Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 27, 2002, 381–390

ON ZEROS OF NORMAL FUNCTIONS

Maria Nowak

Uniwersytet Marii Curie-Skłodowskiej, Instytut Matematyki pl. Marii Curie-Skłodowskiej 1, PL-20-031 Lublin, Poland; nowakm@golem.umcs.lublin.pl

Abstract. We give necessary conditions for zero sets of normal functions, little normal functions and functions of uniformly bounded characteristic.

1. Introduction

A function f meromorphic in the unit disc D = {z ∈ C : |z| < 1} is normal if

(1.1)  ∥f∥ = sup_{z∈D}(1 − |z|^{2})f^{♯}(z) < ∞,

where f^{♯}(z) = |f′(z)|/(1 + |f(z)|^{2}). If

(1.2)  lim_{|z|→1}(1 − |z|^{2})f^{♯}(z) = 0,

then f is called a little normal function. The class of normal and little normal fuctions will be denoted by N and N_{0} , respectively.

The Bloch space ℬ consists of those functions f analytic on D for which

sup_{z∈D} |f′(z)|(1 − |z|^{2}) < ∞,

and the little Bloch space ℬ_{0} consists of those functions f ∈ ℬ for which

lim_{|z|→1} |f′(z)|(1 − |z|^{2}) = 0.

Since |f^{♯}(z)| ≤ |f′(z)|, it is clear that every Bloch function is a normal function. It was also observed by Tse [T] that if f ∈ ℬ, then g = e^{f} ∈ N. There are extensive results about normal and Bloch functions, see, e.g., [L], [ACP] and references given there.

In 1988 D. Ulrich [U] used random series to show that zero sets of elements of ℬ_{0} are different from the zero sets of elements of ℬ. More exactly, he proved that there is a function in ℬ whose zeros cannot be zeros of any function in ℬ_{o}.

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

If {z_{n}} is the sequence of zeros of a normal function f and |z_{1}| ≤ |z_{2}| ≤ ⋯ < 1 , then we call {z_{n}} the ordered zeros of f.

Recently it has been proved in [GNW] that if {z_{n}} are the ordered zeros of a Bloch function nonvanishing at zero, then

(1.3)  ∏_{n=1}^{N} \frac{1}{|z_{n}|} = O((log N)^{1/2}),  as N → ∞,

and if {z_{n}} (z_{n} ≠ 0) are the ordered zeros of a little Bloch function, then

∏_{n=1}^{N} \frac{1}{|z_{n}|} = o((log N)^{1/2}), as N → ∞.

This result has been motivated by Ch. Horowitz’ paper [Hor] on zeros of functions in Bergman space A^{p}. Horowitz proved that if {z_{n}} are ordered zeros of a function in A^{p}, 0 < p < ∞, nonvanishing at zero, then

∏_{n=1}^{N} \frac{1}{|z_{n}|} = O(N^{1/p}), as N → ∞.

Here we apply the above-mentioned result in [U], to show that (1.3) is sharp for the Bloch space in the sense that O((log N)^{1/2}) cannot be replaced by o((log N)^{1/2}).

In 1972 Anderson, Clunie and Pommerenke [ACP] showed that if f is normal, {z_{n}} is the sequence of zeros of f and D_{1} is a disc that touches ∂D from inside, then

∑_{z_{n}∈D_{1}}(1 − |z_{n}|) < ∞.

Here we obtain the following

Theorem 2. If f is a normal function, f(0) ≠ 0, and {z_{n}} are ordered zeros of f, then

(1.4)  ∏_{n=1}^{N} \frac{1}{|z_{n}|} = O(N^{∥f∥^{2}/2}),  as N → ∞.

We also obtain a similar result for little normal functions. In the last section we consider the class of functions of uniformly bounded characeristic (UBC) introduced by Yamashita in [Y].

2. A remark on zeros of Bloch functions

For a function f analytic on D and 0 < r < 1 set

∥f_{r}∥_{0} = exp(\frac{1}{2π} ∫_{0}^{2π} log |f(re^{iθ})| dθ).

It is known (see, e.g., [ACP], [U]) that if f is a Bloch function, then

(2.1)  ∥f_{r}∥_{0} = O((log \frac{1}{1 − r} )^{1/2}),

while

∥f_{r}∥_{0} = o((log \frac{1}{1 − r} )^{1/2}),

if f is in the little Bloch space. Moreover, it has been proved in [U], that there is f ∈ ℬ for which

(2.2)  ∥f_{r}∥_{0} ≠ o((log \frac{1}{1 − r} )^{1/2}).

We will use this result to show

Theorem 1. There is a function f ∈ ℬ with f(0) ≠ 0 whose ordered zeros {z_{n}} satisfy

∏_{n=1}^{N} \frac{1}{|z_{n}|} ≠ o((log N)^{1/2}), as N → ∞.

Proof. Assume that f ∈ ℬ satisfies (2.2) and f(0) ≠ 0. (One can take f(z) = f_{ω}(z)/z^{2} , where f_{ω} is given by (17) in [U].) Then there is a sequence {r_{m}}, 0 < r_{m} < 1, lim_{m→∞} r_{m} = 1, and a positive constant c such that

∥f_{r_{m}}∥_{0} ≥ c(log \frac{1}{1 − r_{m}} )^{1/2}.

This and the Jensen formula give

(2.3)  |f(0)| ∏_{|z_{k}|<r_{m}} \frac{r_{m}}{|z_{k}|} ≥ c(log \frac{1}{1 − r_{m}} )^{1/2}.

Let n(r) denote the number of zeros of f in the disc |z| ≤ r, where each zero is counted according to its multiplicity. Note that (2.3) implies that n(r_{m}) → ∞ as m → ∞. Moreover, by (2.1),

N(r, 0) = ∫_{0}^{r} \frac{n(t)}{t} dt ≤ C log log \frac{1}{1 − r} ,

which implies that (see, e.g., [SS, p. 225])

(2.4)  n(r) ≤ \frac{C log log \frac{1}{1 − r}}{1 − r} ,

or, equivalently,

log \frac{1}{1 − r} (1 + \frac{log log log \frac{1}{1 − r} + log C}{log \frac{1}{1 − r}} ) ≥ log n(r).

So, if ε > 0, then for r sufficiently close to 1,

log \frac{1}{1 − r} ≥ \frac{1}{1 + ε} log n(r).

Consequently, (2.3) yields

∏_{k=1}^{n(r_{m})} \frac{1}{|z_{k}|} > ∏_{k=1}^{n(r_{m})} \frac{r_{m}}{|z_{k}|} ≥ c_{1}(log n(r_{m}))^{1/2}

with some c_{1} > 0.

3. Proof of Theorem 2 Proof of Theorem 2. For 0 < r < 1, and for a function f meromorphic in D

set

A(r, f) = \frac{1}{π} ∫_{0}^{r}∫_{0}^{2π}(f^{♯}(te^{iθ}))^{2}t dθdt.

Then the Ahlfors–Shimizu characteristic T_{0}(r, f) is given by

T_{0}(r, f) = ∫_{0}^{r} \frac{A(t, f)}{t} dt.

By (1.1),

A(r, f) ≤ ∥f∥^{2} ∫_{0}^{r} \frac{2t}{(1 − t^{2})^{2}} dt = ∥f∥^{2} \frac{r^{2}}{1 − r^{2}} .

Consequently,

T_{0}(r, f) ≤ ∥f∥^{2} ∫_{0}^{r} \frac{tdt}{1 − t^{2}} = \frac{∥f∥^{2}}{2} log(\frac{1}{1 − r^{2}} ) ≤ \frac{∥f∥^{2}}{2} log(\frac{1}{1 − r} ).

On zeros of normal functions 385

On the other hand, if f(0) ≠ 0, then by the Ahlfors–Shimizu theorem [Hay, p. 12]

T_{0}(r, f) = N(r, 0) + m_{0}(r, 0) − m_{0}(0, 0),

where

N(r, 0) = ∑_{|z_{n}|<r} log \frac{r}{|z_{n}|}

and

m_{0}(r, 0) = \frac{1}{2π} ∫_{0}^{2π} log \frac{1}{k(f(re^{iθ}), 0)} dθ,

where k(w, a) denotes the cordial distance on the Riemann sphere given by

k(w, a) = \frac{|w − a|}{\sqrt{(1 + |a|^{2})(1 + |w|^{2})}} .

Thus

m_{0}(0, 0) = log \frac{\sqrt{1 + |f(0)|^{2}}}{|f(0)|} = c_{0} > 0.

It is also clear that m_{0}(r, 0) > 0. It then follows that

\begin{aligned}∑_{|z_{n}|<r} log \frac{r}{|z_{n}|} & = T_{0}(r, f) − m_{0}(r, 0) + c_{0} ≤ T_{0}(r, f) + c_{0} \\ & ≤ \frac{∥f∥^{2}}{2} log(\frac{1}{1 − r} ) + c_{0}. \end{aligned}

Now reasoning in a similar way as in [Hor, p. 625] shows that the inequality

∑_{n=1}^{N} log \frac{r}{|z_{n}|} ≤ \frac{∥f∥^{2}}{2} log(\frac{1}{1 − r} ) + c_{0}

actually holds for 0 < r < 1 and for all positive integers N. This in turn implies

∏_{n=1}^{N} \frac{r}{|z_{n}|} ≤ e^{c_{0}}(\frac{1}{1 − r} )^{∥f∥^{2}/2}.

Putting r = 1 − 1/N, we get for N ≥ 2,

∏_{n=1}^{N} \frac{1}{|z_{n}|} ≤ 4e^{c_{0}}N^{∥f∥^{2}/2}.

Remark. We do not know if (1.4) is sharp for normal functions. In [Hor] the author has constructed analytic functions whose ordered zeros 0 < |z_{1}| ≤ |z_{2}| ⋯ satisfy

∏_{n=1}^{N} \frac{1}{|z_{n}|} = O(N^{α}), α > 0.

He has also proved that these functions are in A^{p} with some p > 0. Here we show that, at least in some cases, they are not normal. To this end consider

(3.1)  f(z) = ∏_{k=1}^{∞}(1 − μz^{2^{k}}),  μ > 2.

It follows from [Hor, p. 698] that the ordered zeros of f satisfy

∏_{n=1}^{N} \frac{1}{|z_{n}|} = O(N^{α}) with α = log μ/ log 2.

To see that f given by (3.1) is not normal take 0 < x_{n} < 1 such

(3.2)  x_{n}^{2^{n}} = \frac{1}{μ},  n ≥ 1.

Since

f′(z) = ∑_{n}(−μ)2^{n}z^{2^{n}−1} ∏_{k≠n}(1 − μz^{2^{k}}),

we see that

\begin{aligned}|f′(x_{n})| & = 2^{n}μ^{1/2^{n}} ∏_{k<n}|(1 − μμ^{−2^{k−n}})| ∏_{k>n}(1 − μμ^{−2^{k−n}}) \\ & = 2^{n}μ^{1/2^{n}} ∏_{k=1}^{n−1}(μμ^{−2^{−k}} − 1)f(\frac{1}{μ}). \end{aligned}

By (3.2), 2^{n} = − log μ/ log x_{n}, and

(1 − x_{n})f^{♯}(x_{n}) = (1 − x_{n})|f′(x_{n})| > (1 − x_{n})\frac{− log μ}{log x_{n}}f(\frac{1}{μ}) ∏_{k=1}^{n−1}(μμ^{−2^{−k}} − 1).

Since

lim_{x→1} \frac{(1 − x)}{log x} = −1 and ∏_{k=1}^{∞}(μμ^{−2^{−k}} − 1)

diverges to +∞, and since f(x_{n}) = 0,

lim_{n→∞}(1 − x_{n})f^{♯}(x_{n}) = +∞.

4. Zeros of functions in N_{0}

The following theorem yields a necessary condition for zeros of the little normal functions.

Theorem 3. If f ∈ N_{0} , f(0) ≠ 0, and {z_{n}} are ordered zeros of f, then

∑_{n=1}^{N} log \frac{1}{|z_{n}|} = o(log N), as N → ∞.

Proof. It follows from (1.2) that, given ε > 0, there is r_{ε} such that

|f^{♯}(z)|(1 − |z|^{2}) < ε

for r_{ε} < |z| < 1. In [Y, p. 353] the following formula has been obtained

T_{0}(r, f) = \frac{1}{π} ∫_{|z|<r}(f^{♯}(z))^{2} log \frac{r}{|z|} dx dy.

Thus for 1 > r > r_{ε} we have

T_{0}(r, f) = \frac{1}{π} ∫_{|z|<r_{ε}}(f^{♯}(z))^{2} log \frac{r}{|z|} dx dy + \frac{1}{π} ∫_{r_{ε}<|z|<r}(f^{♯}(z))^{2} log \frac{r}{|z|} dx dy

≤ C + 2ε^{2} ∫_{r_{ε}}^{r} \frac{t}{(1 − t^{2})^{2}} log \frac{1}{t} dt.

Using the inequality log(1/t) ≤ (1 − t)/t, 0 < t < 1, we get

T_{0}(r, f) ≤ C + 2ε^{2} ∫_{r_{ε}}^{r} \frac{dt}{1 − t} ≤ C + 2ε^{2} log \frac{1}{1 − r} .

A reasoning similar to that used in the proof of Theorem 2 shows that the inequality

∑_{n=1}^{N} log \frac{r}{|z_{n}|} ≤ C + 2ε^{2} log \frac{1}{1 − r}

holds for 1 > r > r_{ε} and for all positive integers N. Finally, putting r = 1 − 1/N we see that for N ≥ 2,

∑_{n=1}^{N} log \frac{1}{|z_{n}|} ≤ C + log 4 + 2ε^{2} log N,

which implies the desired result.

5. The class UBC

For w ∈ D, we set

φ_{w}(z) = \frac{w − z}{1 − \overline{w}z} , z ∈ D.

In [Y] the author defined the class of functions of uniformly bounded characteristic as follows: a meromorphic function f in D is said to be of bounded characteristic (f ∈UBC) if and only if

sup_{w∈D} T_{0}(1, f_{w}) < ∞,

where T_{0}(1, f) = lim_{r→1} T_{0}(r, f) and f_{w} = f ∘ φ_{w}. In [Y] the sharp inclusion

UBC ⊂ N

was showed. Moreover, it is also clear that UBC is a subclass of the Nevalinna class (or the class of bounded characteristic, BC). Consequently, each non-zero f ∈ UBC admits the decomposition

(5.1)  f = \frac{b_{1}g}{b_{2}} ,

where b_{1} and b_{2} are the Blaschke products whose zeros are precisely the zeros and poles of f, respectively, and g ∈ BC has neither pole nor zero in D. Yamishita [Y] also proved that if f ∈UBC and (5.1) is satisfied, then g and fb_{2} = gb_{1} are also in UBC. The class UBC can be also considered as a meromorphic analogue of the space BMOA (i.e., the space of analytic functions of bounded mean oscillation, see e.g. [B]). The following characterization of zeros of functions of uniformly bounded characteristic has been motivated by the value distribution theorem for BMOA given in [B] and [Str].

Theorem 4. If {z_{n}} is the sequence of zeros of f ∈ UBC, then

sup\{∑_{n=1}^{∞} log \frac{1}{|φ_{w}(z_{n})|} : w ∈ D, |f(w)| ≥ 1\} < ∞.

In the case when |f(w)| < 1 for all w ∈ D we assume that

∑_{n=1}^{∞} log(1/|φ_{w}(z_{n})|) = 0.

Proof. Let f ∈UBC and let {z_{n}} and {p_{n}} be its zeros and poles, respectively. If T (r, f) denotes the Nevanlinna characteristic, then

T (r, f) = \frac{1}{2π} ∫_{0}^{2π} log^{+} |f(re^{iθ})| dθ + ∑_{|p_{n}|<r} log \frac{r}{|p_{n}|}

≥ \frac{1}{2π} ∫_{0}^{2π} log |f(re^{iθ})| dθ + ∑_{|p_{n}|<r} log \frac{r}{|p_{n}|}

= log |f(0)| + ∑_{|z_{n}|<r} log \frac{r}{|z_{n}|}, 0 < r < 1,

where the last equality follows from Jensen’s formula. We also know that the Nevanlinna characteristic and Ahlfors–Schimizu characteristic differ by a bounded term, that is [Hay, p. 12],

|T (r, f) − T_{0}(r, f) − log^{+} |f(0)|| ≤ \frac{1}{2} log 2.

Thus

T_{0}(r, f) ≥ T (r, f) − log^{+} |f(0)| − \frac{1}{2} log 2.

It then follows that

T_{0}(1, f) ≥ log |f(0)| + ∑_{z_{n}} log \frac{1}{|z_{n}|} − log^{+} |f(0)| − \frac{1}{2} log 2.

Replacing f by f_{w}, we see that if |w| < 1 is such that |f(w)| ≥ 1, then

∑_{z_{n}} log \frac{1}{|φ_{w}(z_{n})|} ≤ T_{0}(1, f_{w}),

which ends the proof.

Note that actually the following generalization of Theorem 4 is true.

Theorem 5. If {z_{n}} is a sequence of zeros of f ∈ UBC, then for any positive δ

sup\{∑_{n=1}^{∞} log \frac{1}{|φ_{w}(z_{n})|} : w ∈ D, |f(w)| ≥ δ\} < ∞.

390 Maria Nowak

References

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Received 1 October 2001