Exercises and Further Results
5. The Nevanlinna Class
Then since|B(z)| ≤1, it is clear that
u(z)≤log|g(z)|.
On the other hand
log|B(z)| =log|f(z)| −log|g(z)| ≤u(z)−log|g(z)|.
By Theorem 2.4, this means that
0≤u(z)−log|g(z)|
and henceu(z)=log|g(z)|. It of course follows thatg(z)∈N.
Theorem 5.3. Let f(z)∈N, f ≡0. Then f(z) has a nontangential limit f(eiθ)almost everywhere, and
log|f(eiθ)| ∈L1(dθ). (5.2)
The least harmonic majorant of log|f(z)|has the form
Pz(θ)dμ(θ), where dμ(θ)=log|f(θ)|dθ/2π+2μs,
(5.3)
with dμ, singular to dθ.
Proof. Letg(z)= f(z)/B(z), where B(z) is the Blaschke product with the same zeros as f(z). We knew from Theorem 5.1 and Lemma 5.2 that
log|g(z)| =
Pz(θ)dμ(θ).
(5.4)
Writedμ=k(θ)(dθ/2π)+dμs, wheredμs, is singular todθ. By (5.4) and by Theorem I.5.3, log|g(z)|has nontangential limitk(θ) almost everywhere. Since
|B(eiθ)| =1 almost everywhere, it follows that log|f(z)| has nontangential limitk(θ) almost everywhere. Therefore (5.2) and (5.3) will be proved once we show there exist nontangential limits for f(z).
By (5.4), log|g(z)| is the difference of two positive harmonic functions:
log|g| =u1−u2,uj ≥0. Letvj(z) be a harmonic conjugate function ofuj(z).
BecauseDis simply connected,vj(z) is well defined, andvj(z) is unique if we setvj(0)=0. Then
logg(z)=(u1+iv1)−(u2+iv2)+ic withca real constant, and hence
g(z)= eice(−u2+iv2) e−(u1+iv1) .
The bounded functionse−(uj+ivj)have nontangential limits almost everywhere, and these limits cannot vanish on a set of positive measure. Consequentlyg and f =Bghave nontangential limits.
If f(z)∈N, f ≡0, then by (5.2)
log|f(θ)|dθ >−∞.
However, the sharper inequality log|f(z)| ≤
log|f(θ)|Pz(θ)dθ 2π, (5.5)
which was proved forHp functions in Section 4, can fail for f(z)∈N. Con- sider the function
g(z)=exp1+z 1−z.
Then log|g(z)| =(1− |z|2)/(|1−z|2)= Pz(1). The functiongis inN, and the measure determined by log|g(z)|is the unit charge ateiθ =1. Since
log|g(0)| =1>0=
log|g(θ)|dθ 2π, (5.5) fails forg(z).
This counterexample contains the only thing that can go wrong with (5.5) for a function in N. The right side of (5.5) is a harmonic function; it ma- jorizes log|f(z)|if and only if it is bigger than the least harmonic majorant of log|f(z)|, which is the Poisson integral of the measuredμ, in (5.3). Com- paring (5.3) with (5.5), we therefore see that (5.5) is true for f(z)∈N if and only if the singular termdμ, in (5.3) is nonpositive.
The functions inNfor whichdμs,≤0 form a subclass ofNcalledN+. We give the classical definition:Let f(z)∈N.We say f(z)∈ N+if
r→1lim
log+|f(r eiθ)|dθ =
log+|f(eiθ)|dθ.
Theorem 5.4. Let f(z)∈N, f ≡0. Then the following are equivalent.
(a) f(z)∈ N+
(b) The least harmonic majorant oflog+|f(θ)|is
log+|f(θ)|Pz(θ)dθ 2π. (c) For all z∈ D,
log|f(z)| ≤
log|f(θ)|Pz(θ)dθ 2π.
(d) The least harmonic majorant oflog|f(z)|is the Poisson integral of dμ=log|f(θ)|dθ
2π +dμs,
where dμs⊥dθand
dμs≤0.
Proof. We have already proved that (c) and (d) are equivalent.
If f(z)∈N, then log+|f(z)|has least harmonic majorant U(z)=
Pz(θ)dν(θ),
where the positive measurevis the weak-star limit of the measures log+|f(r eiθ)|dθ/2π,
by Theorem I.6.7 and the remark thereafter. By Fatou’s lemma,
log+|f(θ)|Pz(θ)dθ 2π ≤lim
r→1
log+|f(r eiθ)|Pz(θ)dθ 2π =
Pz(θ)dν(θ), and hence
log+|f(θ)|dθ 2π ≤dν, (5.6)
because a measure is determined by its Poisson integral. By definition f(z)∈ N+ if and only if the two sides of (5.6), which are positive, have the same integral. Thus f ∈N+if and only if
log+|f(θ)|dθ 2π =dν and (a) and (b) are equivalent.
Finally, a comparison of the least harmonic majorants of log|f(z)| and of log+|f(z)|, as in the proof of Theorem 5.1, shows that (b) and (d) are equivalent.
It follows from (b) or (c) that
N ⊃ N+⊃ Hp, p>0.
It also follows that f(z)∈ Hp if and only f(z)∈ N+and f(eiθ)∈Lp. This fact generalizes Corollary 4.3 and it has the same proof, using Jensen’s in- equality with (c). This fact can be written
N+∩Lp =Hp, p>0.
The example given before Theorem 5.4 shows thatN ∩Lp = Hp.
We return to Theorem 5.3 and use formula (5.3) to obtain an important factorization theorem for functions inN. Let f(z)∈N, f ≡0. Let B(z) be the Blaschke product formed from the zeros of f(z), and letg(z)= f(z)/B(z).
Theng(z)∈N, and log|g(z)|is the Poisson integral of the measureμin (5.3).
By Theorem 5.4, f ∈ N+if and only ifg∈N+.
It is not hard to recoverg, and therefore also f = Bg, from the measureμ.
We actually did that in the proof of Theorem 5.3, but let us now do it again more carefully. Write
dμ=log|f(θ)|dθ/2π −(dμ1−dμ2), (5.7)
wheredμj ≥0 anddμj⊥dθ. The function F(z)=exp eiθ+z
eiθ−zlog|f(eiθ)|dθ 2π
is an outer function on the disc, because F(z)=eu(z)+iv(z),
whereu(z) is the Poisson integral of log|f(eiθ)|andv(z) is a conjugate function ofu(z) normalized byv(0)=0. Among the outer functions associated with log|f(θ)|,F(z) is determined by the conditionF(0)>0.
Similarly, let
Sj(z)=exp −
eiθ+z eiθ−zdμj(θ)
, j =1,2. (5.8)
ThenSj(z) is analytic onD, andSj(z) has the following properties:
(i) Sj(z) has no zeros inD.
(ii) |Sj(z)| ≤1,
(iii) |Sj(eiθ)| =1 almost everywhere, and (iv) Sj(0)>0.
Properties (i) and (iv) are immediate from (5.8). Since log|Sj(z)| = −
Pz(θ)dμj(θ), (5.9)
property (ii) holds because μj ≥0 and property (iii) follows from Lemma I.5.4 becausedμj⊥dθ. A function with properties (i)–(iv) is called asingular function. Every singular function S(t) has the form (5.8) for some positive singular measure. This measure is determined by (5.9).
We now have
log|g(z)| =log|F(z)| +log|S1(z)| −log|S2(z)|
by the decomposition (5.7) ofμ. Since g(z) has no zeros on D,logg(z) is single valued and hence
logg(z)=ic+logF(z)+logS1(z)−logS2(z), ca real constant, so that
g(z)=eicF(z)S1(z)/S2(z).
We have now proved most of the Canonical Factorization theorem:
Theorem 5.5. Let f(z)∈N, f ≡0. Then
f(z)=C B(z)F(z)S1(z)/S2(z), |C| =1, (5.10)
where B(z)is a Blaschke product, F(z)is an outer function, and S1(z)and S2(z)are singular functions. Except for the choice of the constant C,|C| =1, the factorization(5.10)is unique. Every function of the form(5.10)is in N.
Proof. We have already derived the factorization (5.10). There can be no dif- ficulties about the uniqueness of the factors becauseB(z) is determined by the zeros of f(z), and as|B(eiθ)| = |S1(eiθ)| = |S2(eiθ)| =1 almost everywhere, F(z) is determined by log|f(eiθ)|.S1andS2are then determined by the least harmonic majorant of log|f|. If f(z) is a function of the form (5.10), then
log|f(z)| ≤log|F(z)| +log|S1(z)| −log|S2(z)|,
so that log|f(z)| is majorized by a Poisson integral. It now follows from Theorem 5.1 that f(z)∈ N.
Corollary 5.6. Let f(z)∈N, f ≡0. Then in (5.10) the singular factor S2 ≡1if and only if f(z)∈N+.
Proof. S2(z)≡1 if and only if
dμ=log|f(θ)|(dθ/2π)−dμ1, withdμ1 ≥0, and this holds if and only if f(z)∈ N+.
For emphasis we state
Corollary 5.7. If f(z)∈Hp,p>0, then f(z)has a unique decomposition f(z)=C B(z)S(z)F(z),
where|C| =1,B(z)is a Blaschke product, S(z) is a singular function, and F(z)is an outer function in Hp.
Corollary 5.8. Let f(z)∈N+. Then f(z)∈ Hpif and only if the outer factor F(z)is in Hp.
The proofs of these corollaries are left to the reader.