromorphic functions on Δ such that Q_{1}^{#} ⫋ X ⫋ N, where N is the class of normal functions. Notice that this observation gives an affirmative answer to the question in [Wu]. In Sections 4 and 5 we generalize a result in [DG] for Q_{p} spaces and give its counterpart for the subharmonic case.

2. The Green function on Δ with pole at a ∈ Δ is given by g(z, a) = log(1/|φ_{a}(z)|), where φ_{a}(z) = (a − z)/(1 − āz) is a Möbius transformation of Δ. For 0 < r < 1, let Δ(a, r) = {z ∈ Δ : |φ_{a}(z)| < r} be the pseudohyperbolic disk with center a and radius r. For 0 < p < ∞, we define the classes Q_{p}^{#} and Q_{p,0}^{#} of meromorphic functions f on Δ, respectively, for which

(2.1)  sup_{a∈Δ} ∫∫_{Δ}(f^{#}(z))^{2}(g(z, a))^{p} dA(z) < ∞

Quotient decomposition of Q_{p}^{#} functions 285

and

(2.2)  lim_{|a|→1} ∫∫_{Δ}(f^{#}(z))^{2}(g(z, a))^{p} dA(z) = 0,

where f^{#}(z) = |f′(z)|/(1 + |f(z)|^{2}) is the spherical derivative of f. We have that Q_{1}^{#} = UBC and for each p ∈ (1, ∞) the class Q_{p}^{#} is the class of normal meromorphic functions N (cf. [AL]) for which

∥f∥_{N} = sup_{z∈Δ}(1 − |z|^{2})f^{#}(z) < ∞.

Replacing f^{#}(z) by |f′(z)| in the above expressions (2.1) and (2.2) for analytic functions f on Δ, we obtain the spaces Q_{p} and Q_{p,0} (cf. [AL] and [AXZ]).

Let us define for an outer function O two cut-off outer functions (cf. [Al] and [Xi]):

O_{+}(z) = exp[∫_{∂Δ} \frac{ζ + z}{ζ − z} log(max\{|O(ζ)|, 1\})\frac{|dζ|}{2π}]

and

O_{−}(z) = exp[∫_{∂Δ} \frac{ζ + z}{ζ − z} log(min\{|O(ζ)|, 1\})\frac{|dζ|}{2π}].

Then it is easy to see that both O_{−} and 1/O_{+} belong to H^{∞} and O = O_{−}O_{+} .

Recently, we proved the following quotient decompositions of the Q_{p}^{#} and Q_{p,0}^{#} functions.

Theorem A ([AW]). Let f = IO/J ∈ T_{0} , where I, J are inner functions whose greatest common divisor is 1 and O is outer having two cut-off outer functions O_{+} and O_{−}. Let f_{1} = IO_{−} and f_{2} = J/O_{+} . For 0 < p < 1, if f = f_{1}/f_{2} belongs to Q_{p}^{#} , respectively Q_{p,0}^{#} , then both f_{1} and f_{2} lie in H^{∞} ∩Q_{p}, respectively H^{∞} ∩ Q_{p,0} .

We should mention that the converse of Theorem A is not true.

Theorem 1. There are two functions f_{1}, f_{2} ∈ H^{∞} ∩Q_{p} but f = f_{1}/f_{2} ∉ Q_{p}^{#} for 0 < p < ∞.

Proof. Let 0 < β < \frac{1}{2} and take the sequences \{z_{n}^{(1)}\} = \{1 − β^{n}\} and {z_{n}^{(2)}} = {1 − β^{n} − β^{2n}}. Note that

(1 − |z_{n+1}^{(i)}|^{2}) ≤ 2β(1 − |z_{n}^{(i)}|^{2}), n ≥ 1, i = 1, 2.

Consequently, the sequences {z_{n}^{(i)}} (i = 1, 2) are uniformly separated. Consider the Blaschke products B_{i} associated with the sequences {z_{n}^{(i)}}:

B_{i}(z) = ∏_{n=1}^{∞} \frac{z_{n}^{(i)} − z}{1 − z_{n}^{(i)}z} , i = 1, 2.

Clearly, B_{i} ∈ H^{∞} ∩ Q_{p} (i = 1, 2) if 1 ≤ p < ∞.

Take now p ∈ (0, 1). By a simple computation, we see that

∑_{j=k+1}^{∞}(1 − |z_{j}^{(i)}|^{2})^{p} ≤ \frac{2^{p}β^{p}}{1 − (2β)^{p}} (1 − |z_{k}^{(i)}|^{2})^{p}, k = 1, 2, …, i = 1, 2.

Then using the results of [RT], we deduce that dμ_{i}(z) = ∑_{n=1}^{∞}(1 − |z_{n}^{(i)}|^{2})^{p}δ_{z_{n}^{(i)}} is a bounded p-Carleson measure and then, by [EX], B_{i} ∈ Q_{p} ∩ H^{∞}, i = 1, 2.

Finally, B_{1}/B_{2} ∉ Q_{p}^{#} for any p > 0 because B_{1}/B_{2} is not a normal function. Indeed, we have

|\frac{z_{j}^{(2)} − z_{j}^{(1)}}{1 − z_{j}^{(1)}z_{j}^{(2)}} | ≤ \frac{β^{j}}{2 − β^{2j}} → 0, as j → ∞,

and then it follows that {z_{n}^{(1)}}∪{z_{n}^{(2)}} is not interpolating which, using Theorem 2 of [CC], implies B_{1}/B_{2} is not a normal function as announced.

Now we turn to the main result in this paper.

Theorem 2. Let p ∈ (0, ∞). Let f = IO/J ∈ T_{0} , where I, J are inner functions whose greatest common divisor is 1 and O is outer having the two cut-off outer functions O_{+} and O_{−}. Let f_{1} = IO_{−} and f_{2} = J/O_{+} . Then f = f_{1}/f_{2} ∈ Q_{p}^{#} if and only if both f_{1} and f_{2} lie in H^{∞} ∩ Q_{p} and

inf_{z∈Δ}(|f_{1}(z)|^{2} + |f_{2}(z)|^{2}) > 0.

Proof. Our first observation is the following identity:

(2.3)  \begin{aligned}T (1, f_{1}/f_{2}) = & \frac{1}{4π} ∫_{0}^{2π} log(|f_{1}(e^{iθ})|^{2} + |f_{2}(e^{iθ})|^{2}) dθ \\ & − \frac{1}{2} log(|f_{1}(0)|^{2} + |f_{2}(0)|^{2}). \end{aligned}

In fact, let J = B_{J}S_{J}, where B_{J} is a Blaschke product and S_{J} is a singular inner function. By Lemma 4.2 in [Ya2] we have

T (1, f_{1}/f_{2}) = lim_{r→1} \frac{1}{4π} ∫_{0}^{2π} log(1 + |(IO/J)(re^{iθ})|^{2}) dθ

− \frac{1}{2} log(|B_{J}(0)|^{2} + |IO(0)/S_{J}(0)|^{2})

= lim_{r→1} \frac{1}{4π} ∫_{0}^{2π} log(|(J/O_{+})(re^{iθ})|^{2} + |(IO_{−})(re^{iθ})|^{2}) dθ

− lim_{r→1} \frac{1}{4π} ∫_{0}^{2π} log|(B_{J}S_{J}/O_{+})(re^{iθ})|^{2} dθ

− \frac{1}{2} log(|B_{J}(0)|^{2} + |IO(0)/S_{J}(0)|^{2}).

Note that every function f ∈ T_{0} with f ≠ 0 has a non-tangential limit at e^{iθ} a.e. on ∂Δ, denoted by f(e^{iθ}), and log |f(e^{iθ})| belongs to L^{1}[0, 2π]. Bearing in mind that log |S_{J}/O_{+}| is harmonic and that

lim_{r→1} ∫_{0}^{2π} log |B_{J}(re^{iθ})| dθ = 0,

we have

T (1, f_{1}/f_{2}) = lim_{r→1} \frac{1}{4π} ∫_{0}^{2π} log(|(J/O_{+})(re^{iθ})|^{2} + |(IO_{−})(re^{iθ})|^{2}) dθ

− \frac{1}{2} log |S_{J}(0)/O_{+}(0)|^{2} − \frac{1}{2} log(|B_{J}(0)|^{2} + |IO(0)/S_{J}(0)|^{2})

= \frac{1}{4π} ∫_{0}^{2π} log(|f_{1}(e^{iθ})|^{2} + |f_{2}(e^{iθ})|^{2}) dθ − \frac{1}{2} log(|f_{1}(0)|^{2} + |f_{2}(0)|^{2}).

Thus (2.3) is proved. Next, putting f_{ϱ}(z) = f(ϱz) for 0 < ϱ ≤ 1 in (2.3) we get

(2.4)  \begin{aligned}T (ϱ, f_{1}/f_{2}) & = \frac{1}{4π} ∫_{0}^{2π} log(|f_{1}(ϱe^{iθ})|^{2} + |f_{2}(ϱe^{iθ})|^{2}) dθ \\ & − \frac{1}{2} log(|f_{1}(0)|^{2} + |f_{2}(0)|^{2}). \end{aligned}

For a ∈ Δ, by replacing f in (2.4) by f ∘ φ_{a}, we obtain

(2.5) \begin{aligned}T (ϱ, f_{1} ∘ φ_{a}/f_{2} ∘ φ_{a}) & = \frac{1}{4π} ∫_{0}^{2π} log(|f_{1} ∘ φ_{a}(ϱe^{iθ})|^{2} + |f_{2} ∘ φ_{a}(ϱe^{iθ})|^{2}) dθ \\ & − \frac{1}{2} log(|f_{1}(a)|^{2} + |f_{2}(a)|^{2}). \end{aligned}

Proof of necessity. Suppose f_{1}/f_{2} ∈ Q_{p}^{#} .

Case 1. If 0 < p ≤ 1, by Theorem A and the fact that H^{∞} ⊂ Q_{1} we know that both f_{1} and f_{2} belong to H^{∞} ∩ Q_{p}. Since Q_{p}^{#} ⊂ UBC for 0 < p ≤ 1, f_{1}/f_{2} ∈ UBC, which is equivalent to

sup_{a∈Δ} T (1, f_{1} ∘ φ_{a}/f_{2} ∘ φ_{a}) < ∞.

Because the first term in the right side of (2.5) is increasing on ϱ, we have

0 < sup_{a∈Δ} lim_{ϱ→1} \frac{1}{4π} ∫_{0}^{2π} log(|f_{1} ∘ φ_{a}(ϱe^{iθ})|^{2} + |f_{2} ∘ φ_{a}(ϱe^{iθ})|^{2}) dθ

≤ \frac{1}{2} log(∥f_{1}∥_{∞}^{2} + ∥f_{2}∥_{∞}^{2}) < ∞.

Hence

sup_{a∈Δ} T (1, f_{1} ∘ φ_{a}/f_{2} ∘ φ_{a}) < ∞ ⟺ inf_{a∈Δ}(|f_{1}(a)|^{2} + |f_{2}(a)|^{2}) > 0.

Case 2. If 1 < p < ∞, we have that f_{1}/f_{2} ∈ N since Q_{p}^{#} = N for all p ∈ (1, ∞). It is clear that both f_{1} and f_{2} belong to H^{∞} ∩ Q_{p} since f_{1}, f_{2} ∈ H^{∞} which is contained in Q_{p} for all p ∈ (1, ∞). To complete the proof of necessity we claim that the following statement is true.

Statement. Let f be a meromorphic function on Δ. Then f ∈ N if and only if there exists a ϱ ∈ (0, 1) such that sup_{a∈Δ} T (ϱ, f ∘ φ_{a}) < ∞.

Once the statement is proved just as in the case 1 above we have that f_{1}/f_{2} ∈ N implies that

inf_{a∈Δ}(|f_{1}(a)|^{2} + |f_{2}(a)|^{2}) > 0.

Now we give the proof of the statement. Assume first that f ∈ N. Then for a fixed ϱ ∈ (0, 1) we have

T (ϱ, f ∘ φ_{a}) = \frac{1}{π} ∫_{0}^{ϱ} \frac{dt}{t} ∫∫_{|z|<t}(f ∘ φ_{a})^{#}(z)^{2} dA(z)

= \frac{1}{π} ∫_{0}^{ϱ} \frac{dt}{t} ∫∫_{Δ(a,t)}(f^{#}(z))^{2} dA(z)

≤ \frac{1}{π} ∥f∥_{N}^{2} ∫_{0}^{ϱ} \frac{dt}{t} ∫∫_{Δ(a,t)} \frac{dA(z)}{(1 − |z|^{2})^{2}}

= \frac{1}{2}∥f∥_{N}^{2} log \frac{1}{1 − ϱ^{2}} ,

which shows that

sup_{a∈Δ} T (ϱ, f ∘ φ_{a}) < ∞.

Conversely, suppose that ϱ ∈ (0, 1) and sup_{a∈Δ} T (ϱ, f ∘ φ_{a}) < ∞. Choose ϱ_{0} with 0 < ϱ_{0} < ϱ such that

sup_{a∈Δ} T (ϱ, f ∘ φ_{a})(log \frac{ϱ}{ϱ_{0}} )^{−1} < 1.

Now,

\begin{aligned}T (ϱ, f ∘ φ_{a}) & ≥ \frac{1}{π} ∫_{ϱ_{0}}^{ϱ} \frac{dt}{t} ∫∫_{Δ(a,t)}(f^{#}(z))^{2} dA(z) \\ & ≥ \frac{1}{π} (log \frac{ϱ}{ϱ_{0}} ) ∫∫_{Δ(a,ϱ_{0})}(f^{#}(z))^{2} dA(z). \end{aligned}

It follows that

sup_{a∈Δ} ∫∫_{Δ(a,ϱ_{0})}(f^{#}(z))^{2} dA(z) ≤ sup_{a∈Δ} T (ϱ, f ∘ φ_{a})π(log \frac{ϱ}{ϱ_{0}} )^{−1} < π,

which shows that f ∈ N by Lemma 3.2 in [Ya2].

Proof of sufficiency. Let

δ := inf_{z∈Δ}(|f_{1}(z)|^{2} + |f_{2}(z)|^{2}) > 0,

and let f_{1}, f_{2} ∈ H^{∞} ∩ Q_{p} with 0 < p < ∞. We obtain

sup_{a∈Δ} ∫∫_{Δ}((f_{1}/f_{2})^{#}(z))^{2}(g(z, a))^{p} dA(z)

= sup_{a∈Δ} ∫∫_{Δ} \frac{|f′_{1}(z)f_{2}(z) − f_{1}(z)f′_{2}(z)|^{2}}{(|f_{1}(z)|^{2} + |f_{2}(z)|^{2})^{2}} (g(z, a))^{p} dA(z)

≤ \frac{2}{δ^{2}} (∥f_{1}∥_{∞}^{2} + ∥f_{2}∥_{∞}^{2}) sup_{a∈Δ} ∫∫_{Δ}(|f′_{1}(z)|^{2} + |f′_{2}(z)|^{2})(g(z, a))^{p}dA(z)

< ∞,

which shows that f_{1}/f_{2} ∈ Q_{p}^{#} . The proof of Theorem 2 is complete.

We have the following result, similar to Theorem 2.

Theorem 3. Let f = IO/J ∈ T_{0} , where I, J are inner functions whose greatest common divisor is 1 and O is outer having the two cut-off outer functions O_{+} and O_{−}. Let f_{1} = IO_{−} and f_{2} = J/O_{+} . Then, for 0 < p < 1, the following are equivalent.

(a) f = f_{1}/f_{2} ∈ Q_{p,0}^{#} .

(b) inf_{z∈Δ}(|f_{1}(z)|^{2} + |f_{2}(z)|^{2}) > 0 and both f_{1} and f_{2} lie in H^{∞} ∩ Q_{p,0} .

Bearing in mind that Q_{p,0} ⊂ VMOA if 0 < p ≤ 1 and that the only Blaschke products in VMOA are the finite Blaschke products [Se] and using Theorem 3, we obtain the following result.

Corollary 1. If 0 < p ≤ 1, B_{1} and B_{2} are two Blaschke products without common zeros, then B_{1}/B_{2} ∈ Q_{p,0}^{#} if and only if B_{i} ∈ Q_{p,0} , i = 1, 2.

3. Note that Q_{p}^{#} = N for all p ∈ (1, ∞) and Q_{1}^{#} = UBC ⫋ N. It is natural to ask whether there exists a class X of meromorphic functions on Δ such that UBC ⫋ X ⫋ N. We describe such a class X which in fact also gives an affirmative answer to the question in [Wu] (see Remark 3.1).

For 0 < p < ∞, let X_{p} be the family of meromorphic functions f on Δ such that

(3.1)  sup_{a∈Δ} ∫∫_{Δ}(f^{#}(z))^{p}(1 − |z|^{2})^{p−2}g(z, a) dA(z) < ∞.

Obviously, X_{p} = Q^{#}(p, 1) (cf. [Wu]).

Theorem 4. We have UBC ⫋ X_{p} ⫋ N for 2 < p < ∞.

Proof. By Theorem 3.2.1(ii) in [Wu] we need only show that there exists a normal function f such that f ∉ X_{p}. By [LX] there are two functions f_{1} and f_{2} in N such that

M_{0} := inf_{z∈Δ}(1 − |z|^{2})(f_{1}^{#}(z) + f_{2}^{#}(z)) > 0.

Therefore,

sup_{a∈Δ} ∫∫_{Δ}{(f_{1}^{#}(z))^{p} + (f_{2}^{#}(z))^{p}}(1 − |z|^{2})^{p−2}g(z, a) dA(z)

≥ sup_{a∈Δ} 2^{−p} ∫∫_{Δ}(f_{1}^{#}(z) + f_{2}^{#}(z))^{p}(1 − |z|^{2})^{p−2}g(z, a) dA(z)

≥ sup_{a∈Δ} \frac{M_{0}^{p}}{2^{p}} ∫∫_{Δ}(1 − |z|^{2})^{−2}g(z, a) dA(z)

= sup_{a∈Δ} \frac{M_{0}^{p}}{2^{p}} ∫∫_{Δ}(1 − |w|^{2})^{−2} log \frac{1}{|w|} dA(w) = ∞.

Hence, f_{1} ∉ X_{p} or f_{2} ∉ X_{p}. Thus, f_{1} ∈ N \ X_{p} or f_{2} ∈ N \ X_{p}.

4. Dyakonov and Girela [DG] gave a new characterization of Q_{p} functions as follows.

Theorem B ([DG]). Let 0 < p < 1. An analytic function f belongs to Q_{p} if and only if

(4.1)  sup_{a∈Δ} ∫∫_{Δ} |f′(z)|^{2} \frac{(1 − |φ_{a}(z)|^{2})^{p}}{|φ_{a}(z)|^{2p}} dA(z) < ∞.

In fact, Theorem B can be generalized.

Theorem 5. Let 0 < p < ∞ and 0 ≤ s < 1. An analytic function f belongs to Q_{p} if and only if

(4.2)  sup_{a∈Δ} ∫∫_{Δ} |f′(z)|^{2} \frac{(1 − |φ_{a}(z)|^{2})^{p}}{|φ_{a}(z)|^{2s}} dA(z) < ∞.

Note that if s = 0 it has been proved in [ASX] and if 0 < p = s < 1 it is just Theorem B. The following is the counterpart of Theorem 5 for the meromorphic case and we omit the proof of Theorem 5 here since it is similar to that of Theorem 6 below.

Theorem 6. Let 0 < p < ∞ and 0 < s < 1. Then a meromorphic function f belongs to Q_{p}^{#} if and only if

(4.3)  sup_{a∈Δ} ∫∫_{Δ}(f^{#}(z))^{2} \frac{(1 − |φ_{a}(z)|^{2})^{p}}{|φ_{a}(z)|^{2s}} dA(z) < ∞.

Proof. We suppose that f ∈ Q_{p}^{#} . By Theorem 2.2.2 in [Wu] we know that f ∈ N ∩ M_{p}^{#} , where

M_{p}^{#} = {f : f meromorphic on Δ, sup_{a∈Δ} ∫∫_{Δ}(f^{#}(z))^{2}(1 − |φ_{a}(z)|^{2})^{p} dA(z) < ∞}.

Thus for a fixed r, 0 < r < 1 and 0 < s < 1, we have

∫∫_{Δ}(f^{#}(z))^{2} \frac{(1 − |φ_{a}(z)|^{2})^{p}}{|φ_{a}(z)|^{2s}} dA(z) = ∫∫_{Δ(a,r)}(f^{#}(z))^{2} \frac{(1 − |φ_{a}(z)|^{2})^{p}}{|φ_{a}(z)|^{2s}} dA(z)

+ ∫∫_{Δ\Δ(a,r)}(f^{#}(z))^{2} \frac{(1 − |φ_{a}(z)|^{2})^{p}}{|φ_{a}(z)|^{2s}} dA(z)

≤ ∫∫_{Δ(a,r)}(f^{#}(z))^{2} \frac{(1 − |φ_{a}(z)|^{2})^{p}}{|φ_{a}(z)|^{2s}} dA(z) (4.4)

+ \frac{1}{r^{2s}} ∫∫_{Δ\Δ(a,r)}(f^{#}(z))^{2}(1 − |φ_{a}(z)|^{2})^{p} dA(z).

Since f ∈ M_{p}^{#} , the supremum over all a ∈ Δ of the second term in the right-hand side of (4.4) is finite. The change of variables w = φ_{a}(z) in the first term of (4.4) yields

sup_{a∈Δ} ∫∫_{Δ(a,r)}(f^{#}(z))^{2} \frac{(1 − |φ_{a}(z)|^{2})^{p}}{|φ_{a}(z)|^{2s}} dA(z)

≤ ∥f∥_{N}^{2} ∫∫_{Δ(a,r)}(1 − |z|^{2})^{−2} \frac{(1 − |φ_{a}(z)|^{2})^{p}}{|φ_{a}(z)|^{2s}} dA(z)

= ∥f∥_{N}^{2} ∫∫_{|w|<r}(1 − |w|^{2})^{p−2}|w|^{−2s} dA(w)

= 2π∥f∥_{N}^{2} ∫_{0}^{r}(1 − t^{2})^{p−2} t^{1−2s} dt < ∞.

Hence, (4.3) follows.

Conversely, suppose that the supremum in (4.3) is K with 0 < K < ∞. It is easy to see that this implies that f ∈ M_{p}^{#} since |φ_{a}(z)| < 1. Fix an r ∈ (0, 1) such that r^{2s}K/(1 − r^{2})^{p} < \frac{1}{2}π. Thus we obtain

\begin{aligned}∫∫_{Δ(a,r)}(f^{#}(z))^{2}d A(z) & ≤ \frac{r^{2s}}{(1 − r^{2})^{p}} ∫∫_{Δ(a,r)}(f^{#}(z))^{2} \frac{(1 − |φ_{a}(z)|^{2})^{p}}{|φ_{a}(z)|^{2s}} dA(z) \\ & ≤ \frac{r^{2s}K}{(1 − r^{2})^{p}} < \frac{π}{2} . \end{aligned}

By Lemma 3.2 in [Ya2] we obtain that f ∈ N, and thus f ∈ N ∩ M_{p}^{#} . By Theorem 2.2.2 in [Wu] again we conclude that f ∈ Q_{p}^{#} . The proof is now complete.

5. An example in [AWZ] shows that for s = 0 Theorem 6 is not true. That is, unlike the analytic case, for a meromorphic function f, (f^{#}(z))^{2} may not be subharmonic in general and the Green function g(z, a) in the definition of Q_{p}^{#} , sometimes, can not be replaced by the expression 1 − |φ_{a}(z)|^{2} . Moreover, we have

Theorem 7. Let u ≥ 0 be subharmonic on Δ, 0 < p < ∞ and 0 ≤ s < 1. Then the following are equivalent.

(i)  sup_{a∈Δ} ∫∫_{Δ} u(z)(g(z, a))^{p} dA(z) < ∞.

(ii)  sup_{a∈Δ} ∫∫_{Δ} u(z)\frac{(1 − |φ_{a}(z)|^{2})^{p}}{|φ_{a}(z)|^{2s}} dA(z) < ∞.

Corollary 2. Let u ≥ 0 be subharmonic in Δ and p > 1. Then

sup_{z∈Δ} u(z)(1 − |z|^{2})^{2} < ∞

if and only if

sup_{a∈Δ} ∫∫_{Δ} u(z)(g(z, a))^{p} dA(z) < ∞.

Remark. Theorem 7 and Corollary 2 generalize some results in [AL] and [ASX] since |f′(z)|^{2} is subharmonic in Δ for an analytic function f.

Acknowledgements. This research is supported in part by a grant from the National Natural Science Foundation of China (No. 10371069) and a grant from the NSFG (No. 010446) of China. I would like to thank the referee for helpful comments.

References

[Al] Aleman, A.: A conjecture of L. Carleson and applications. - Michigan Math J. 39, 1992, 537–549.

[AL] Aulaskari, R., and P. Lappan: Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal. - In: Complex Analysis and its Applications, Pitman Res. Notes Math. Ser. 305, Longman Scientific & Technical, Harlow, 1994, pp. 136–146.

[ASX] Aulaskari, R., D. Stegenga, and J. Xiao: Some subclasses of BMOA and their characterization in terms of Carleson measures. - Rocky Mountain J. Math. 26, 1996, 485–506.

[AW] Aulaskari, R., and H. Wulan: Outer functions in Q_{p}^{#} , Q_{p,0}^{#} . - J. Funct. Spaces Appl. 2, 2004, 17–24.

[AXZ] Aulaskari, R., J. Xiao, and R. Zhao: On subspaces and subsets of BMOA and UBC. - Analysis 15, 1995, 101–121.

[Ca] Carleson, L.: On a class of meromorphic functions and its associated exceptional sets. - Thesis, University of Uppsala, 1950.

[CC] Cima, J. A., and P. Colwell: Blaschke quotients and normality. - Proc. Amer. Math. Soc. 19, 1968, 796–798.

[DG] Dyakonov, K. M., and D. Girela: On Q_{p} spaces and pseudoanalytic extension. - Ann. Acad. Sci. Fenn. Math. 25, 2000, 477–486.

[EX] Essén, M., and J. Xiao: Some results on Q_{p} -spaces, 0 < p < 1. - J. Reine Angew. Math. 485, 1997, 173–195.

[LX] Lappan, P., and J. Xiao: Q_{α}^{#} -bounded composition maps on normal classes. - Note Mat. 20, 2000/01, 65–72.

[Ne] Nevanlinna, R.: Analytic Functions. - Springer-Verlag, Berlin–Heidelberg–New York, 1970.

[RT] Reséndiz, O., and L. M. Tovar: Carleson measures, Blaschke products and Q_{p} spaces. In: Complex Analysis and Differential Equations (Uppsala, 1997), pp. 296–305, Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist. 64, Uppsala Univ., Uppsala, 1999.

[Se] Stegenga, D.: Bounded Toeplitz operators on H^{1} and applications of the duality between H^{1} and the functions of bounded mean osillation. - Amer. J. Math. 98, 1976, 573–589.

[Wu] Wulan, H.: On some classes of meromorphic functions. - Ann. Acad. Sci. Fenn. Math. Diss. 116, 1998, 1–57.

[Xi] Xiao, J.: Biholomorphically invariant families amongst Carleson class . - Indag. Math. (N.S.) 12, 2001, 597–607.

[Ya1] Yamashita, S.: Criteria for a Blaschke quotient to be of uniformly bounded characteristic. - Proc. Amer. Math. Soc. 93, 1985, 618–620.

[Ya2] Yamashita, S.: Functions of uniformly bounded characteristic . - Ann. Acad. Sci. Fenn. Ser. A I Math. 7, 1982, 349–367.

Received 27 May 2003