Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 27, 2002, 69–90

ON THE GROWTH OF ENTIRE FUNCTIONS WITH ZERO SETS HAVING INFINITE EXPONENT OF CONVERGENCE

Joseph Miles

University of Illinois, Department of Mathematics 1409 West Green Street, Urbana, IL 61801, U.S.A.; joe@math.uiuc.edu

Abstract. Let Z be a sequence of complex numbers tending to infinity having infinite exponent of convergence. For r > 0, let n(r) denote the number of members of Z of modulus at most r. If

μ = \overline{lim}_{r→∞} \frac{log log n(r)}{log log r} ,

it is shown that

inf_{f} \underline{lim}_{r→∞} \frac{log M(r, f)}{n(r)} ≤ min (π + log 2, log \frac{μ + 1}{μ − 1} ) ,

where the infimum is over all entire f vanishing precisely on Z and M(r, f) denotes the maximum of |f(z)| on |z| = r. This bound strengthens earlier results of Bergweiler.

1. Introduction

Let Z = {z_{j} : j = 1, 2, 3, …} be a sequence of complex numbers, not necessarily distinct, tending to infinity and ordered so that |z_{1}| ≤ |z_{2}| ≤ · · ·. We consider entire functions with zero set precisely Z, i.e., entire f with a zero of multiplicity m at a complex number β provided that z_{j} = β for exactly m different values of j. We compare the rate of growth of the maximum modulus function

M(r, f) = max_{|z|=r} |f(z)|

of such f with n(r), the number of distinct values of j such that |z_{j}| ≤ r.

The exponent of convergence σ of the sequence Z is defined to be the infimum of all α > 0 for which

∑_{z_{j}≠0} |z_{j}|^{−α} < ∞,

or equivalently by

(1.1)  σ = \overline{lim}_{r→∞} \frac{log n(r)}{log r}.

2000 Mathematics Subject Classification: Primary 30D15; Secondary 30D20, 30D35.

It is an easy consequence of Jensen’s theorem that for any entire f vanishing precisely on Z, the order ρ of f, defined by

ρ = \overline{lim}_{r→∞} \frac{log log M(r, f)}{log r},

satisfies ρ ≥ σ. We shall be concerned exclusively with sequences Z with infinite exponent of convergence, and thus all entire f we consider will be of infinite order.

One measure of the growth of n(r) for sequences with infinite exponent of convergence is

(1.2)  μ = \overline{lim}_{r→∞} \frac{log log n(r)}{log log r}.

It is elementary to verify that μ ≥ 1 if σ > 0 and that σ = ∞ if μ > 1.

Bergweiler has studied the growth of entire functions vanishing precisely on a sequence Z with infinite exponent of convergence. Let

L(μ) := sup_{Z} inf_{f} \underline{lim}_{r→∞} \frac{log M(r, f)}{n(r)},

where the supremum is over all sequences Z satisfying (1.2) and the infimum is over all entire f vanishing precisely on Z. (In the case μ = 1, we additionally require of Z that σ = ∞.) Bergweiler [2] proved the following two theorems.

Theorem A. For every Z satisfying (1.2), there exists an entire f vanishing precisely on Z such that

\underline{lim}_{r→∞} \frac{log M(r, f)}{n(r)} ≤ α(μ) := \{ \begin{matrix}\frac{3μ}{4(μ − 1)}, & 1 < μ < ∞, \\ \frac{3}{4}, & μ = ∞. \end{matrix}

Theorem B. For 1 < μ < ∞ there exists Z satisfying (1.2) such that for every entire f vanishing precisely on Z,

\underline{lim}_{r→∞} \frac{log M(r, f)}{n(r)} ≥ β(μ) := (μ − 1) log \frac{μ − 1}{μ} + (μ + 1) log \frac{μ + 1}{μ}.

We note that Theorem A implies that L(μ) ≤ α(μ) and Theorem B implies L(μ) ≥ β(μ). We also note that α(μ) tends to infinity as μ → 1^{+} and that Theorem A gives no information about L(1). It is elementary that β(μ) is a decreasing function of μ on (1, ∞) with β(μ) → 2 log 2 as μ → 1^{+} and β(μ) → 0 as μ → ∞. Bergweiler [1] has asked (i) whether L(μ) is a bounded function for 1 ≤ μ ≤ ∞ and in particular if L(1) is finite and (ii) whether L(μ) → 0 as μ → ∞.

We answer these questions by proving the following two theorems. We remark that our results give information about the minimum modulus of f as well as the maximum modulus.

Theorem 1. If Z satisfies (1.2) for 1 < μ ≤ ∞, then

(1.3)  inf_{f} \underline{lim}_{r→∞} \frac{∥log |f(re^{iθ})|∥_{∞}}{n(r)} ≤ \{ \begin{matrix}log \frac{μ + 1}{μ − 1}, & 1 < μ < ∞, \\ 0, & μ = ∞, \end{matrix}

where the infimum is over all entire f vanishing precisely on Z.

Theorem 2. If Z is any sequence with infinite exponent of convergence, then

inf_{f} \underline{lim}_{r→∞} \frac{∥log |f(re^{iθ})|∥_{∞}}{n(r)} ≤ π + log 2,

where the infimum is over all entire f vanishing precisely on Z.

We note that the combination of Theorems 1 and 2 yields

L(μ) ≤ \{ \begin{matrix}min(π + log 2, log \frac{μ + 1}{μ − 1}), & 1 ≤ μ < ∞, \\ 0, & μ = ∞. \end{matrix}

We further note that Theorem 1, in conjunction with Theorem B, shows that

(1.4)  \frac{1 − o(1)}{μ} ≤ L(μ) ≤ \frac{2 + o(1)}{μ},  μ → ∞.

By a refinement of his proof of Theorem B, Bergweiler was able to show [2, p. 103] that lim inf_{μ→1} L(μ) > 1.6. This result together with Theorem 2 yields

(1.5)  1.6 < sup_{Z} inf_{f} \underline{lim}_{r→∞} \frac{log M(r, f)}{n(r)} ≤ π + log 2,

where the supremum is over all Z with infinite exponent of convergence and the infimum is over all entire f vanishing precisely on Z. We have been unable to narrow the gap between the upper and lower bounds in either (1.4) or (1.5).

Our results involve upper bounds for log M(r, f) in terms of n(r) on a sequence tending to infinity. It is observed in [2] that in general it is not possible to obtain such bounds on a large set of r-values, for example on a set of positive logarithmic density. Suppose, for example, that Z has infinite exponent of convergence, all members of Z are positive real numbers, and

\overline{lim}_{{r→∞ \atop r∈E}} \frac{log n(r)}{log r} < ∞

for some set E ⊂ [1, ∞) of upper logarithmic density 1. It is well known [5] that any entire f vanishing precisely on Z has infinite lower order. Thus

lim_{{r→∞ \atop r∈E}} \frac{log M(r, f)}{n(r)} = ∞

for any such f.

It is perhaps worth remarking that no results such as Theorems 1 and 2 are possible for the ratio log M(r, f)/N(r), where

N(r) = ∫_{0}^{r} \frac{n(t)}{t} dt

is the integrated counting function of value distribution theory. For in Bergweiler’s examples in Theorem B, n(r)/N(r) → ∞ as r → ∞ and thus log M(r, f)/N(r) → ∞ for every entire f vanishing precisely on Z.

There is a vast literature concerning comparisons of the growth of log M(r, f) for an entire function f to the distribution of its zeros. An excellent collection of references appears in [2].

2. Preliminaries

The following lemma, used in the proof of Theorem 2, is due in its essential form to Newman [8]. (See also [3].) We are indebted to J. Fournier for bringing the lemma to our attention and for several helpful communications. For completeness we include Fournier’s proof of the lemma in the precise form required for our purposes.

Lemma 1. Suppose 0 < ε < 1 and that

P (θ) = ∑_{k=−M}^{M} c_{k}e^{ikθ}

is a real trigonometric polynomial with

(2.1)  |c_{k}| ≤ \frac{1}{2}(\frac{1}{M + 1 − ε − |k|}),  1 ≤ |k| ≤ M,

and

(2.2)  |c_{0}| ≤ \frac{1}{M + 1 − ε}.

Then there exists a real trigonometric polynomial

Q(θ) = ∑_{k=−L}^{L} d_{k}e^{ikθ}

where L > M such that d_{k} = c_{k} for |k| ≤ M and

∥Q∥_{∞} < \frac{π}{1 − ε} + ε.

Proof. Let

R(θ) = ∑_{k=−M}^{−1} c_{k}e^{ikθ} + \frac{c_{0}}{2}

and let b_{j} = c_{j−M−1} for 1 ≤ j ≤ M and b_{M+1} = \frac{1}{2}c_{0} . For 0 ≤ j ≤ M − 1, we have by (2.1)

(2.3)  |b_{j+1}| = |c_{j−M}| ≤ \frac{1}{2}(\frac{1}{M + 1 − ε + j − M} ) < \frac{1}{2(1 − ε)(j + 1)}.

From (2.2) we conclude

(2.4)  |b_{M+1}| = \frac{|c_{o}|}{2} ≤ \frac{1}{2(M + 1 − ε)} < \frac{1}{2(1 − ε)(M + 1)}.

We define a linear functional T: H^{1} → 𝒞 by

T (f) = T (∑_{j=0}^{∞} a_{j}z^{j}) = ∑_{j=0}^{M} a_{j}\overline{b_{j+1}}.

By a theorem of Hardy [4, p. 48],

|T (f)| ≤ ∑_{j=0}^{M} |a_{j}| |\overline{b_{j+1}}| ≤ ∑_{j=0}^{M} \frac{|a_{j}|}{2(1 − ε)(j + 1)} ≤ \frac{π∥f∥_{1}}{2(1 − ε)},

where we have used (2.3) and (2.4). The Hahn–Banach theorem implies that T extends to a bounded linear functional on L^{1}[−π, π] with norm at most π/2(1−ε), and thus there exists a function h on [−π, π] bounded by π/2(1 − ε) such that

T (f) = \frac{1}{2π} ∫_{−π}^{π} f(θ)\overline{h(θ)} dθ

74 Joseph Miles for all f ∈ H^{1} , where f(θ) = lim_{r→1} f(re^{iθ}).

Letting f(θ) = e^{i(n−1)θ} ∈ H^{1} for n ≥ 1, we get

T (f) = \frac{1}{2π} ∫_{−π}^{π} e^{i(n−1)θ}\overline{h(θ)} dθ = \{ \begin{matrix}\overline{b_{n}}, & 1 ≤ n ≤ M + 1, \\ 0, & n > M + 1. \end{matrix}

Taking conjugates we obtain

\frac{1}{2π} ∫_{−π}^{π} e^{−inθ}H(θ) dθ = \{ \begin{matrix}b_{n}, & 1 ≤ n ≤ M + 1, \\ 0, & n > M + 1, \end{matrix}

where

H(θ) = e^{iθ}h(θ).

Setting H^{∗}(θ) = e^{−i(M+1)θ}H(θ), we have

(2.5)  \frac{1}{2π} ∫_{−π}^{π} e^{−i(n−M−1)θ}H^{∗}(θ) dθ = \{ \begin{matrix}b_{n}, & 1 ≤ n ≤ M + 1, \\ 0, & n > M + 1. \end{matrix}

Using c_{n−M−1} = b_{n} for 1 ≤ n ≤ M and \frac{1}{2}c_{o} = b_{M+1} , we rewrite (2.5) as

(2.6)  \frac{1}{2π} ∫_{−π}^{π} e^{−ikθ}H^{∗}(θ) dθ = \{ \begin{matrix}c_{k}, & −M ≤ k ≤ −1, \\ \frac{1}{2}c_{o}, & k = 0, \\ 0, & k > 0. \end{matrix}

Clearly ∥H^{∗}∥_{∞} = ∥h∥_{∞} ≤ π/2(1 − ε) and we note by (2.6) that H^{∗} has the same Fourier coefficients as does R for k ≥ −M. We set

G(θ) = H^{∗}(θ) + \overline{H^{∗}(θ)}.

Thus ∥G∥_{∞} ≤ π/(1 − ε) and G has the same Fourier coefficients as does P for |k| ≤ M.

For all integers k, let

B_{k} = \frac{1}{2π} ∫_{−π}^{π} e^{−ikθ}G(θ) dθ.

Consider the L^{th} Cesàro mean σ_{L} of G, i.e., let

σ_{L}(θ, G) = ∑_{k=−L}^{L}(1 − \frac{|k|}{L + 1})B_{k}e^{ikθ}.

Because Fejér’s kernel is positive, we have

∥σ_{L}∥_{∞} ≤ ∥G∥_{∞} < \frac{π}{1 − ε}.

Finally, set

Q(θ) = σ_{L}(θ, G) + ∑_{k=−M}^{M} \frac{|k|}{L + 1}c_{k}e^{ikθ}.

Since |c_{k}| < 1/(1 − ε) for |k| ≤ M, we have

∥Q∥_{∞} ≤ ∥σ_{L}∥_{∞} + \frac{M(M + 1)}{(L + 1)(1 − ε)} < \frac{π}{1 − ε} + ε

provided that L + 1 > M(M + 1)/ε(1 − ε). Furthermore, for |k| ≤ M, since B_{k} = c_{k}, we see that

\frac{1}{2π} ∫_{−π}^{π} e^{−ikθ}Q(θ) dθ = c_{k},

proving Lemma 1.

The following elementary growth lemma is used in the proof of Theorem 1.

Lemma 2. Suppose g: [0, ∞) → [0, ∞) is a nondecreasing function continuous from the right and suppose that 0 < α < \overline{lim}_{x→∞} g(x)/x. Then there exists a nondecreasing sequence x_{j} → ∞ such that

g(x) − g(x_{j}) ≤ α(x − x_{j}), 0 ≤ x ≤ x_{j}.

Proof. Let b_{j} → ∞ be an increasing sequence and let

x_{j} = inf{x ≥ 0 : g(x) ≥ αx + b_{j}}.

Clearly x_{j} is a well-defined sequence of real numbers such that x_{j+1} ≥ x_{j} → ∞. The continuity of g from the right ensures that g(x_{j}) ≥ αx_{j} + b_{j}. The definition of x_{j} implies that

g(x) < αx + b_{j}, 0 ≤ x < x_{j},

and hence

g(x) − g(x_{j}) < α(x − x_{j}), 0 ≤ x < x_{j},

completing the proof of Lemma 2.

The proofs of both Theorems 1 and 2 are based on an analysis of the Fourier series of log |H(re^{iθ})| where H is entire. Suppose that H is entire with |H(0)| = 1. Suppose that H has zero sequence {z_{ν}} with due regard to multiplicity and that |z_{ν}| ≠ r for ν = 1, 2, 3, …. Let n(t) be the number of zeros of H counting multiplicity of modulus no more than t. We write

log |H(re^{iθ})| = ∑_{m=−∞}^{∞} c_{m}(r, H)e^{imθ},

where the Fourier coefficients, first studied by F. Nevanlinna [7] (see also [6]), are given by the following formulas:

(2.7)  c_{0}(r, H) = N(r) = ∫_{0}^{r} \frac{n(t)}{t} dt,
(2.8)  c_{m}(r, H) = \frac{β_{m}}{2} r^{m} + \frac{1}{2m} ∑_{|z_{ν}|<r} (\frac{r}{z_{ν}} )^{m} − (\frac{\overline{z_{ν}}}{r} )^{m},  m ≥ 1,

where

(2.9)  log H(z) = ∑_{m=1}^{∞} β_{m}z^{m}

near 0 for some branch of the logarithm, and

c_{−m}(r, H) = \overline{c_{m}(r, H)}, m ≥ 1.

We collect certain estimates (see (2.14) and (2.15)) involving the Fourier series of log |H(re^{iθ})| which are common to the proofs of Theorems 1 and 2. Suppose z_{ν} ≠ 0, z_{ν} → ∞, and H is the convergent product

(2.10)  H(z) = ∏_{ν=1}^{∞} E(\frac{z}{z_{ν}} , ν − 1),

where we recall that the logarithm vanishing at z = 0 of the usual Weierstrass factor E(z, p) is given by

(2.11)  log E(z, p) = −\frac{z^{p+1}}{p + 1} − \frac{z^{p+2}}{p + 2} − ⋯,  |z| < 1.

Suppose that H has no zero of modulus r. From (2.8), (2.9), and (2.11) we conclude that if |z_{m}| < r, then

(2.12)  c_{m}(r, H) = \frac{1}{2m} ∑_{{ν>m \atop |z_{ν}|<r}} (\frac{r}{z_{ν}} )^{m} − \frac{1}{2m} ∑_{|z_{ν}|<r}(\frac{\overline{z_{ν}}}{r} )^{m},

and if |z_{m}| > r then

(2.13)  c_{m}(r, H) = −\frac{1}{2m} ∑_{{ν≤m \atop r<|z_{ν}|}}(\frac{r}{z_{ν}} )^{m} − \frac{1}{2m} ∑_{|z_{ν}|<r}(\frac{\overline{z_{ν}}}{r} )^{m}.

Since the modulus of each term on the right side of (2.13) is at most 1/2m and there are at most m such terms, we see that if |z_{m}| > r, then

(2.14)  |c_{m}(r, H)| ≤ \frac{1}{2}.

We now suppose that |z_{m}| < r. Taking into account the possibility of H having more than one zero of modulus |z_{m}|, we see from (2.12) that

|c_{m}(r, H)| ≤ \frac{1}{2m} ∫_{|z_{m}|}^{r}((\frac{r}{t} )^{m} − (\frac{t}{r} )^{m}) dn(t)

+ \frac{1}{2m}(n(|z_{m}|) − m)((\frac{r}{|z_{m}|})^{m} − (\frac{|z_{m}|}{r})^{m}) + \frac{1}{2m} ∑_{ν≤m}(\frac{|z_{ν}|}{r})^{m} (2.15)

≤ \frac{1}{2} ∫_{|z_{m}|}^{r}((\frac{r}{t} )^{m} + (\frac{t}{r} )^{m})\frac{n(t)}{t} dt + \frac{1}{2},

where we have used integration by parts and the fact that the summation in the middle expression contains at most m terms, each of modulus no more than 1/2m. A critical role in the proof of both of our theorems is played by (2.15).

3. Proof of Theorem 1

We first suppose 1 < μ < ∞. With no loss of generality, we may presume that |z_{1}| > e, for multiplication of f by a polynomial leaves (1.3) unaffected. Let ε in (0, 1) be such that (1 − ε)^{2}μ > 1 + 2ε. With the convention log^{+} 0 = 0, we apply Lemma 2 for t ≥ e with x = log log t, g(x) = log^{+} log^{+} n(e^{e^{x}}), and α = (1 − ε)μ to conclude that there exists an increasing sequence r_{j}^{∗} such that

log^{+} log^{+} n(t) − log log n(r_{j}^{∗}) ≤ (1 − ε)μ(log log t − log log r_{j}^{∗}), e ≤ t ≤ r_{j}^{∗}.

By continuity it is immediate that there exists r_{j} > r_{j}^{∗} such that n(r_{j}) = n(r_{j}^{∗}) and

(3.1)

log^{+} log^{+} n(t) − log log(1 + ε)n(r_{j})
≤ (1 − ε)μ(log log t − log log r_{j}), e ≤ t ≤ r_{j}.

We rearrange (3.1) to obtain

(3.2)  \frac{log^{+} n(t)}{log(1 + ε)n(r_{j})} ≤ (\frac{log t}{log r_{j}} )^{(1−ε)μ},  e ≤ t ≤ r_{j}.

Hence

\frac{log^{+} n(t)}{log t} ≤ (\frac{log t}{log r_{j}} )^{(1−ε)μ−1}(\frac{log(1 + ε)n(r_{j})}{log r_{j}}), e ≤ t ≤ r_{j},

or

(3.3)  \frac{log^{+} n(t)}{log t} ≤ \frac{log(1 + ε)n(r_{j})}{log r_{j}},  e ≤ t ≤ r_{j}.

To simplify notation, we define

(3.4)  u_{j} := \frac{log(1 + ε)n(r_{j})}{log r_{j}}

and rewrite (3.3) as

(3.5)  n(t) ≤ t^{u_{j}},  0 ≤ t ≤ r_{j}.

Since Z has infinite exponent of convergence we observe from (1.1) and (3.3) that u_{j} → ∞ as j → ∞.

For a fixed α > 1, consider the function g(y) = y^{α} − 1, 0 ≤ y ≤ 1. Clearly g(0) = −1, g(1) = 0, and g is increasing and convex on (0, 1). If y_{0} in (0, 1) is determined by y_{0}^{α−1} = 1 − ε, then

\frac{g(y) − (−1)}{y − 0} ≤ \frac{g(y_{0}) − (−1)}{y_{0} − 0} = 1 − ε, 0 < y < y_{0},

and

\frac{g(y) − 0}{y − 1} ≥ \frac{g(y_{0}) − 0}{y_{0} − 1} = αỹ_{0}^{α−1} > α(1 − ε), y_{0} < y < 1,

for some ỹ_{0} in (y_{0}, 1).

Recalling that α = (1 − ε)μ > 1 and setting y = (log t)/ log r_{j}, we conclude that there exists r̃_{j} in [1, r_{j}) with log r̃_{j}/ log r_{j} = y_{0} such that

(3.6)  (\frac{log t}{log r_{j}} )^{(1−ε)μ} < (1 − ε)\frac{log t}{log r_{j}} ,  1 < t < r̃_{j},

and

(3.7)  (\frac{log t}{log r_{j}} )^{(1−ε)μ} − 1 < (1 − ε)^{2}μ(\frac{log t}{log r_{j}} − 1),  r̃_{j} < t < r_{j}.

From (3.2) and (3.6) we have

\frac{log^{+} n(t)}{log(1 + ε)n(r_{j})} < (1 − ε)\frac{log t}{log r_{j}} , e ≤ t < r̃_{j},

or

(3.8)  n(t) < t^{(1−ε)u_{j}},  0 < t < r̃_{j}.

From (3.2) we have

(3.9) log^{+} n(t) − log(1 + ε)n(r_{j}) < (log(1 + ε)n(r_{j}))((\frac{log t}{log r_{j}} )^{(1−ε)μ} − 1),

for e ≤ t ≤ r_{j}. The combination of (3.7) and (3.9) yields

log n(t) − log(1 + ε)n(r_{j}) < (1 − ε)^{2}μu_{j}(log t − log r_{j}),  r̃_{j} < t < r_{j},

or

(3.10)  n(t) < (1 + ε)n(r_{j})(\frac{t}{r_{j}} )^{(1−ε)^{2}μu_{j}},  r̃_{j} < t < r_{j}.

We define the entire function H by (2.10) and proceed to estimate |c_{m}(r_{j}, H)| from above. First we note from (2.7), (3.4), and (3.5) that

(3.11)  \begin{aligned}|c_{0}(r_{j}, H)| & = ∫_{0}^{r_{j}} \frac{n(t)}{t} dt ≤ ∫_{0}^{r_{j}} t^{u_{j}−1} dt \\ & = \frac{r_{j}^{u_{j}}}{u_{j}} = \frac{(1 + ε)n(r_{j})}{u_{j}} = o(n(r_{j})). \end{aligned}

For |z_{m}| > r_{j}, i.e., for m > n(r_{j}), we have (2.14) with r = r_{j}. Thus we consider m such that |z_{m}| < r_{j}. First suppose m > (1 + ε)u_{j}. From (2.15) and (3.5) we have

(3.12)  \begin{aligned}|c_{m}(r_{j}, H)| & ≤ ∫_{|z_{m}|}^{r_{j}}(\frac{r_{j}}{t} )^{m} \frac{n(t)}{t} dt + \frac{1}{2} \\ & ≤ r_{j}^{m} ∫_{|z_{m}|}^{r_{j}} t^{u_{j}−m−1}dt + \frac{1}{2} ≤ r_{j}^{m}\{\frac{|z_{m}|^{u_{j}−m}}{m − u_{j}} + \frac{1}{2r_{j}^{m}} \} \\ & < r_{j}^{m}\{\frac{1}{εu_{j}|z_{m}|^{m(1−(1/(1+ε))}} + \frac{1}{2r_{j}^{m}} \}. \end{aligned}

80 Joseph Miles Writing

(3.13) c_{m}(r_{j}, H) = r_{j}^{m}β_{m}(j), (1 + ε)u_{j} < m ≤ n(r_{j}),

we see from (3.12) that there exists δ_{j} → 0 such that

(3.14) |β_{m}(j)|^{1/m} < δ_{j}, (1 + ε)u_{j} < m ≤ n(r_{j}).

We now suppose that m ≤ [(1 + ε)u_{j}]. We have from (2.15) that

(3.15)  \begin{aligned}|c_{m}(r_{j}, H)| & ≤ \frac{1}{2} ∫ r̃_{j} ; |z_{m}|((\frac{r_{j}}{t} )^{m} + (\frac{t}{r_{j}} )^{m})\frac{n(t)}{t} dt \\ & + \frac{1}{2} ∫ r_{j} ; r̃_{j}((\frac{r_{j}}{t} )^{m} + (\frac{t}{r_{j}} )^{m})\frac{n(t)}{t} dt + \frac{1}{2} \\ & = c_{m}^{a}(r_{j}, H) + c_{m}^{b}(r_{j}, H) + \frac{1}{2}, \end{aligned}

where of course the first term is omitted if |z_{m}| ≥ r̃_{j}. From (3.10) we have

\begin{aligned}c_{m}^{b}(r_{j}, H) & ≤ \frac{(1 + ε)n(r_{j})}{2} ∫ r_{j} ; r̃_{j}((\frac{r_{j}}{t} )^{m} + (\frac{t}{r_{j}} )^{m})(\frac{t}{r_{j}} )^{(1−ε)^{2}μu_{j}} \frac{dt}{t} \\ & ≤ \frac{(1 + ε)n(r_{j})}{2}(\frac{1}{(1 − ε)^{2}μu_{j} − m} + \frac{1}{(1 − ε)^{2}μu_{j} + m}). \end{aligned}

Estimating the sums by the corresponding integrals we obtain for large j

(3.16) \begin{aligned}∑_{m=1}^{[(1+ε)u_{j}]} c_{m}^{b}(r_{j}, H) & ≤ \frac{(1 + ε)n(r_{j})}{2}(\frac{1}{(1 − ε)^{2}μu_{j} + 1} \\ & + log \frac{(1 − ε)^{2}μu_{j} + (1 + ε)u_{j}}{(1 − ε)^{2}μu_{j} − (1 + 2ε)u_{j}} ) \\ & ≤ \frac{(1 + ε)n(r_{j})}{2}(log \frac{(1 − ε)^{2}μ + 1 + ε}{(1 − ε)^{2}μ − (1 + 2ε)}) + o(n(r_{j})). \end{aligned}

By (3.15) we may write

(3.17)  c_{m}(r_{j}, H) = c_{m}^{α}(r_{j}, H) + c_{m}^{β}(r_{j}, H)

where |c_{m}^{α}(r_{j}, H)| ≤ c_{m}^{a}(r_{j}, H) and |c_{m}^{β}(r_{j}, H)| ≤ c_{m}^{b}(r_{j}, H) + \frac{1}{2} . Thus by (3.4) and (3.16)

(3.18) ∑_{m=1}^{[(1+ε)u_{j}]} |c_{m}^{β}(r_{j}, H)| ≤ \frac{(1 + ε)n(r_{j})}{2} log(\frac{(1 − ε)^{2}μ + 1 + ε}{(1 − ε)^{2}μ − (1 + 2ε)}) + o(n(r_{j})).

We next consider c_{m}^{a}(r_{j}, H) for (1 − \frac{1}{2}ε)u_{j} < m ≤ [(1 + ε)u_{j}]. Writing
(3.19)  c_{m}^{a}(r_{j}, H) = r_{j}^{m}β_{m}(j),  (1 − \frac{1}{2}ε)u_{j} < m ≤ [(1 + ε)u_{j}],

we have from (3.8) and (3.15) that

\begin{aligned}|β_{m}(j)| & ≤ ∫ r̃_{j} ; |z_{m}| \frac{n(t)}{t^{m+1}} dt ≤ ∫ r̃_{j} ; |z_{m}| t^{(1−ε)u_{j}−m−1} dt \\ & < \frac{|z_{m}|^{(1−ε)u_{j}−m}}{m − (1 − ε)u_{j}} < \frac{2}{εu_{j}|z_{m}|^{m(1−(1−ε)/(1−ε/2))}} . \end{aligned}

Thus there exists ε_{j} → 0 such that

(3.20)  |β_{m}(j)|^{1/m} < ε_{j},  (1 − \frac{1}{2}ε)u_{j} < m ≤ [(1 + ε)u_{j}].

Finally, suppose m ≤ [(1 − \frac{1}{2}ε)u_{j}]. Let m′ = [(1 − \frac{1}{4}ε)u_{j}]. Using (3.4) and (3.8) we have for large j

c_{m}^{a}(r_{j}, H) ≤ ∫ r̃_{j} ; |z_{m}|(\frac{r_{j}}{t} )^{m} \frac{n(t)}{t} dt ≤ ∫ r̃_{j} ; |z_{m}|(\frac{r_{j}}{t} )^{m′} \frac{n(t)}{t} dt

= r_{j}^{m′} ∫ r̃_{j} ; |z_{m}| \frac{n(t)}{t^{m′+1}} dt ≤ r_{j}^{m′} ∫ r̃_{j} ; |z_{m}| t^{(1−ε)u_{j}−m′−1} dt

< r_{j}^{m′} ∫_{1}^{∞} t^{(1−ε)u_{j}−m′−1} dt < r_{j}^{m′}

< (1 + ε)^{(1−ε/4)}(n(r_{j}))^{1−ε/4}.

Thus

(3.21) ∑_{m=1}^{[(1−ε/2)u_{j}]} c_{m}^{a}(r_{j}, H) < (1 + ε)^{(1−ε/4)}(n(r_{j}))^{1−ε/4}(1 − \frac{1}{2}ε)u_{j} = o(n(r_{j})).

From elementary considerations, because H has no zeros on |z| = r_{j}, there exists M_{j} > (1 + ε)u_{j} such that

(3.22)  ∑_{|m|>M_{j}} |c_{m}(r_{j}, H)| < εn(r_{j}).

We define κ_{m}(j) for (1 − \frac{1}{2}ε)u_{j} < m ≤ M_{j} by

(3.23)  κ_{m}(j) = \{ \begin{matrix}\frac{−2c_{m}^{α}(r_{j}, H)}{r_{j}^{m}}, & (1 − \frac{1}{2}ε)u_{j} < m ≤ [(1 + ε)u_{j}], \\ \frac{−2c_{m}(r_{j}, H)}{r_{j}^{m}}, & (1 + ε)u_{j} < m ≤ M_{j}. \end{matrix}

From (2.14), (3.13), (3.14), (3.19), and (3.20) we note that there exists γ_{j} > 0 such that whether |z_{m}| < r_{j} or |z_{m}| > r_{j},

(3.24)  |κ_{m}(j)|^{1/m} < γ_{j} → 0,  (1 − \frac{1}{2}ε)u_{j} < m ≤ M_{j}.

We consider a subsequence r_{j_{k}} such that

(3.25)  u_{j_{k+1}} > 2M_{j_{k}},  1 ≤ k < ∞,

and

γ_{j_{k+1}} < \frac{1}{2r_{j_{k}}} , 1 ≤ k < ∞,

ensuring by (3.24) that for k ≥ 1

(3.26)  |κ_{m}(j_{p})| < (\frac{1}{2r_{j_{k}}} )^{m},  (1 − \frac{1}{2}ε)u_{j_{p}} < m ≤ M_{j_{p}},  p > k.

We define

(3.27)  T_{j}(z) = ∑_{m=1+[(1−ε/2)u_{j}]}^{M_{j}} κ_{m}(j)z^{m},  T (z) = ∑_{k=1}^{∞} T_{j_{k}}(z),

and note by (3.25) that the powers of z appearing in the various T_{j_{k}} are distinct. We also note by (3.24) that T is entire. We set

f_{k}(z) = H(z)e^{T_{j_{k}} (z)}

and

f(z) = H(z)e^{T (z)}.

From (2.7), (2.8), (3.17), (3.23), and (3.27) we see that (3.28)

c_{m}(r_{j_{k}}, f_{k}) = \{ \begin{matrix}c_{o}(r_{j_{k}}, H), & m = 0, \\ c_{m}^{α}(r_{j_{k}}, H) + c_{m}^{β}(r_{j_{k}}, H), & 1 ≤ m ≤ [(1 − \frac{1}{2}ε)u_{j_{k}}], \\ c_{m}^{β}(r_{j_{k}}, H), & (1 − \frac{1}{2}ε)u_{j_{k}} < m ≤ [(1 + ε)u_{j}], \\ 0, & (1 + ε)u_{j_{k}} < m ≤ M_{j_{k}}, \\ c_{m}(r_{j_{k}}, H), & m > M_{j_{k}}. \end{matrix}

From (3.11), (3.18), (3.21), (3.22), and (3.28) we see that (3.29)

∑_{m=−∞}^{∞} |c_{m}(r_{j_{k}}, f_{k})| < (1 + ε)n(r_{j_{k}}) log(\frac{(1 − ε)^{2}μ + (1 + ε)}{(1 − ε)^{2}μ − (1 + 2ε)}) + (ε + o(1))n(r_{j_{k}}).

Since |κ_{m}(j_{k})| < 1, it follows from (3.25) that

∥∑_{p<k} T_{j_{p}}(r_{j_{k}}e^{iθ})∥_{∞} ≤ M_{j_{k−1}}r_{j_{k}}^{M_{j_{k−1}}} < (\frac{1}{2}u_{j_{k}})((1 + ε)n(r_{j_{k}}))^{1/2} = o(n(r_{j_{k}})).

Also by (3.26)

∥∑_{p>k} T_{j_{p}}(r_{j_{k}}e^{iθ})∥_{∞} ≤ ∑_{m>M_{j_{k}}} \frac{(r_{j_{k}})^{m}}{(2r_{j_{k}})^{m}} < 1,

implying that

(3.30)  ∥log|e^{∑_{p≠k} T_{jp} (r_{j_{k}} e^{iθ})}| ∥_{∞} = o(n(r_{j_{k}})).

The combination of (3.29) and (3.30) yields

∥log |f(r_{j_{k}}e^{iθ})|∥_{∞} < (1 + ε)n(r_{j_{k}}) log(\frac{(1 − ε)^{2}μ + 1 + ε}{(1 − ε)^{2}μ − (1 + 2ε)}) + (ε + o(1))n(r_{j_{k}}).

Letting ε → 0^{+} , we obtain (1.3) for 1 < μ < ∞.

If μ = ∞, it is elementary from Lemma 2 that for every μ′ < ∞ there exists a sequence satisfying (3.1) with μ′ in place of μ. Thus the conclusion of Theorem 1 holds with μ′ in place of μ for every μ′ in (1, ∞), proving the theorem in the case μ = ∞.

4. Proof of Theorem 2

In view of Theorem 1, we may restrict our attention to sequences Z for which μ < \frac{3}{2} . With no loss of generality we may presume that |z_{1}| > 1. Suppose 0 < ε < \frac{1}{3} .

Let m_{j} be an increasing sequence of positive integers. Let

r_{j}^{∗} = inf\{t ≥ |z_{1}| : \frac{log n(t)}{log t} ≥ m_{j} + 1 − ε\}.

Because Z has infinite exponent of convergence, it is elementary that r_{j}^{∗} is a nondecreasing sequence of real numbers tending to infinity. It is immediate that

(4.1)  n(t) < t^{m_{j}+1−ε},  t < r_{j}^{∗}.

Because n(t) is continuous from the right, for each j there exists ν such that |z_{ν}| = r_{j}^{∗} and

(4.2)  n(r_{j}^{∗}) ≥ (r_{j}^{∗})^{m_{j}+1−ε}.

We wish to focus on a sequence of circles containing no members of Z and thus choose r_{j} > r_{j}^{∗} such that n(r_{j}) = n(r_{j}^{∗}) and

(4.3)  (\frac{r_{j}}{r_{j}^{∗}} )^{n(r_{j})} < 1 + ε.

For j ≥ 1 we define

r̃_{j} = inf\{t ≥ |z_{1}| : \frac{log n(t)}{log t} ≥ \frac{1}{2}(m_{j} + 1 − ε)\},

and note trivially that r̃_{j} ≤ r_{j}^{∗} and

(4.4)  n(t) < t^{(m_{j}+1−ε)/2},  t < r̃_{j}.

Since log n(t) is continuous from the right, we see from the definition of r̃_{j} that

\frac{log n(r̃_{j})}{log r̃_{j}} ≥ \frac{m_{j} + 1 − ε}{2}.

Since μ < \frac{3}{2} , for large j we have log n(r̃_{j}) < (log r̃_{j})^{3/2} and thus

(4.5)  \frac{log r̃_{j}}{m_{j}} > \frac{log r̃_{j}}{m_{j} + 1 − ε} ≥ \frac{(log r̃_{j})^{2}}{2 log n(r̃_{j})} > \frac{(log r̃_{j})^{1/2}}{2} → ∞

as j → ∞.

We form the product (2.10) and now estimate |c_{m}(r_{j}, H)|. By (2.7), (4.1), and (4.3) we have

(4.6)  c_{o}(r_{j}, H) = ∫_{0}^{r_{j}^{∗}} \frac{n(t)}{t} dt + ∫_{r_{j}^{∗}}^{r_{j}} \frac{n(t)}{t} dt < \frac{(r_{j}^{∗})^{m_{j}+1−ε}}{m_{j} + 1 − ε} + log(1 + ε).

We next suppose 1 ≤ m ≤ m_{j}. We note from (4.2) for large j that n(r_{j}) > 2^{m_{j}} > m_{j} ≥ m and thus |z_{m}| < r_{j}. From (2.15), (4.1), and (4.3) we conclude

|c_{m}(r_{j}, H)| ≤ \frac{1}{2} ∫_{|z_{m}|}^{r_{j}^{∗}}((\frac{r_{j}}{t} )^{m} + (\frac{t}{r_{j}} )^{m})\frac{n(t)}{t} dt

+ \frac{1}{2} ∫_{r_{j}^{∗}}^{r_{j}}((\frac{r_{j}}{t} )^{m} + (\frac{t}{r_{j}} )^{m})\frac{n(t)}{t} dt + \frac{1}{2}

< \frac{(1 + ε)(r_{j}^{∗})^{m_{j}+1−ε}}{2} (\frac{1}{m_{j} + 1 − ε − m} + \frac{1}{m_{j} + 1 − ε + m})

+ (1 + ε) log(1 + ε) + \frac{1}{2}.

We conclude for 1 ≤ m ≤ m_{j} that there are expressions c_{m}^{α}(r_{j}) and c_{m}^{β}(r_{j}) satisfying

(4.7)  c_{m}(r_{j}, H) = c_{m}^{α}(r_{j}) + c_{m}^{β}(r_{j}),
(4.8)  |c_{m}^{α}(r_{j})| ≤ \frac{(1 + ε)(r_{j}^{∗})^{m_{j}+1−ε}}{2(m_{j} + 1 − ε − m)},

and

(4.9)  |c_{m}^{β}(r_{j})| ≤ \frac{(1 + ε)(r_{j}^{∗})^{m_{j}+1−ε}}{2(m_{j} + 1 − ε + m)} + 1.

We set c_{−m}^{α}(r_{j}) = \overline{c_{m}^{α}(r_{j})} for 1 ≤ m ≤ m_{j} and also set c_{0}^{α}(r_{j}) = c_{0}(r_{j}, H). Now

∑_{m=1}^{m_{j}} \frac{1}{m_{j} + 1 − ε + m} < log 2,

and thus by (4.9) for large j

(4.10)  \begin{aligned}∑_{m=1}^{m_{j}} |c_{m}^{β}(r_{j})| & < \frac{(1 + ε)(log 2)(r_{j}^{∗})^{m_{j}+1−ε}}{2} + m_{j} \\ & < \frac{(1 + 2ε)(log 2)(r_{j}^{∗})^{m_{j}+1−ε}}{2} . \end{aligned}

We now suppose m_{j} < m ≤ n(r_{j}) and note that |z_{m}| < r_{j}. From (2.15), (4.1), (4.3), and (4.4) we have (with the omission of an obvious term if |z_{m}| > r̃_{j})

(4.11)  \begin{aligned}|c_{m}(r_{j}, H)| & ≤ ∫ r̃_{j} ; |z_{m}|(\frac{r_{j}}{t} )^{m} \frac{n(t)}{t} dt + ∫ r_{J}^{∗} ; r̃_{j} (\frac{r_{j}}{t} )^{m} \frac{n(t)}{t} dt \\ & + ∫_{r_{j}^{∗}}^{r_{j}}(\frac{r_{j}}{t} )^{m} \frac{n(t)}{t} dt + \frac{1}{2} \\ & < \frac{r_{j}^{m}}{|z_{m}|^{m−(m_{j}+1−ε)/2}(m − \frac{1}{2}(m_{j} + 1 − ε))} \\ & + \frac{r_{j}^{m}}{r̃_{j}^{m−m_{j}−1+ε}(m − m_{j} − 1 + ε)} \\ & + (1 + ε) log(1 + ε) + \frac{1}{2} \\ & = r_{j}^{m}(β_{m}^{a}(j) + β_{m}^{b}(j) + β_{m}^{c}(j) + β_{m}^{d}(j)). \end{aligned}

Trivially

(4.12)  \begin{aligned}(β_{m}^{c}(j) + β_{m}^{d}(j))^{1/m} & = ((1 + ε) log(1 + ε) + \frac{1}{2})^{1/m}/r_{j} \\ & < \frac{1}{r_{j}} → 0, m_{j} < m ≤ n(r_{j}), j → ∞. \end{aligned}

Likewise for m_{j} < m ≤ n(r_{j}) we have for large j

(4.13)  (β_{m}^{a}(j))^{1/m} < \frac{1}{|z_{m}|^{1−(m_{j}+1−ε)/2m}} < \frac{1}{|z_{m_{j}}|^{1/4}} → 0

as j → ∞.

For m_{j} < m ≤ n(r_{j}) we have

β_{m}^{b}(j) ≤ \frac{1}{εr̃_{j}^{m−m_{j}−1+ε}}

and thus

(4.14)  log(β_{m}^{b}(j))^{1/m} ≤ −\frac{log ε}{m} − \frac{m − m_{j} − 1 + ε}{m} log r̃_{j}.

We consider two cases. If m_{j} < m ≤ 2m_{j}, then

(4.15)  log(β_{m}^{b}(j))^{1/m} < −\frac{log ε}{m_{j}} − \frac{ε}{2m_{j}} log r̃_{j} → −∞,  j → ∞,

by (4.5). For m > 2m_{j} we have by (4.14)

(4.16)

log(β_{m}^{b}(j))^{1/m} ≤ −\frac{log ε}{m} + (−1 + \frac{m_{j}}{m} + \frac{1 − ε}{m}) log r̃_{j}

< −\frac{log ε}{2m_{j}} + (−\frac{1}{2} + \frac{1 − ε}{2m_{j}}) log r̃_{j} → −∞, j → ∞.

For m > m_{j}, including those for which |z_{m}| > r_{j}, the combination of (2.14), (4.11), (4.12), (4.13), (4.15), and (4.16) implies the existence of δ_{j} → 0 such that if

(4.17)  c_{m}(r_{j}, H) = β_{m}(j)r_{j}^{m},

then

(4.18)  |β_{m}(j)|^{1/m} < δ_{j} → 0,  j → ∞.

Using (4.6) and (4.8) we apply Lemma 1 with M = m_{j} to conclude for large j that there exists a trigonometric polynomial Q_{j}(θ) with

Q_{j}(θ) = ∑_{m=−L_{j}}^{L_{j}} d_{m}(r_{j})e^{imθ},

where L_{j} > m_{j}, such that

(4.19)  d_{m}(r_{j}) = c_{m}^{α}(r_{j}),  |m| ≤ m_{j},

and

(4.20)  ∥Q_{j}∥_{∞} < (\frac{π}{1 − ε} + ε)(1 + ε)(r_{j}^{∗})^{m_{j}+1−ε}.

We write

(4.21)  d_{m}(r_{j}) = γ_{m}(r_{j})r_{j}^{m},  m_{j} < m ≤ L_{j},

yielding by (4.20)

|γ_{m}(r_{j})| ≤ (\frac{π}{1 − ε} + ε)(1 + ε)r_{j}^{m_{j}+1−ε−m}

and hence

log |γ_{m}(r_{j})|^{1/m} ≤ \frac{log(π/(1 − ε) + ε) + log(1 + ε)}{m} − \frac{m + ε − m_{j} − 1}{m} log r_{j}.

For m_{j} < m ≤ min(L_{j}, 2m_{j}), we have

(4.22) log |γ_{m}(r_{j})|^{1/m} < \frac{log(π/(1 − ε) + ε) + log(1 + ε)}{m_{j}} − \frac{ε}{2m_{j}} log r_{j} → −∞

as j → ∞ by (4.5). If L_{j} > 2m_{j}, for 2m_{j} < m ≤ L_{j} we have (4.23)

log |γ_{m}(r_{j})|^{1/m} < \frac{log(π/(1 − ε) + ε) + log(1 + ε)}{2m_{j}} + (−\frac{1}{2} + \frac{1 − ε}{2m_{j}}) log r_{j} → −∞

as j → ∞. Combining (4.22) and (4.23), we conclude there exists κ_{j} > 0 such that

(4.24)  |γ_{m}(r_{j})|^{1/m} < κ_{j} → 0,  m_{j} < m ≤ L_{j},

as j → ∞.

By elementary considerations, there exists M_{j} > L_{j} such that

(4.25)  ∑_{|m|>M_{j}} |c_{m}(r_{j}, H)| < εn(r_{j}).

For m_{j} < m ≤ M_{j}, we define

(4.26)  b_{m}(j) = \{ \begin{matrix}\frac{−2c_{m}(r_{j}, H) + 2d_{m}(r_{j})}{r_{j}^{m}}, & m_{j} < m ≤ L_{j}, \\ \frac{−2c_{m}(r_{j}, H)}{r_{j}^{m}}, & L_{j} < m ≤ M_{j}. \end{matrix}

From (4.17), (4.18), (4.21), and (4.24), we conclude that there exists η_{j} > 0 such that

(4.27)  |b_{m}(j)|^{1/m} < η_{j} → 0,  m_{j} < m ≤ M_{j},  j → ∞.

Let

T_{j}(z) = ∑_{m_{j}<m≤M_{j}} b_{m}(j)z^{m}

and

f_{j}(z) = H(z)e^{T_{j}(z)}.

By (2.8), (4.7), and (4.26)

(4.28)  c_{m}(r_{j}, f_{j}) = \{ \begin{matrix}c_{o}(r_{j}, H) = c_{o}^{α}(r_{j}), & m = 0, \\ c_{m}^{α}(r_{j}) + c_{m}^{β}(r_{j}), & 1 ≤ m ≤ m_{j}, \\ d_{m}(r_{j}), & m_{j} < m ≤ L_{j}, \\ 0, & L_{j} < m ≤ M_{j}, \\ c_{m}(r_{j}, H), & m > M_{j}. \end{matrix}

We also of course have c_{−m}(r_{j}, f_{j}) = \overline{c_{m}(r_{j}, f_{j})} for m ≥ 1.

From (4.2), (4.10), (4.19), (4.20), (4.25), and (4.28), we have for all θ in [−π, π] and large j that

(4.29)  \begin{aligned}|log |f_{j}(r_{j}e^{iθ})|| & ≤ | ∑_{|m|≤L_{j}} d_{m}(r_{j})e^{imθ}| \\ & + ∑_{1≤|m|≤m_{j}} |c_{m}^{β}(r_{j})| + ∑_{|m|>M_{j}} |c_{m}(r_{j}, H)| \\ & < (\frac{π}{1 − ε} + ε)(1 + ε)n(r_{j}) + (log 2 + 3ε)n(r_{j}). \end{aligned}

We consider a subsequence m_{j_{k}} satisfying

(4.30)  m_{j_{k+1}} > M_{j_{k}},  k ≥ 1,

and

(4.31)  |b_{m}(j_{p})| < (\frac{1}{2r_{j_{k}}} )^{m},  k ≥ 1, p ≥ k + 1, m_{j_{p}} < m ≤ M_{j_{p}},

where we have used (4.27).

We set

T (z) = ∑_{k=1}^{∞} T_{j_{k}}(z),

and note by (4.30) that the powers of z in the various T_{j_{k}} are distinct. From (4.27) we see that T is entire and from (4.31) it follows that the Maclaurin coefficients of T have modulus less than 1. Thus by (4.2), (4.3), and the fact that μ < \frac{3}{2} , for −π ≤ θ ≤ π and large k we have

(4.32)  ∑_{p=1}^{k−1} |T_{j_{p}}(r_{j_{k}}e^{iθ})| ≤ M_{j_{k−1}}r_{j_{k}}^{M_{j_{k−1}}} < m_{j_{k}}r_{j_{k}}^{m_{j_{k}}} < εn(r_{j_{k}}).

From (4.31) we have for −π ≤ θ ≤ π that

(4.33)  \begin{aligned}|∑_{p>k} T_{j_{p}}(r_{j_{k}}e^{iθ})| & ≤ ∑_{p=k+1}^{∞} ∑_{m_{jp} <m≤M_{jp}} |b_{m}(j_{p})|r_{j_{k}}^{m} \\ & ≤ ∑_{p=k+1}^{∞} ∑_{m_{jp} <m≤M_{jp}}(\frac{1}{2})^{m} < 1. \end{aligned}

We set

f(z) = H(z)e^{T (z)}

and note by (4.32) and (4.33) for large k and −π ≤ θ ≤ π that

|log |f(r_{j_{k}}e^{iθ})| − log |f_{j_{k}}(r_{j_{k}}e^{iθ})|| < εn(r_{j_{k}}) + 1.

We conclude by (4.29) that for large k

∥log |f(r_{j_{k}}e^{iθ})|∥_{∞} < ((\frac{π}{1 − ε} + ε)(1 + ε) + log 2 + 5ε)n(r_{j_{k}}).

Since ε in (0, \frac{1}{3}) is arbitrary, we obtain the conclusion of Theorem 2.

90 Joseph Miles

References

[1] Beardon, A.F., and C.C. Yang: Open problems for the workshop on value distribution theory and its applications. - Hong Kong, July 1996. J. Contemp. Math. Anal. 32, 1997, 9–36.

[2] Bergweiler, W.: Canonical products of infinite order. - J. Reine Angew. Math. 430, 1992, 85–107.

[3] Boas, R.P.: Majorant problems for trigonometric series. - J. Anal. Math. 10, 1962, 253– 271.

[4] Duren, P.L.: Theory of H^{p} Spaces. - Academic Press, New York, 1970.

[5] Gol’dberg, A.A.: Meromorphic functions with separated zeros and poles. - Izv. Vysš. Učebn. Zaved. Mat. 17:4, 1960, 67–72 (Russian).

[6] Miles, J., and D.F. Shea: An extremal problem in value distribution theory. - Quart. J. Math. Oxford (2) 24, 1973, 377–383.

[7] Nevanlinna, F.: Bemerkungen zur Theorie der ganzen Funktionen endlicher Ordnung. - Soc. Sci. Fenn. Comment. Phys.-Math. 2:4, 1923, 1–7.

[8] Newman, D.J.: Problem 4874. - Amer. Math. Monthly 66, 1959, 817; 67, 1960, 811–812.

Received 10 August 2000