Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 97–108

ON NEVANLINNA’S SECONDARY DEFICIENCY

Zhuan Ye

Northern Illinois University, Department of Mathematics DeKalb, IL 60115, U.S.A.; ye@math.niu.edu

Abstract. Analogies between the Nevanlinna theory and the theory of heights in number theory have motivated to the determination the precise error term of the second fundamental theorem of the Nevanlinna theory. The Nevanlinna secondary deficiency, introduced by P.M. Wong, gives the error term in a certain sense. We first prove that, from a topological point of view, almost all meromorphic functions have secondary deficiency negative infinity and that the set of meromorphic functions having maximum secondary deficiency is dense in a standard topology. Then we give an improved upper bound on the secondary deficiency for meromorphic functions of finite order and show that the upper bound is sharp.

1. Introduction

Recently there have been a number of papers concerning Nevanlinna’s error terms in value distribution theory because of an analogy between the second fundamental theorem and Roth’s theorem of number theory. Based on Serge Lang’s conjecture concerning Roth’s theorem in [5] and [4] and P. Vojta’s Nevanlinnatheory and number-theory dictionary in [11], Lang conjectured, broadly speaking, in the one variable case, that for any nonconstant meromorphic function f in C and q distinct points a_{1}, a_{2}, …, a_{q} in C ∪ {∞}, we have

(1) qT (f, r)−∑_{j=1}^{q} N(f, a_{j}, r)+N_{Ram}(f, r) ≤ 2T (f, r)+log T (f, r)+o(log T (f, r))

for all large r outside a set of finite Lebesgue measure. This conjecture was proved to be correct by Lang [6] and Wong [12], while Z. Ye [13] showed that the upper bound is sharp. Moreover, Lang in [6] raised a question of determining whether the inequality (for all large r)

(2) qT (f, r)−∑_{j=1}^{q} N(f, a_{j}, r)+N_{Ram}(f, r) ≥ 2T (f, r)+log T (f, r)−o(log T (f, r))

holds for almost all (in a suitable sense) meromorphic functions.

The coefficient 2 of the term T (f, r) in (1) is well known as an upper bound of sums of deficient values and indexes of multiplicity of f in Nevanlinna theory.

1991 Mathematics Subject Classification: Primary 30D35.

Similarly, P.M. Wong [12] introduced the concept of secondary deficiency, which identifies the coefficient of the term log T (f, r) in (1), or the power index, as its counterpart in Roth theorem. An up-to-date account of these matters appears in [6]. The concept of “almost all” in function spaces was considered by A. Offord (e.g. [9]) from a probabilistic point of view and by P. Gauthier and W. Hengartner in [1] from a topological point of view.

In this paper, we first prove the set of entire (meromorphic) functions having the secondary deficiency one to be dense in a space of entire (meromorphic) functions with τ -topology (τ_{χ} -topology). Then we show that almost all functions have the secondary deficiency negative infinity. Thus almost all functions fail to satisfy inequality (2). Moreover, we obtain an improved upper bound of Nevanlinna’s secondary deficiency for meromorphic functions of finite order and show that the upper bound is sharp.

I would like to thank the referee for helpful comments.

2. Preliminaries

Let M(C) and H(C) be the sets of meromorphic functions and entire functions in the complex plane C, respectively. Now we are going to assign topologies to M(C), and hence to H(C), such that M(C) and H(C) become complete metric spaces with the topologies. A detailed discussion can be found in [1].

For any K a compact set in C, any ε > 0 and any f ∈ M(C), we denote

O(f, K, ε) = {g ∈ M(C) : ∥f − g∥_{K} ≡ sup_{z∈K} |f(z) − g(z)| ≤ ε and f − g ∈ H(K)}.

Clearly all these O(f, K, ε)’s consist of a subbasis of M(C) and generate what we call the τ -topology of uniform convergence on compact sets. It is straightforward to show that M(C) with the τ -topology is a complete metric space. Therefore (M(C), τ) is of second Baire category. As usual, we regard H(C) as a subspace of M(C). Thus H(C) is a complete metric space, too, since H(C) is closed in the τ -topology.

Another topology on M(C) is called spherically uniform convergence on compact sets, denoted by τ_{χ}. For f and g in M(C), set, for any n ∈ Z^{+} ,

d_{n}(f, g) = sup_{|z|≤n} χ(f(z), g(z)),

where χ denotes the spherical metric on the Riemann sphere \overline{C}, with \overline{C} = C ∪ {∞}. Hence, in the usual way, the

d(f, g) = ∑_{n=1}^{∞} \frac{1}{2^{n}} \frac{d_{n}(f, g)}{1 + d_{n}(f, g)}

is an induced metric on M(C). Clearly d(f, g) ≤ d_{n}(f, g) for any n. If f_{n}(z) = n, then f_{n} is a Cauchy sequence in the τ_{χ} topology and is not Cauchy in the τ topology. It follows that (M(C), d) is not a complete metric space. However, it is not hard to prove that (M(C) ∪ {∞}, d) is a complete metric space. So (M(C) ∪ {∞}, d) is of second Baire category.

Finally we give the definition of “almost all” in the topological sense. Let X be a space of functions. We say that almost all functions in X have a certain property if X is of second category while the set of functions in X not possessing the said property is of category one.

As for Nevanlinna theory, standard references are [2] and [8].

3. Results

Let f be a meromorphic function in C, q be any positive integer and a_{1}, a_{2}, …, a_{q} be any q (> 0) points in \overline{C}. We write

(3)  S(f, {a_{j}}_{1}^{q}, r) = (q − 2)T (f, r) − ∑_{j=1}^{q} N(f, a_{j}, r) + N_{Ram}(f, r),

where N_{Ram}(f, r) = N(f′, 0, r) + 2N(f, ∞, r) − N(f′, ∞, r). Thus the definition of secondary deficiency in [12] can be written as

(4)  δ_{2}(f, \{a_{j}\}) = lim inf_{r→∞} \frac{S(f, \{a_{j}\}_{1}^{q}, r)}{log T (f, r)}.

It is known (e.g. [12]) that δ_{2}(f, \{a_{j}\}) ≤ 1 if all a_{j} ’s are distinct in \overline{C}. In this paper we are going to prove the following theorems.

Theorem 1. The set

H_{1} = \{f ∈ H(C) : ∃ distinct \{a_{1}, a_{2}, …, a_{q}\} ⊂ \overline{C} such that δ_{2}(f, \{a_{j}\}) = 1\}

is dense in H(C) in the τ -topology.

Theorem 2. The set

H = \{f ∈ H(C) : ∃ distinct \{a_{1}, a_{2}, …, a_{q}\} ⊂ \overline{C} such that δ_{2}(f, \{a_{j}\}) > −∞\}

is of first category in H(C) in the τ -topology.

Remark. In 1972, P. Gauthier and W. Hengartner [1] proved that, for almost all (as defined in Section 2) meromorphic functions f, ∑_{a∈C} δ(f, a) = 0, where δ(f, a) is the Nevanlinna deficiency. On the other hand, the proof of our Theorem 1 tells us that the set of entire functions having maximum Nevanlinna deficiency, i.e., ∑_{a∈C} δ(f, a) = 2, is dense in a space of entire functions in the τ -topology. Clearly, H_{1} ⊂ H. Hence, H is dense in (H(C), τ) by Theorem 1 and H_{1} is of first category by Theorem 2. Furthermore, Theorem 2 states that for almost all (as defined in the previous section) entire functions f and for any q distinct points \{a_{1}, a_{2}, …, a_{q}\} ⊂ \overline{C}, δ_{2}(f, \{a_{j}\}) = −∞. Thus, summarizing this discussion, we have 100 Zhuan Ye

Corollary. For almost all functions in (H(C), τ), we have

δ_{2}(f, \{a_{j}\}) = −∞ for any \{a_{j}\}_{1}^{q} ⊂ \overline{C}.

The set of entire functions having

δ_{2}(f, \{a_{j}\}) > −∞ for some \{a_{j}\}_{1}^{q} ⊂ \overline{C}

is dense in (H(C), τ).

Similarly, for a space of meromorphic functions we have

Theorem 3. The set

M_{1} = \{f ∈ M(C) : ∃ distinct \{a_{1}, a_{2}, …, a_{q}\} ⊂ \overline{C} such that δ_{2}(f, \{a_{j}\}) = 1\}

is dense in M(C) in the τ_{χ} -topology.

Theorem 4. The set

M = \{f ∈ M(C) : ∃ distinct \{a_{1}, a_{2}, …, a_{q}\} ⊂ \overline{C} such that δ_{2}(f, \{a_{j}\}) > −∞\}

is of first category in M(C) in the τ_{χ} -topology.

The following theorem is about an upper bound of meromorphic functions of finite order.

Theorem 5. Let f be any transcendental meromorphic function of finite order ϱ. Then for any distinct points \{a_{j}\}_{1}^{q} ⊂ \overline{C}, we have

\begin{aligned}δ_{2}(f, \{a_{j}\}) & ≤ 1 − \frac{1}{ϱ}  if ϱ > 0, \\ δ_{2}(f, \{a_{j}\}) & = −∞  if ϱ = 0. \end{aligned}

Moreover, for any given ϱ ≥ 0, there is an entire function f of order ϱ and {a_{j}}_{j=1}^{q} ⊂ C such that f assumes this maximum secondary deficiency.

4. Proof of Theorems We begin by constructing a class of entire functions which play an important

role in the proof of our theorems.

Let ϱ > 0 and r_{n} = (n^{2}/ϱ)^{1/ϱ}, n = 1, 2, 3, …, and set

(5)  E_{ϱ}(z) = ∏_{n=1}^{∞}(1 + (\frac{z}{r_{n}} )^{n}).

On Nevanlinna’s secondary deficiency 101

Let r > 0, with r ∈ [r_{k}, r_{k+1}). Then

(6)  \begin{aligned}n(E_{ϱ}, 0, r) & = k(k + 1)/2(1 + o(1))ϱr^{ϱ}/2 and \\ N(E_{ϱ}, 0, r) & = (1 + o(1))r^{ϱ}/2 \end{aligned}

for all large r. Moreover, for |z| = r ∈ [r_{k}, r_{k+1}), we have

(7) log |E_{ϱ}(z)| = ∑_{n=1}^{k} n log|\frac{z}{r_{n}} | + ∑_{n=1}^{k} log|(\frac{r_{n}}{z} )^{n} + 1| + ∑_{n=k+1}^{∞} log|1 + (\frac{z}{r_{n}} )^{n}|

= N(E_{ϱ}, 0, r) + I_{1} + I_{2}.

It is obvious from (7) and k = \sqrt{ϱ} r_{k}^{ϱ/2} that, for |z| = r ∈ [r_{k}, r_{k+1}),

(8)  |I_{1}| ≤ k log 2 ≤ \sqrt{ϱ} r^{ϱ/2} log 2.

Since log(1 + x) ≤ x for all 0 < x ≤ 1 and there is a constant C = C(ϱ) > 0 such that

(9) (2 + k)^{2/ϱ} − (1 + k)^{2/ϱ} ≥ \{ \begin{matrix}2(1 + k)^{−1+2/ϱ}/ϱ & if −1 + 2/ϱ ≥ 0 \\ 2(2 + k)^{−1+2/ϱ}/ϱ & if −1 + 2/ϱ ≤ 0 \end{matrix} ≥ Ck^{−1+2/ϱ},

for some constant C > 0, we have from (9) that, for |z| = r ∈ [r_{k}, r_{k+1}),

(10)

|I_{2}| ≤ log 2 + ∑_{n=k+2}^{∞}(\frac{r_{k+1}}{r_{k+2}} )^{n} = log 2 + \frac{r_{k+2}}{r_{k+2} − r_{k+1}} (\frac{r_{k+1}}{r_{k+2}} )^{k+2}

≤ log 2 + \frac{(k + 2)^{2/ϱ}}{(k + 2)^{2/ϱ} − (k + 1)^{2/ϱ}} ≤ log 2 + \frac{(2k)^{2/ϱ}}{Ck^{−1+2/ϱ}}

≤ log 2 + Ck ≤ Cr^{ϱ/2}.

It follows from (6), (7), (8) and (10) that E is an entire function and

(11) N(E_{ϱ}, 0, r) ≤ T (E_{ϱ}, r) ≤ log M(E_{ϱ}, r) = log E_{ϱ}(r) ≤ (1 + o(1))N(E_{ϱ}, 0, r)

for all large r, and so

(12) m(E_{ϱ}, 0, r) = o(T (E_{ϱ}, r)) and log n(E_{ϱ}, 0, r) = log T (E_{ϱ}, r) + O(1).

Proof of Theorem 1. Given f ∈ H(C), a compact set K ⊂ C and an ε > 0, we will find an f_{∗} ∈ H_{1} such that ∥f − f_{∗}∥_{K} ≤ ε.

In fact, there is a polynomial P such that ∥f − P ∥_{K} ≤ ε/2. Set

(13)  f_{λ}(z) = P (z) exp(λ ∫_{0}^{z} E(ξ) dξ),

where λ ∈ C is a parameter and E = E_{1} is as in (5) with ϱ = 1. Clearly f_{λ} is an entire function of z. Furthermore, for |λ| < 1,

(14)  \begin{aligned}∥f − f_{λ}∥_{K} & ≤ ∥f − P ∥_{K} + ∥P ∥_{K} ∥1 − exp(λ ∫_{0}^{z} E(ξ) dξ)∥_{K} \\ & ≤ \frac{ε}{2} + ∥P ∥_{K}|λ| exp(∥∫_{0}^{z} E(ξ) dξ∥_{K}). \end{aligned}

It follows from (14) that there exists a small real number ε_{0} > 0 such that for any λ ∈ C with |λ| ≤ ε_{0} we have

(15)  ∥f − f_{λ}∥_{K} ≤ ε.

On the other hand, by [8, p. 276], there exists c ∈ {λ ∈ C \ {0} : |λ| ≤ ε_{0}} with

lim_{r→∞} \frac{m(P′/(P E), −c, r)}{T (P′/(P E), r)} = 0.

Since T (P′/(P E), r) = (1 + o(1))T (E, r) for all large r,

(16)  lim_{r→∞} \frac{m(P′/(P E), −c, r)}{T (E, r)} = 0.

Now set f_{∗} = f_{c}, where c is from (16) and f_{c} is defined as in (13). Thus (11) implies that

(17) log T (f_{∗}, r) ≤ log T (P, r) + log r + log M(E, r) + log 2 = (1 + o(1))T (E, r)

for all large r. Hence, set q = 2, a_{1} = 0 and a_{2} = ∞; we then obtain from (13), (17), (12) and (16) that

(18)

S(f_{∗}, {a_{j}}_{1}^{q}, r) = −N(f_{∗}, 0, r) − N(f_{∗}, ∞, r) + N_{Ram}(f_{∗}, r)
= O(log r) + N(P′ + cP E, 0, r)
= T (P′ + cP E, 0, r) − m(P′ + cP E, 0, r) + O(log r)
≥ T (E, r) − m(P′/(P E), −c, r) − m(E, 0, r) + O(log r)
≥ (1 + o(1)) log T (f_{∗}, r)

for all large r. It follows from (18) and (15) that δ_{2}(f_{∗}, {a_{j}}) = 1 and ∥f −f_{∗}∥_{K} ≤ ε. Thus Theorem 1 is proved.

Proof of Theorem 2. Let k, n and q be any positive integers. Set

(19)

S_{k} = {f ∈ H(C) : T (f, r) ≤ k for r ≥ k}  and
B_{qnk} = {f ∈ H(C) : ∃{a_{j}}_{1}^{q} ⊂ C with |a_{j}| ≤ n, and
S(f, {a_{j}}_{1}^{q}, r) ≥ −n log T (f, r), for r ≥ k; or
S(f, {a_{j}}_{1}^{q} ∪ {∞}, r) ≥ −n log T (f, r) for r ≥ k},

where some a′_{j} s may be the same point in C. Clearly ⋃_{k} S_{k} is the set of all constant functions. By the definition of H in Theorem 2,

(20)  H ⊂ ⋃_{q=1}^{∞} ⋃_{n=1}^{∞} ⋃_{k=1}^{∞} (B_{qnk} ∪ S_{k}) .

Now we are going to prove that B_{qnk} ∪ S_{k} is closed and has an empty interior. In fact, for any convergent sequence {f_{p}}_{p=1}^{∞} ⊂ B_{qnk}∪S_{k} in the τ -topology with, say, f its limit function, we prove that B_{qnk} ∪ S_{k} is closed by showing f ∈ B_{qnk} ∪ S_{k}. If f ∉ S_{k}, then f_{p} ∉ S_{k} for all large p since T (f_{p}, r) converges to T (f, r). Hence f_{p} ∈ B_{qnk} for all large p since f_{p} ∈ B_{qnk} ∪ S_{k}. Thus for each f_{p} there are q complex numbers {a_{j}^{p}}_{j=1}^{q}, with |a_{j}^{p}| ≤ n, such that

\frac{S(f_{p}, \{a_{j}^{p}\}_{j=1}^{q}, r)}{log T (f_{p}, r)} ≥ −n or \frac{S(f_{p}, \{a_{j}^{p}\}_{j=1}^{q} ∪ \{∞\}, r)}{log T (f_{p}, r)} ≥ −n

for all r ≥ k. Since {a_{j}^{p}}_{p=1}^{∞} is bounded, for each fixed 0 < j ≤ q, by the diagonal method there is a sequence {p_{s}}_{s=1}^{∞} such that, for each j, {a_{j}^{p_{s}}}_{s=1}^{∞} converges to a_{j}^{∗}, where some of the a_{j}^{∗} ’s may coincide. Since f_{p} is uniformly convergent to f on any compact set in C, f′_{p} uniformly converges to f′ and f_{p_{s}} − a_{j}^{p_{s}} uniformly converges to f − a_{j}^{∗} for each j. Hence, by Hurwitz’s theorem and noting f ∉ S_{k} for any fixed j, we have for any r with r ≥ k

lim_{s→∞} N(f_{p_{s}}, a_{j}^{p_{s}}, r) = N(f, a^{∗}, r) and lim_{s→∞} T (f_{p_{s}}, r) = T (f, r).

It turns out that, for any r with r ≥ k,

−n ≤ lim_{s→∞} \frac{S(f_{p_{s}}, \{a_{j}^{p_{s}}\}_{j=1}^{q}, r)}{log T (f_{p_{s}}, r)} = \frac{S(f, \{a_{j}^{∗}\}_{j=1}^{q}, r)}{log T (f, r)}.

We can similarly consider the case {a_{j}^{p}}_{j=1}^{q} ∪ {∞}. Thus f ∈ B_{qnk} ∪ S_{k}, i.e., B_{qnk} ∪ S_{k} is closed.

To prove int(B_{qnk} ∪ S_{k}) = ∅, we show that the complement of B_{qnk} ∪ S_{k} is dense in H(C). For any given f ∈ H(C), a compact set K ⊂ C and an ε > 0, let O(f, K, ε) be an open neighborhood of f; then there is a polynomial P_{n} of degree n ≥ 1 such that ∥f − P_{n}∥_{K} ≤ ε. Since T (P_{n}, r) = n log r + O(1) and

S(P_{n}, {a_{j}}_{1}^{q}, r) = ∑ m(P_{n}, a_{j}, r)−2T (P_{n}, r)+N_{Ram}(P_{n}, r)+O(1) = −2T (P_{n}, r),

δ_{2}(P_{n}, {a_{j}}) = −∞ for any q with {a_{j}}_{1}^{q} ⊂ C, i.e., P_{n} ∉ B_{qnk} ∪ S_{k}. This proves B_{qnk} ∪ S_{k} has an empty interior.

It follows from (20) that H is contained in a countable union of nowhere dense sets. Therefore H is of first category and Theorem 2 is proved.

Proof of Theorem 3. It suffices to prove that for any g ∈ M(C) and any ε > 0 there exists g_{∗} such that d(g, g_{∗}) ≤ ε and g_{∗} ∈ M_{1} .

Set K = {z ∈ C; |z| ≤ n_{0}} for some positive integer n_{0} . Thus there is a polynomial P_{2} such that

gP_{2} ∈ H(K) and |g(z)P_{2}(z)| + |P_{2}(z)| ≠ 0, for z ∈ K.

Let m = min_{z∈K}(|g(z)P_{2}(z)|^{2} + |P_{2}(z)|^{2})^{1/2} > 0. There is a polynomial P_{1} such that

|gP_{2} − P_{1}| < εm/2 for all z ∈ K.

Set

g_{λ}(z) = \frac{P_{1}(z)}{P_{2}(z)} exp(λ ∫_{0}^{z} E(ξ) dξ),

where λ ∈ C and E is from (13). Thus, for |λ| ≤ 1,

(21)

d(g, g_{λ}) ≤ d_{n_{0}}(g, g_{λ}) ≤ max_{z∈K} \frac{|g − g_{λ}|}{(1 + |g|^{2})^{1/2}(1 + |g_{λ}|^{2})^{1/2}}

≤ max_{z∈K} \frac{|gP_{2} − P_{1}e^{λ ∫_{0}^{z} E(ξ) dξ}|}{(|P_{2}|^{2} + |gP_{2}|^{2})^{1/2}}

≤ max_{z∈K}(|gP_{2} − P_{1}| + |P_{1}||1 − e^{λ ∫_{0}^{z} E(ξ) dξ}|)/m

≤ \frac{ε}{2} + \frac{|λ|}{m}∥P_{1}∥_{K} exp(∥∫_{0}^{z} E(ξ) dξ∥_{K}).

It follows from the proof of Theorem 1 that there exists g_{∗} such that d(g, g_{∗}) ≤ ε and g_{∗} ∈ M_{1} . Thus Theorem 3 is proved.

Proof of Theorem 4. Since f_{n} may converge in (M(C) ∪ {∞}, τ_{χ}) to infinity uniformly in any compact set of C, we let

S_{k} = {f ∈ M(C) : T (f, r) ≤ k for r ≥ k} ∪ {∞}.

The rest of the proof of Theorem 2 can be carried over here sentence by sentence if H(C), with entire functions and polynomials replaced by M(C), with meromorphic functions and rational functions.

Proof of Theorem 5. For any r and R, 1 < r < R < ∞, we have (e.g., [3, Lemma 7])

(22)  S(f, \{a_{j}\}_{1}^{q}, r) ≤ log\{\frac{R}{r} \frac{T (f, R)}{R − r}\} + O(1).

Furthermore, by a growth lemma (e.g. [3, Lemma 4]), there exists for any constant C > 1 and any ε > 0 a sequence {r_{j}} going to infinity such that

T (f, R_{j}) ≤ CT (f, r_{j}) for all j,

where R_{j} = r_{j} + r_{j}/T^{ε}(f, r_{j}). Applying (22) to r_{j} and R_{j}, we get

S(f, {a_{j}}_{1}^{q}, r_{j}) ≤ log(1 + T^{−ε}(f, r_{j})) + (1 + ε) log T (f, r_{j}) − log r_{j} + O(1).

It follows from log T (f, r_{j}) ≤ (ϱ + ε) log r_{j} that

δ_{2}(f, \{a_{j}\}) ≤ 1 + ε − (lim inf_{j→∞} \frac{log T (f, r_{j})}{log r_{j}})^{−1} ≤ 1 + ε − (ϱ + ε)^{−1}.

By the arbitrariness of ε, the first part of the theorem is proved.

Now we construct some entire functions which indicate the sharpness of the secondary deficiency.

It suffices to prove the second part of the theorem for the case 0 < ϱ < +∞. Let f(z) = E_{ϱ}(z) in (5). Then, for r ∈ [r_{k}, r_{k+1}),

(23)  \begin{aligned}\frac{zf′(z)}{f(z)} & = ∑_{n=1}^{k} n + ∑_{n=1}^{k−1} \frac{−n}{1 + (z/r_{n})^{n}} + \frac{−k}{1 + (z/r_{k})^{k}} \\ & + \frac{(k + 1)(z/r_{k+1})^{k+1}}{1 + (z/r_{k+1})^{k+1}} + ∑_{n=k+2}^{∞} \frac{n(z/r_{n})^{n}}{1 + (z/r_{n})^{n}} \\ & = n(f, 0, r) + J_{1} + \frac{−k}{1 + (z/r_{k})^{k}} + \frac{(k + 1)(z/r_{k+1})^{k+1}}{1 + (z/r_{k+1})^{k+1}} + J_{2}. \end{aligned}

Since, for 1 ≤ n ≤ k − 1 and r ∈ [r_{k}, r_{k+1}), we have from the definition of r_{k}

|\frac{z}{r_{n}} |^{n} ≥ |\frac{k}{n}|^{2n/ϱ} ≥ 1 + \frac{2n}{ϱ} \frac{k − n}{n} if n ≥ ϱ/2,

then, using (23) and (6) and noting k ≤ \sqrt{ϱ} r^{ϱ/2} for all large r,

(24)  \begin{aligned}|J_{1}| & ≤ O(1) + ∑_{n=[ϱ/2]}^{k−1} \frac{n}{(r/r_{n})^{n} − 1} ≤ O(1) + ∑_{n=1}^{k−1} \frac{ϱn}{2(k − n)} \\ & = O(k log k) = o(n(f, 0, r)). \end{aligned}

Since, for n ≥ k + 2 and r ∈ [r_{k}, r_{k+1}), we obtain from (1 − x) ≤ e^{−x}

|\frac{z}{r_{n}} |^{n} ≤ |\frac{r_{k+1}}{r_{n}} |^{n} = |\frac{k + 1}{n}|^{2n/ϱ} ≤ exp(−\frac{n − k − 1}{n} \frac{2n}{ϱ}) = e^{−2(n−k−1)/ϱ},

106 Zhuan Ye then, again using (23) and (6) for all large r,

(25) |J_{2}| ≤ ∑_{n=k+2}^{∞} \frac{n}{(r_{n}/r)^{n} − 1} ≤ ∑_{n=k+2}^{∞} \frac{n}{e^{2(n−k−1)/ϱ} − 1} = O(k) = o(n(f, 0, r)).

Moreover, if |a| ≥ 1 and a ≠ −1, then Re{1/(1 + a)} ≤ 1/2; thus, for r ∈ [r_{k}, r_{k+1}),

(26) Re{1/(1 + (z/r_{k})^{k})} ≤ 1/2 if (z/r_{k})^{k} ≠ −1.

To estimate the remaining term in (23), set T_{k} = (r_{k} + r_{k+1})/2. Since (1 + x)^{k+1} ≥ 1 + (k + 1)x

for x > 0, and (9), we find a constant C > 0 such that, for r ∈ [r_{k}, T_{k}],

(27)  \begin{aligned}(\frac{r_{k+1}}{r} )^{k+1} & ≥ (\frac{2r_{k+1}}{r_{k} + r_{k+1}} )^{k+1} = (1 + \frac{(k + 1)^{2/ϱ} − k^{2/ϱ}}{(k + 1)^{2/ϱ} + k^{2/ϱ}} )^{k+1} \\ & ≥ 1 + (k + 1)\frac{Ck^{−1+2/ϱ}}{2(k + 1)^{2/ϱ}} ≥ 1 + \frac{C}{2^{2/ϱ}} \end{aligned}

for all large k. It follows from (23)–(27) that, for r ∈ [r_{k}, T_{k}] and (z/r_{k})^{k} ≠ −1,

|\frac{zf′(z)}{f(z)}| ≥ Re\{\frac{zf′(z)}{f(z)}\}

≥ n(f, 0, r) − |J_{1}| − k Re \frac{1}{1 + (z/r_{k})^{k}} − |J_{2}| − |\frac{(k + 1)(z/r_{k+1})^{k+1}}{1 + (z/r_{k+1})^{k+1}} |

≥ n(f, 0, r) + o(n(f, 0, r)) − \frac{k}{2} + o(n(f, 0, r)) − \frac{k + 1}{|r_{k+1}/z|^{k+1} − 1}

≥ (1 + o(1))n(f, 0, r).

It follows from (6) that, for r ∈ [r_{k}, T_{k}],

(28) \frac{1}{2π} ∫_{0}^{2π} log|\frac{f′}{f} (re^{iθ})| dθ ≥ log n(f, 0, r) − log r + O(1) ≥ (ϱ − 1) log r + O(1).

For any r ∈ [T_{k}, r_{k+1}), applying Jensen’s formula to f′/f and noting N(f, 0, r) = N(f, 0, T_{k}), we have from (28), (6), (12) and

1 ≥ T_{k}/r ≥ T_{k}/r_{k+1} = (1 + (r_{k}/r_{k+1}))/2 → 1 (as k → ∞) that

(29)

\frac{1}{2π} ∫_{0}^{2π} log|\frac{f′}{f} (re^{iθ})| dθ = N(f′/f, 0, r) − N(f′/f, ∞, r) + O(1)

= N(f′, 0, r) − N(f, 0, r) + O(1) ≥ N(f′, 0, T_{k}) − N(f, 0, T_{k}) + O(1) ≥ N(f′/f, 0, T_{k}) − N(f′/f, ∞, T_{k}) + O(1)

= \frac{1}{2π} ∫_{0}^{2π} log|\frac{f′}{f} (T_{k}e^{iθ})| dθ + O(1)

≥ (ϱ − 1) log T_{k} + O(1)

≥ (ϱ − 1) log r + (ϱ − 1) log(T_{k}/r) + O(1) ≥ (ϱ − 1) log r + O(1).

It follows from (28) and (29) that, for all large r,

(30)  \frac{1}{2π} ∫_{0}^{2π} log|\frac{f′}{f} (re^{iθ})| dθ ≥ (ϱ − 1) log r + O(1).

Thus for all large r we have from Theorem 3, Jensen’s formula and (30)

S(f, {0, ∞}, r) = N_{Ram}(f, r) − N(f, 0, r) − N(f, ∞,)

= \frac{1}{2π} ∫_{0}^{2π} log |f′/f|(re^{iθ}) dθ ≥ (ϱ − 1) log r + O(1).

It turns out from log T (f, r) = ϱ log r + O(1) that δ_{2}(f, {0, ∞}) = 1 − 1/ϱ. Thus Theorem 5 is completely proved.

Remark. Functions similar to (5) were considered in [13], [3] and [7]. However, our functions here make (30) hold for all large r rather than some large r. With ϱ = ∞ we have shown in Theorem 1 that the set of entire functions having Nevanlinna’s secondary deficiency one is dense in H(C).

5. Examples

In this section we calculate Nevanlinna’s second deficiency for some classical functions, based on a recent work of L. Sons and Z. Ye in [10].

Example 1. Let f be any rational function; then, by Theorem 5, δ_{2}(f, {a_{j}}) = −∞ for any {a_{j}} ⊂ C.

Example 2. Let f(z) = e^{z}. As we showed in [10], S(e^{z}, {a_{j}}_{1}^{q}, r) = O(1) for any {a_{j}}_{1}^{q} ⊂ C \ {0}. Since T (e^{z}, r) = r/π, we have δ_{2}(f, {a_{j}}_{1}^{q}) = 0. Therefore the coefficient of term log T (e^{z}, r) in (1) is 0. In fact, if the order of f is less than one, then, by Theorem 5, S(f, {a_{j}}_{1}^{q}, r) < 0 for at least a sequence of {r_{j}} which tends to infinity.

108 Zhuan Ye

Example 3. Let ℘ be a Weierstrass function. Since the order of ℘ is two and S(℘, {a_{j}}_{1}^{q}, r) = O(1) for any {a_{j}}_{1}^{q} ⊂ C as we have shown in [10, Theorem 1], it follows that δ_{2}(℘, {a_{j}}_{1}^{q}) = 0.

Example 4. Let ζ(z) = ζ(z; ω_{1}, ω_{2}) be a Weierstrass ζ -function with Imω_{1}/ω_{2} > 0. Since S(ζ, {a_{j}} ∪ {0, ∞}, r) = − log r + O(1) and log T (ζ, r) = 2 log r + O(1) (see [10, Theorem 2]), then δ_{2}(ζ, {a_{j}} ∪ {0, ∞}) = −1/2, for any {a_{j}} ⊂ C \ {0}. Furthermore, we know from [10, Theorem 2] that the L-type of ζ is ψ(r) = r^{−1/2} . Hence we see the secondary deficiency of ζ to be exactly equal to the power index in its L-type.

Example 5. Let ϑ(z) be a Weierstrass ϑ-function. Then from [10, Theorem 4] we have S(ϑ, {a_{j}} ∪ {0, ∞}, r) = log r + O(1) and log T (ϑ, r) = 2 log r + O(1); thus δ_{2}(ϑ, {a_{j}} ∪ {0, ∞}) = 1/2 for any {a_{j}} ⊂ C \ {0}. Again the 1/2 identifies the power index in the L-type of ϑ function.

There are other functions which were considered in [10]. Similarly, their secondary deficiencies can be easily calculated.

References

[1] Gauthier, P.M., and W. Hengartner: The value distribution of most functions of one or several complex variables. - Ann. of Math. 96, 1972, 31–52.

[2] Hayman, W.: Meromorphic functions. - Clarendon Press, 1975.

[3] Hinkkanen, A.: A sharp form of Nevanlinna’s second fundamental theorem. - Invent. Math. 108, 1992, 549–574.

[4] Lang, S.: Report on diophantine approximations. - Proc. Nat. Acad. Sci. U.S.A. 55, 1966, 31–34.

[5] Lang, S.: The error term in Nevanlinna theory. - Duke Math. J. 56, 1988, 193–218.

[6] Lang, S., and W. Cherry: Topics on Nevanlinna theory. - Lecture Notes in Math. 1433, Springer-Verlag, 1990.

[7] Miles, J.: A sharp form of the lemma on the logarithmic derivative. - J. London Math. Soc. 45, 1992, 243–254.

[8] Nevanlinna, R.: Analytic functions. - Springer-Verlag, 1970.

[9] Offord, A.C.: The distribution of values of an entire function whose coefficients are independent random variables. - Proc. London Math. Soc. 14a, 1965, 199–238.

[10] Sons, L.S., and Zhuan Ye: The best error terms of classical functions. - To appear.

[11] Vojta, P.: Diophantine approximations and value distribution theory. - Lecture Notes in Math. 1239, Springer-Verlag, 1987.

[12] Wong, P.M.: On the second main theorm of Nevanlinna theory. - Amer. J. Math. 111, 1989, 549–583.

[13] Ye, Z.: On Nevanlinna’s error terms. - Duke Math. J. 64, 1991, 243–260.

Received 16 August 1993