ON NEVANLINNA’S INVERSE PROBLEM

David Drasin

August 5, 1997

1. The monograph [9] appeared in the early 1970s, and in winter 1972, Allen Weitsman and I, ignorant of the Russian alphabet, resolved to master Chapter 7 of [9]. This is how we learned about (concrete) Riemann surfaces and quasi–conformal mappings, and our paper [6] reflected the impact of this source. Reading [9] was the most important single event of my scientific career. Since it has never been translated into English, for many years I considered it a “secret weapon.”

In the following year, I was able to pursue these ideas further, and obtained [3] a solution to the inverse problem of Rolf Nevanlinna:

Theorem. Given 0 ≤ δ_{j}, θ_{j} ≤ 1, ∑_{j} δ_{j} + θ_{j} ≤ 2, and a sequence {a_{j}} of distinct elements of ℂ̂, there exists a meromorphic function f with

δ(a_{j}, f) = δ_{j}; θ(a_{j}, f) = θ_{j}

and δ(a_{j}, f) = θ(a_{j}, f) = 0 if a ∉ {a_{j}}.

Here δ(a_{j}, f) and θ(a_{j}, f) are the usual deficiency and branching terms of the theory; see [9, 10, 19] for a full account.

Developments since the publication of [3] have allowed some simplifications to be made, and it is possible to see what are the key ideas:

(1) the theory of functions of completely regular growth, due to the late B. Ja. Levin and A. Pfluger, for which the essential reference remains [15];

These arguments were developed in conversations with José Fernández and Antonio Sanchez-Calle in Spring 1996 while I was at the Universidad Autónoma de Madrid on sabbatical from Purdue University. I thank them, their colleagues, and the Minesterio de Educación y Ciencia, Dirección General de Investigación Cientifica y Técnica (España) for their hospitality and support.

Misha Sodin and Jang-Mei Wu made important suggestions concerning exposition and history.

2 DAVID DRASIN

(2) approximation of δ–subharmonic functions u by those of the special form v(z) = log |f(z)|, where f is meromorphic in the plane; Yulmukhametov [20] now presents the state–of–art;

(3) the elementary quasi–conformal mappings which were used in [3], and a weak form of the Teichmüller-Belinskii theorem.

Chapter 7 of [9] was devoted to the solution to the inverse problem in many important special cases. For all these reasons, it seems appropriate to present here a shorter proof of our theorem. The main ideas can be be seen in the special case that we ignore the θs in the inverse problem, and work only with the δs; in the general case, the auxiliary function U(z) of (2.8) below has to be modified. As in [3] we note that this method also can produce functions of finite order; the scope of the method is determined by suitable δ–subharmonic functions such as U(z). In particular, this gives a simpler way to solve the full inverse problem (which involves the branching terms θ(a_{j}, f)).

Much as in [3], the theory of Riemann surfaces is avoided, in contrast to the exposition in [9]. Step (2) has no antecedent in [3], and (1) was performed in a much more cumbersome manner. Here we use the explicit δ–subharmonic function U of (2.7); [3] required many stages of ad hoc construction to obtain a similar function, which was constructed by a zero–distribution, rather than directly from the desired asymptotic expansion as in (2.7). Step (3) remains much as in [3].

2. A smooth model. The fundamental reference function is one which is of order λ(r), mean type. For r > 0 set

(2.1)  S(r) = exp ∫_{1}^{r} λ(t)\frac{dt}{t},

where λ(r) = 1 (0 ≤ r ≤ 1) and

(2.2) rλ′(r) → 0 (r → ∞).

The value of (2.1) and (2.2) is that

S(t) ∼ (\frac{t}{r} )^{λ(r)} S(r) | log \frac{t}{r} | < A(r) → ∞,

where A(r) → ∞ at a rate governed by (2.1). Thus each r is a Pólya peak of order λ(r). It is covenient and consistent with (2.1) that require in addition that

(2.3)  r^{2}λ′′(r) → 0 (r → ∞);

ON NEVANLINNA’S INVERSE PROBLEM 3

this is simply that d^{2}λ/d(log r)^{2} = o(1). This is an analogue of the classical proximate order comparison function ρ(r) [2], but because of the simple restriction (2.2) (with (2.3) here) is far more flexible; see [4] for a general exposition. For example, when dealing with ρ(r), (2.2) is replaced by the less natural condition

rρ′(r) log r → 0 (r → ∞)

In our application here,

(2.4)  λ(r) ↑ ∞ (r → ∞),

but that is not required in general. Independent of the particular λ(r), S ↑ ∞ in (2.1) and in fact

(2.5)  S(t) ≥ S(s)(\frac{t}{s})^{λ(s)} ≥ S(s)\frac{t}{s}  (t > s).

Also, we observe that

(2.6)  S′(t) = t^{−1}λ(t)S(t),  t > 0.

As in [3], we construct a formal quasi–regular solution G to the inverse problem, and by controlling the rate at which (2.4) occurs, ensure that the formal Nevanlinna data transfer from G to the associated meromorphic function f. The method requires in addition to (2.2) and (2.3) that, if we define a sequence {r_{n}} by

(2.7)  λ(r_{n}) = inf\{t; λ(t) = c_{n} ≡ n + \frac{1}{4}\},

then the r_{n} → ∞ rather slowly. (This limitation, however, ensures that the function f can be of as ‘small’ an infinite order as desired.) These conditions are displayed in (3.7), (4.21) and (5.14) below.

Given S from (2.1), write z = re^{iθ}, |θ| ≤ π and define

(2.8)  U(z) = U(re^{iθ}) = S(r) cos λ(r) θ.

Lemma 2.9. U belongs to the Sobolev class W _{loc}^{1,1}. The Laplacian of U is a signed measure (charge) μ whose action on a test-function φ ∈ C_{0}^{∞} is

(2.10) (ΔU, φ) = −2 ∫_{−∞}^{0} φ(x)λ(−x)S(−x) sin πλ(−x) dx + ∫_{ℂ} φ(z)R(z) dx dy,

4 DAVID DRASIN

where the ‘error term’ R(z) satisfies

(2.11)  |R(z)| ≤ Cλ(r)S(r)r^{−2} (rλ′(r) + r^{2}|λ′′(r)|)

for some absolute constant C.

Proof. If | arg z| < π we may compute ΔU using the first equality in (2.10) directly from (2.8). Thus

R(z) = r^{−2}S(r) [{cos λθ − (2λ + 1) sin λθ θ − cos λθ θ^{2}(rλ′)}(rλ′) − sin λθ θr^{2}λ′′] ,

and an appeal to (2.2) and (2.3) shows that R can be estimated as in (2.11). Note that if λ(r) is constant then U is harmonic off the negative axis. To compute ΔU for negative z, we take a small disk D with center at r_{0} < 0. Then as a distribution ΔU acts on φ ∈ C_{0}^{∞}(D) by

∫ ∫_{D} φΔU dA = ∫ ∫_{D} UΔφ dA.

To compute the latter integral, let ℝ divide D into D^{+} and D^{−} in the obvious way. Then we may apply Green’s identities to Ω = D^{+} or D^{−}, since U is smooth on each of these regions. If U^{+} and U^{−} are these restrictions, then routine computations give that

∫ ∫_{D} UΔφ dA = ∫_{D∩ℝ} φ(x)(\frac{∂U^{−}}{∂y} − \frac{∂U^{+}}{∂y}) dx
= ∫_{D∩ℝ} φ(x) 2λ(|x|)S(|x|) sin πλ(|x|) x^{−1}dx,

which completes the proof.

Corollary 2.12. Let ΔU = μ and write

(2.13)  μ = μ^{+} − μ^{−};  |μ| = μ^{+} + μ^{−}

in terms of the Hahn decomposition. Then

(2.14))  |μ|({|z| < r}) ≡ |μ|(r, U) = {| sin πλ(r)| + o(1)}S(r).

Proof. We have from (2.10) and (2.11) that

|μ|(r, U) = O{∫_{0}^{r} λ(t)S(t)| sin πλ(t)|t^{−1} dt + ∫_{0}^{r} λ(t)S(t)ε(t)t^{−2} tdt dϕ},

with ε(t) = o(1), using (2.2) and (2.3). Thus, if we take K > 1, and recall (2.2) and (2.6), we have

|μ|(r, U) = O( ∫ ^{r} | sin πλ(t)|S′(t) dt + o(1) ∫ ^{r} S′(t) dt)
= O( ∫_{0}^{r/K} S′(t) dt + (| sin πλ(r)| + o(1)) ∫_{r/K}^{r} S′(t) dt + o(1)S(r))
= O(S(r/K)) + (| sin πλ(r)| + o(1))S(r),

and by taking K appropriately large, (2.14) follows from this and (2.5).

3. Approximation. We pass from U to a meromorphic function g in what is now a standard way. Recall from [15, p. 90] that a C^{0}–set E is one which can be covered by disks {|z − z_{j}| < r_{j}} with

∑_{|z_{j}|<r} r_{j} = o(r) (r → ∞).

Theorem 3.1. [20] There exists a function g(z) meromorphic in the plane and a C^{0}–set E so that

(3.2)  |U(z) − log |g(z)|| = o(\frac{S(r)}{λ(r)})  (z ∉ E, |z| = r).

In addition, if n(r, U) is the positive part of ΔU in B(r) and n(r, g) the counting– function of the poles of g, then

(3.3)  |n(r, U) − n(r, g)| = o(S(r))  (r → ∞).

Remarks. I know of no exact formulation which is Theorem 3.1, but the arguments of [20] readily adapt to this purpose, so only a sketch is given, along with copious reference to [20]. That paper is the culmination work from several distinct schools: Kjellberg [13], Kennedy–Hayman–Al-Katifi [12, §10.5], and Azarin [1]. All deal with approximation of subharmonic functions, rather than δ–subharmonic functions, but as we will see, the essential idea is to approximate a general mass distribution by ones which are composed of point–masses. This very natural idea has been used in an ad hoc manner for some years, although for some time it was only used in situations in which the mass or charge is supported on a finite family of nice curves. I

6 DAVID DRASIN

thank M. Sodin for pointing out Lemma 29.1 of N. Levinson [16], especially its proof on p. 165, as well as the argument in [18, p. 49], for which Mergelyan apportions some priority of M. Keldysh (cf. p. 32). The general question of approximating general subharmonic functions u by those of the form v(z) = log |f(z)|, f entire, with no restriction on the support of Δu was seriously addressed by V. Azarin some years later [1], although he considered only functions of finite order. This concept is essential here, since an inspection of (2.10) and (2.11) shows that the support of ΔU is quite large.

While (3.2) is not best possible, it is adequate for our purposes.

Proof. Let ΔU = μ = μ^{+} − μ^{−} as in (2.13). The technique is to approximate these measures by point masses, which then become the zeros and poles of g.

The results of [20] are stated in terms of what are called (β, s)–normal points relative to a given measure ν: z is (β, s)–normal (with respect to ν) if

ν({|ζ − z| < t}) < βt

for all 0 < t < s. For n ≥ 0 let A_{n} be the annulus A_{n} = {z; 2^{n} < |z| < 2^{n+1}}. We follow [20, §§4 and 5] which measures approximation in terms of a sequence {h_{n}} → 0 chosen in (3.6); these h_{n} retard the rate at which (2.4) occurs. For z ∈ A_{n}, |z| = r take

(3.4)  β = S(r)(λ(r)h_{n}^{2}r)^{−1},  s = r^{1/2}.

Lemma 3.5. Let τ be a signed measure, |τ| the measure associated to the total variation of τ, and assume that

|τ|({|z| < r}) < AS(r),

where S is from (2.1).

Choose h_{n} → 0 in (3.4) so that

(3.6)  2^{(n−1)/4}h_{n} ↑ ∞,

and assume in addition to (2.2) and (2.3) that that λ satisfies

(3.7)  2h_{n}λ(2^{n+1})2^{λ(2^{n+2})} ≤ 1.

ON NEVANLINNA’S INVERSE PROBLEM 7

Define the set E by: E = ∪_{n}E_{n} where E_{n} is the non–(β, s)–normal points of A_{n} with respect to the measure |τ|. Then E may be covered by a network of balls B_{j} = {|z − z_{j}| < t_{j}} such that

(3.8)  ∑ t_{j} < A_{0}h_{n}r  (r > 2^{n}),

where in (3.8) the sum is over all B_{j} which meet {|z| < r} and A_{0} = A_{0}(A).

Proof. Now |τ|(A_{n}) < ∞ for each n. Each point z of E_{n} ⊂ A_{n} is the center of some disk B(a, t) ≡ {|ζ − z| < t}, t < s, such that |τ|(B(z, t)) ≥ βt. By a lemma of Besicovitch type (cf. [17, p. 30]) we may choose a sequence of balls B_{j}(= B_{j}^{n}) = B(z_{j}, t_{j}) such that E_{n} ⊂ ∪_{j}B(z_{j}, t_{j}) and such that no point of E_{n} lies in more than p of the balls B_{j}, p an absolute constant. Thus by Corollary 2.12 and our choice of s,

(3.9)  ∑_{B_{j}^{n}} β(z_{j})t_{j} ≤ ∑ |μ|(B(z_{j}, t_{j}) ≤ p|μ|({|z| < 2^{n+2}}) ≤ A_{0}S(2^{n+2}).

However, with our choices of β = β_{n} from (3.4) and (3.6),

β ≥ S(2^{n})(λ(2^{n+1})2^{n+1}h_{n}^{2})^{−1},

and if we sum over our covering of E_{n}, it follows that from (2.1), (3.7) and (3.9) that

∑ t_{j} ≤A_{0}h_{n}^{2}2^{n+1}λ(2^{n+1})\frac{S(2^{n+2})}{S(2^{n})} < A_{0}h_{n}^{2}2^{n+1}λ(2^{n+1})e^{λ(2^{n+2}) log 2}
≤A_{0}h_{n}^{2}2^{n+1}λ(2^{n+1})2^{λ(2^{n+2})} < A_{0}h_{n}2^{n} < A_{0}h_{n}r  (2^{n} < r < 2^{n+1}).

Now let E be as in the statement of Theorem 3.1. Since E ∩ A_{n} is contained in the union of these disks, it follows readily that E is a C^{0}–set. Any point z ∈ E can be covered by such disks with center in A_{k} for at most two values k, so (3.8) follows on summing a geometric series. □

If ν is any measure of bounded support, let

P^{ν}(z) = ∫ log |1 − z/ζ| dν(ζ)

8 DAVID DRASIN

be its (logarithmic) potential. We choose μ from Lemma 2.9, and note that Lemma 3.5 applies. Then for each n ∈ ℤ we let μ_{n} = μ_{|_{{|z|<2^{n}}}} ≡ μ_{n}^{+} − μ_{n}^{−}. According to [20, Theorems 2 and 3] there are polynomials P_{n}^{+} (= P^{μ_{n}^{+}}) and P_{n}^{−} (= P^{μ_{n}^{−}}) of the form (here ^{∗} can be either ^{+} or ^{−}) P_{n}^{∗}(z) = Π(1 − z/a_{j}^{∗}) such that, in view of the choices in (3.4) and (3.6),

|P^{μ_{n}^{∗}}(z) − log |P_{n}^{∗}(z)|| < A log s + B log |z| + 3βs(| log s| + 1) + C

< A_{0} log r + 3\frac{S(r)r^{1/2}}{λ(2^{n})h_{n}^{2}r} (log r + 1) + C

= o(\frac{S(r)}{λ(r)}) (3.10)

for all |z| = r, z ∈ A_{n} which are (β, s)–normal for the measures |μ| and {δ_{(a_{j}^{∗})}}; the set for which (3.10) fails satisfies (3.8). Let η_{n} = η_{n}^{+} − η_{n}^{−} be the measure with positive and negative point masses at the points {a_{j}^{∗}} which are the zeros of the P_{n}^{∗}. To compress notation, define

P^{μ}(z) = \frac{P^{μ_{n}^{+}}}{P^{μ_{n}^{−}}} , P^{η}(z) = \frac{P^{η_{n}^{+}}}{P^{η_{n}^{−}}} ,

where we ignore dependence on n. Since U −P^{μ} is harmonic in {|z| < 2^{n}} it follows that

U(z) − [(P^{μ} − P^{η})(z)] = log |g_{n}(z)|,

where g_{n} is meromorphic in {|z| < 2^{n}}.

Now let m be fixed and choose any z with 2^{m−1} < |z| < 2^{m}. Then (3.8) implies that z is at the center of a disk D = D(z) of radius t|z| with t < Ah_{m−1} such that (3.10) (with n = m) holds at each point of ∂D. Thus the {g_{n}} are uniformly bounded in a fixed neighborhood of z, and in this neighborhood the number of zeros and poles of the {g_{n}} is uniformly bounded By standard normality considerations, there exists a subsequence of the {g_{n}} which converges normally (in the spherical metric) to a function g which is meromorphic in the plane. Of course log |g| is harmonic except for possible limit points of the point masses of the corresponding log |g_{n_{k}}|.

Finally we show (3.3). Let μ^{+} be as in (2.13). Let r > 1 be fixed, and choose any n with 2^{n} > r. In the approximation of μ_{|{|z|<2^{n}}}^{+} by point–masses, [20] produces a certain network {P_{k}} of rectangles with sides parallel to the coordinate axes, and

ON NEVANLINNA’S INVERSE PROBLEM 9

writes μ^{+} = ∑ μ_{k}^{+} where the support of μ_{k}^{+} is contained in P_{k}; one point mass is then assigned to each P_{k}. Of course this network depends on n, but we ignore this since the estimates below do not depend on n, and so are preserved by weak limits. The properties of these rectangles (cf. [20, p. 259]) ensure ([20, p. 265]), that if b > 0 is given, then the number of P_{k} of diameter at least bt which meet any disk B(ζ, t) is at most K(1 + b^{−2}), K an absolute constant. (Of course, similar observations will apply to μ^{−}.)

Fix r > 1, take b = 2 and cover the annulus {(1 − η)r < |z| < (1 + η)r} by O(η^{−1}) disks D(ζ, 3ηr). Now the left side of (3.3) is bounded by the number of such rectangles which meet both {|z| = r} and one of {|z| = (1 ± η)r}. Consider the number of these rectangles which meet any one of the D(ζ, 3ηr). By the estimate in [20], only O(η^{−2}) rectangles can meet such a D(ζ, 3ηr), and then summing, we have that the left side of (3.3) is O(η^{−3}). If we choose η so that

η(r) → 0; η(r)^{3}S(r) → ∞,

we immediately obtain (3.3).

Remark. A simple computation using Lemma 2.9 shows that in fact all sufficiently large z are normal with respect to the measure |ΔU|, with U from (2.8), but the exceptional set E of Lemma 3.5 must occur when considering each |Δ log |P_{n}^{∗}|| or |Δ log |g||.

Now let

(3.11)  Z = {z; U(z) = 0, ℑz ≠ 0}.

Then Z = ∪_{k}Z_{k}, a union of components, and each circle {|z| = r} is divided into about 2λ(r) almost–congruent arcs by Z. Write Z_{k} = {re^{iθ_{k}(r)}; r ≥ r_{k}}. The next Lemma shows that each Z_{k} can be approximated well by curves which avoid the exceptional set E of Theorem 3.1, and on which | log |g|| is relatively large: |g| is close to zero on γ_{k}^{−} and large on γ_{k}^{+}.

Lemma 3.12. Let ε(r) → 0 (r → ∞). Then there exists r′_{k} ≥ r_{k} which depends only on the {h_{n}} of Lemma 3.5, so that to each k correspond arcs γ_{k}^{+}, γ_{k}^{−}, with γ_{k}^{±} = {re^{iθ_{k}^{±}(r)}; r ≥ r′_{k}} with

(1) θ_{k}^{−}(r) < θ_{k}(r) < θ_{k}^{+}(r),

10 DAVID DRASIN

(2) (γ_{k}^{+} ∪ γ_{k}^{−}) ∩ E = ∅,

(3) | cos λ(r)θ_{k}^{±}(r)| > 2ε(r),

(4) λ(r)|θ_{k}^{+}(r) − θ_{k}^{−}(r)| = o(1) (r → ∞).

Proof. Let k be given and Z_{k} the corresponding component of Z. We may suppose that ε(r) is continuous and that

(3.13)  ε(2^{n})h_{n−1}^{−1} ↑ ∞.

Now let ω^{+} = {re^{iθ_{0}(r)}} satisfy θ_{0}(r) > θ_{k}(r), |θ_{0}(r) − θ_{k}(r)| = o(λ(r)^{−1}) and

| cos λ(r)θ_{0}(r)| ≡ 3ε(r);

since ε(r) = o(1), this is possible for large r, and |θ_{0} − θ_{k}| ∼ 3ε(r)λ(r)^{−1}. Using this and (3.8), we may produce r′_{k} = r_{k}(1 + o(1)) and a γ_{k}^{+} which is disjoint from E such that if r ≥ r′_{k} and 2^{n} < r < 2^{n+1} then |θ_{k}^{+}(r) − θ_{0}(r)| < 3h_{n}. Thus, we have (1) and (2), and a glance at (3.7) and (3.13) shows that γ_{k}^{+} satisfies (3). The construction of γ_{k}^{−} is similar and (4) follows.

4. Value-distribution. Near each circle {|z| = r}, our function g of Theorem 3.1 behaves as a Lindelöf function of order λ(r). Thus, this section may be considered routine. However, one advantage of basing an analysis on (2.8) rather than obtaining (2.8) as a consequence of an a priori distribution of zeros and poles becomes clear: the distinction between functions of integral order or non–integral order disappears (cf. [9, §2.5]).

Lemma 4.1. The Nevanlinna characteristic of U (cf [11, Ch. 3]) satisfies

T (r, U) = \frac{1}{2π} ∫_{0}^{2π} U^{+}(re^{iθ}) dθ + N(r, U)

= \{ \begin{matrix}2[q + | sin(πλ(r))| + o(1)]λ(r)^{−1}S(r) & q ≡ [λ(r)] ≤ λ(r) ≤ q + \frac{1}{2}, \\ 2[q + 1 + o(1)]λ(r)^{−1}S(r) & q + \frac{1}{2} ≤ λ(r) < q + 1, \end{matrix} (4.2)

with N(r, U) = ∫^{r} n(u)u^{−1} du and n(u) the positive part of ΔU in (2.9).

The characteristics of U and g satisfy

(4.3)  |T (r, g) − T (r, U)| = o(\frac{S(r)}{λ(r)}),

so that T (r, g) also satisfies (4.2).

Proof. That n(r, U) + n(r, −U) = {| sin πλ(r)| + o(1)}S(r) is the content of (2.13).

Simple calculations using (2.1), (2.2), (2.4) and (2.6) show that

(4.4)  (\frac{S(r)}{λ(r)})′ = \frac{S′(r)}{λ(r)} − \frac{S(r)λ′(r)}{λ^{2}(r)} = (1 + o(1))\frac{S(r)}{r},

and since (2.14) clearly implies that |n|(r, U) ≡ n(r, U) + n(r, −U) = O(S(r)), this means that

(4.5)  |N|(r, U) ≡ ∫ ^{r} |n|(t, U)t^{−1} dt = O(\frac{S(r)}{λ(r)}).

Now |n|(r, U) = |n|(r, g) ≡ 0 for r < 1, so we may obtain in a similar manner, using (3.3) and (4.4), that

(4.6)  ||N|(r, g) − |N|(r, U)| = o(1) ∫ ^{r} \frac{S(t)}{t} dt = o(1)\frac{S(r)}{λ(r)}.

Now if (3.2) holds on all of {|z| = r}, we certainly have

|m(r, g) − m(r, U)| = o(\frac{S(r)}{λ(r)}),

and this with (4.5) and (4.6) would give (4.3) upon a short computation of m(r, U) using (2.8); thus (4.3) holds for all r except perhaps for those of density zero. But T (r, g) is log–convex, and so (4.3) now holds for all r.

It now follows from Lemma 4.1 and the form (2.1) of S(r) that the refined calculus developed for functions of completely regular growth can be used here; (cf [9, §2.5] [15, Ch.3] and, in a situation closer to ours, [7]). For completeness, we sketch one attack, but do not seek the most precise estimates.

Lemma 4.7. Let g be from Theorem 3.1.

(A) Suppose q = [λ(r)] is even. Then

(4.8)  n(r, ∞) = o(S(r)),  n(r, 0) < (1 + o(1))S(r);

if q = [λ(r)} is odd, then the roles of 0 and ∞ are interchanged in (4.8).

12 DAVID DRASIN

(B) For all a ∈ ℂ^{∗} = ℂ \ {0}

(4.9)  n(r, a, g) < 2(λ(r) + 1 + o(1))S(r)

while

(4.10)  |n(r, a, g) − 2λ(r)S(r)| < (2 + o(1))S(r)

where the limit in (4.10) is uniform as r → ∞ in each set {| log |a|| < M}.

Proof. Estimate (4.8) follows at once from Corollary 2.12 and the relation between the mass of U and log |g|. The proof of (B) depends on the argument principle. In order to effectively use this, it is necessary to supplement (3.2) and control arg g.

Associated to g is a function ε_{1}(r) = o(1) such that

(4.11)  rλ′(r) + r^{2}|λ′′(r)| < ε_{1}(r).

Choose A = A(r) → ∞ so that

(4.12)  Aε_{1}(r/A) < λ(r)^{−1} = o(1)  (r → ∞).

Given r_{0}, set r′ = A^{−1}r_{0}. Then if Ω = {r′ < |z| < r_{0}, | arg z| < π}, let

A = {a_{n}}, {a_{n′}}

be the zeros and poles of g which lie in Ω (so these do not include those on the negative axis), and so are an error term in the sense (2.11). By (2.6), (2.10), (2.11), (4.11) and (4.12), the number of points selected satisfies

n(A, Ω) ≤ A ∫ ∫_{Ω} λ(r)S(r){rλ′(r) + r^{2}|λ′′(r)|}r^{−1} drdθ

≤ Aε_{1}(r′) ∫_{r′}^{r_{0}} S′(r) dr < Aε_{1}(r′)S(r_{0}). (4.13)

We set

(4.14)  h(z) = g(z)p(z)

where the zeros and poles of the quotient p(z) = π(1 − z/a_{n})/π(1 − z/a_{n′}) are over A, so that h ≠ 0, ∞ in Ω. It follows from (2.1), (4.11), (4.13), Theorem 3.2 and standard estimates [15, Ch. 1 §§7, 8] that there is a C^{0}–set E_{1} so that

log |h| = log |g| + o(S(r_{0}))

= S(r_{0}) cos λ(r_{0})θ + R_{1}, (4.15)

ON NEVANLINNA’S INVERSE PROBLEM 13

with, for some function ε(r) = o(1) (r → ∞),

(4.16)  |R_{1}| < ε(r_{0})S(r_{0}),  (r_{0} → ∞, z ∉ Ω \ E_{1}),

and (cf. (3.2))

(4.17)  | log |g(z)| − U(z)| < ε(r)S(r)  (r → ∞; r ∉ E_{1}).

To be specific, suppose that in the notation (3.8) we have

(4.18)  ∑ r_{j} ≤ \frac{1}{2}rδ_{1}(r),

for some fixed function δ_{1}(r) → 0 (r → ∞), where the sum in (4.18) is over all balls B(z_{j}, r_{j}) which cover E_{1} in {|z| < r}. With Ω, r′ = A^{−1}r_{0} and A from (4.12), choose δ = δ(r) to satisfy (4.19)–(4.21) below. We first require that

(4.19)  δ = o(1) (r → ∞);  δ(r_{0}) ≥ inf_{[r′,r_{0}]} δ_{1}(t),

where δ_{1} is from (4.18). Then given ε(r) of (4.16) and (4.17), we suppose that

(4.20)  ε(r)δ(r)^{−2} = o(1)  (r → ∞).

Finally, we retard the rate at which λ(r) → ∞ and A(r) → ∞ in (4.12) so that

(4.21) δ(r′)λ(r_{0}) < \frac{1}{2}π;  δ(r′)λ(r_{0}) log \frac{1}{δ(r′)λ(r_{0})} = o(\frac{1}{λ(r_{0})}) (r_{0} → ∞).

Set Ω_{δ} = Ω ∩ {| arg z| < π − δ(r_{0})}. Then with h from (4.14) (cf [5, p. 45]) we introduce in Ω a branch of

(4.22)  H(z) = S(r_{0})^{−1} log(g(z)p(z))

normalized so that | arg H(r_{0})| ≤ π. From (4.14), (4.15), (4.20) and Borel-Carathéo-dory, it follows that H(z) = e^{iλ(r)θ} + o(1) (z ∈ Ω_{δ}), and, on differentiating and recalling (4.20), that

(4.23)  H′(z) = λ(r_{0})e^{iλ(r_{0})θ} + o(1)  (z ∈ Ω_{δ}).

Now recall Z = ∪_{k}Z_{k} from (3.11), as well as the curves γ_{k}^{±} of Lemma 3.12 which surround each Z_{k} ∩ Ω. With ε(r) from (4.17) (or (3.2)) and (4.20), we observe from Lemma 3.12 (3) that for r > r′′ (which does not depend on k)

| log |g(z)|| > U(z) − |U(z) − log |g(z)||
> 2ε(r)S(r) − ε(r)S(r)

> ε(r)S(r) (z ∈ γ_{k}^{±} ∩ Ω). (4.24)

14 DAVID DRASIN

Now we can obtain assertion (B) of Lemma 4.7. By making small changes in r′, r_{0}, we may assume that the boundary components of Ω avoid the exceptional sets E and E_{1}. The γ_{k}^{±} partition Ω into several components; let Ω_{0} be those which are disjoint from Z and Ω_{1} = Ω \ Ω_{0}. Then | log |g|| is large on each boundary ∂ω ⊂ ∂Ω_{0} so by (4.8), (4.24) and Rouché’s theorem we have uniformly in a

n(r, a, Ω_{0}) = o(S(r)) (r′ < r < r_{0}).

The computation on the components of Ω_{1} is a bit more delicate. For any γ ⊂ ∪γ_{k} which meets both boundary components of Ω we see from (4.23) and (4.12)–(4.14) that

|Δ_{γ} arg g| =(1 + o(1))S(r_{0}) − (1 + o(1))S(r′)
(4.25)  =(1 + o(1))S(r_{0}).

By (4.11) there is at most one pair γ_{k}^{±} which fails to meet both boundary components of Ω; for this pair the bound in (4.25) is at most 2(1 + o(1))S(r_{0}). According to Lemma 3.12((1),(4)), |g| is close to zero on one from each pair γ_{k}^{±}, namely that for which cos λ(r)θ < 0, and so if | log |a|| < M for some fixed M and r_{0} is sufficiently large, (Δ_{γ_{k}^{+}} + Δ_{γ_{k}^{−}})(arg g(z) − a) = (1 + o(1))S(r_{0}) for each k. The number of components of Ω_{1} is 2λ(r_{0}) + O(1), and so

(4.26)  ∑_{k} Δ_{γ_{k}^{±}∩Ω} arg(g(z) − a) = (2λ(r_{0}) + O(1))S(r_{0}),

uniformly for a in each set | log |a|| < M.

Finally, let {I_{k}^{0}}, {I′_{k}} be the components of ∂Ω_{1} ∩ {|z| = r_{0}}, ∂Ω_{1} ∩ {|z| = r′} respectively. Then Lemma 4.7 follows from the estimate

(4.27)  ∑(|Δ_{I′_{k}}| + |Δ_{I_{k}^{0}}|) arg(g(z) − a) = o(S(r_{0})),

which is a consequence of a method due to my teacher, W. H. J. Fuchs [9]. Each term in (4.27) is of the form (using the notation of [9, (17)]):

(4.28)  S < \frac{4Rr}{(R − r)^{2}} δT (R, g) + C[1 + log \frac{κR}{δr}]\frac{δ}{κ}m + 2\frac{R}{r},

where r(= r′ or r_{0}) < R, δ is the angular measure of the set I being considered, m = n(r_{0}, 0, g) + n(r_{0}, ∞, g) = O(S(r_{0})) and κr_{0} is an upper bound for the sum

ON NEVANLINNA’S INVERSE PROBLEM 15

of radii of the exceptional set as in (4.18), so that κ = O(rδ_{1}(r′)). We let p ↓ 1 as r → ∞ such that p^{λ(r)} → 1, and take in (4.28) R = pr, δ = o(1)λ(r) (this by Lemma 3.12 (4)). These choices with (4.21) now yield (4.27) and complete the proof of Lemma 4.8.

5. Completion of proof. Given the data {a_{j}}, {δ_{j}} in the statement of our (restricted) inverse problem, we first construct a quasiregular function G which is a formal solution to the inverse problem, and then use a principle which goes back to Teichmüller to obtain our final function f of the Theorem. This principle is reflected in the manner at which the dilatation

μ_{G}(z) = Gz̄/G_{z}(z)

of G tends to zero at infinity; see Lemma 5.8 below, and this will depend on the choice of the {r_{n}} of (2.7).

The function G is what we call a quasi-conformal modification of the function g of Theorem 3.1. A key role is played by the following elementary lemma, which is essentially [3, Lemma 8] and shows the flexibility of small qc changes of variables.

Lemma 5.1. Let a ≠ b and k > 0 be given, and Möbius transformations T and V have T 0 = a and V ∞ = b.

Then there exists ε > 0 and a quasi–conformal homeomorphism ω of the sphere with ∥μ_{ω}∥_{∞} < k such that

ω(z) = \{ \begin{matrix}T (z), & |z| < ε, \\ V (z), & |z| > ε^{−1}. \end{matrix}

The construction will depend on sequences {ε_{j}}, {k_{j}}, to use the notation suggested by Lemma 5.1, and whose significance will be clear from Lemma 5.8.

We introduce a sequence {b_{k}}(k ∈ ℤ) with b_{−k} = b_{k}, b_{k} ≠ b_{k+1} which repeats the {a_{j}} so that

(5.2)  n^{−1} card\{k; |k| ≤ n, b_{k} = a\} → \{ \begin{matrix}δ(a), & (a ∈ \{a_{j}\}) \\ 0 & otherwise; \end{matrix}

It is routine to obtain these {b_{k}} from the data {a_{j}}, {δ_{j}}. Note that in case ∑ δ_{i} ⪇ 2, it is necessary to introduce bs which are not among the original a_{j}s.

16 DAVID DRASIN

For each n ≥ 1 let c_{n} = n + \frac{1}{4} (cf (2.7)), and define the curve Γ′_{n} in the z–plane to be the component of the level–set {U(z) = U(−r_{n})} which contains the point z = −r_{n}, where r_{n} is also from (2.7). Thus U(r_{n}) ≠ 0, and so each Γ′_{n} lies well inside one component of the {U ≠ 0}, and each Γ′_{n} is symmetric with respect to the real axis. Notice that by retarding the rate at which (2.4) occurs, we may make each S(r_{n}) as large as desired, by (2.5). The Γ′_{n} divide the plane into regions Ω_{n}(n ≥ 0) where Ω_{0} contains the positive real axis and is bounded by Γ′_{1}; for n ≥ 1, Ω_{n} is bounded by Γ′_{n} and Γ′_{n+1}. And given any sequence K_{n} → ∞ we may allow (2.4) to occur so slowly that log S(r_{1}) > K_{1}, log S(r_{n+1}) > log S(r_{n}) + K_{n+1}. According to (3.8), we may slightly perturb each Γ′_{n} to obtain a Γ_{n} which avoids the exceptional set E of Theorem 3.1, and so log |g(z)| ∼ U(−r_{n}) (z ∈ Γ_{n}); cf. the proof of Lemma 3.12.

Let g be the meromorphic function obtained from U by Theorem 3.1, and let ω_{0} = T_{0} be a Möbius transformation with T 0 = b_{1}, T ∞ = b_{0}. We then set

(5.3)  G(z) = T_{0} ∘ g  (z ∈ Ω_{0}).

In order to extend G to Ω_{1} we apply Lemma 5.1 and choose a qc homeomorphism ω_{1} with ∥μ_{ω_{1}}∥_{∞} < k_{1} such that ω_{1}(B(0, ε_{1})) = ω_{0}(B(0, ε_{1})) = T_{0}(B(0, ε_{1})) and ω_{1}(∞) = b_{2}. Note from Lemma 5.1 that ε_{1} > 0 will depend on k_{1}. If we take

(5.4)  G(z) = ω_{1} ∘ g(z)  (z ∈ Ω_{1}),

G will be K–quasi–regular in Ω_{0} ∪Ω_{1} if it is continuous on Γ_{1}. Now, in the notation (2.7), cos c_{1}π < 0, and U(−r_{1}) ∼ S(r_{1}) cos c_{1}π on Γ_{1}. Thus (2.5) and (2.7) ensure that given any ε_{1} > 0, we can guarantee that |g| < ε_{1} (z ∈ Γ_{1}) if r_{1} is sufficiently large. In this way we can ensure that G is K–qr in Ω_{0} ∪ Ω_{1}.

In general, suppose G has been defined in ∪_{1=1}^{n−1}Ω_{i}, and to be concrete let us suppose that cos c_{n}π > 0. Then, given the data k_{n} < k_{n−1} ⋯ < k_{1} < 1, b_{n} and b_{n+1} and the q–c homeomorphisms ω_{n−1}, …, ω_{1} we choose ω_{n} with ∥μ_{ω_{n}}∥_{∞} < k_{n} from Lemma 5.1, with ε = ε_{n}, a = b_{n} and b = b_{n+1}. By taking r_{n} sufficiently large, we may then ensure that |g(z)| > ε_{n}^{−1} (z ∈ Γ_{n}) (this is because cos c_{n}π > 0; were cos c_{n}π < 0, we would have that |g(z)| < ε_{n} (z ∈ Γ_{n})). Hence if we take

(5.5)  G(z) = ω_{n} ∘ g(z)  (z ∈ Ω_{n}),

ON NEVANLINNA’S INVERSE PROBLEM 17

then G is q − r and continuous on ∪_{i=1}^{n}Ω_{1}. By repeating this procedure, G becomes K–qr in the plane.

Since (2.8) and (3.2) adequately describe g and our connection between g and G is so explicit, we have from Lemma 4.8 and (5.2)–(5.5)

Lemma 5.6. Let G be constructed as described. Then for each a we have

n(r, a, G) = (2 + o(1))λ(r)S(r) (r → ∞, a ∉ {a_{j}}),
n(r, a, G) = 2(1 − δ(a) + o(1))λ(r)S(r) (r → ∞, a = a_{j}).

Of course, Lemma 5.6 shows that G is a formal solution to our inverse problem in the natural way (cf. [3, p. 92])..

Now f is constructed from G by a well–known procedure. This was first applied to the inverse problem by Le–Van Thiem [14], and an account of the situation before 1970 appears in [9, Chapter 7]. In [6] Weitsman and I introduced comparison functions modeled on the Lindelöf functions, and soon after that came [3]. My late colleague K. V. Rajeswara Rao urged that all references to Riemann surface language be purged in [3], and that viewpoint now seems to have general acceptance. Thus consider the equation

ϕz̄(z) = μ_{G}(z)ϕ_{z}(z)

and let ϕ be the homeomorphic solution which fixes 0, 1, and ∞. We define

(5.7)  f ≡ G ∘ ϕ^{−1},

and compute readily (Cauchy–Riemann equations!) that f is meromorphic in the plane. In order to check that the data of Lemma 5.6 pass to f, we need a generalization of Teichmüller’s theorem, which Weitsman and I obtained in [6] following [9].

Lemma 5.8. [3, Lemma 2, p. 91] Let G be quasiregular in the plane such that the dilatation

μ_{G}(z) ≡ \frac{Gz̄}{G_{z}}

satisfies

(5.9)  ∥μ_{G}∥_{∞} ≤ k_{0} < 1.

Set

(5.10)  L(r) = \frac{1}{2π} ∫_{0}^{2π} |μ_{G}(re^{iθ})| dθ,

and suppose that L(r) → 0 (r → ∞). Let Z = ϕ(z) be the quasi–conformal homeomorphism of the plane with

(5.11)  ϕ(0) = 0, ϕ(1) = 1, ϕ(∞) = ∞

such that μ_{ϕ} = μ_{G}. Then as z_{0} → ∞ the normalized functions

Φ_{z_{0}}(z) = \frac{ϕ(zz_{0})}{ϕ(z_{0})}

tend normally to the identity function Φ(z) = z in {0 < |z| < ∞}.

This can be made rather precise: there exists M < ∞ depending only on (5.9) and (5.11) such that to each ϵ > 0, A < ∞ corresponds η > 0, C < ∞ such that if

(5.12)  L(r) < η (r > R),

then for A^{−1} < |z| < A and |z_{0}| = r_{0} we have that

(5.13)  ω(r_{0}) ≡ sup_{|ζ|≥r_{0}} |log \frac{ϕ(zζ)}{zϕ(ζ)}| < ϵ  (r_{0} > CR^{M}).

Conditions (5.12) and (5.13) yield our key restrictions on the sequence {r_{n}} of (2.7); i. e., the growth of the basic comparison function λ(r):

(5.14)  nω(r_{n}) = o(1)  (n → ∞),

where the {r_{n}} are from (2.7). According to (5.12) and (5.13), (5.14) can be assured by the following plan: given a sequence ϵ_{n} ≡ ω(r_{n}) from (5.14), choose η_{n} so that (5.13) (with ϵ = ϵ_{n}) is a consequence of

(5.15)  L(r) < η_{n}  (r ≥ r′_{n});

here the {r′_{n}} are taken so that C_{n}(r′_{n})^{M} = r_{n}.

Lemma 5.16. Suppose G is constructed as in Lemma 5.6 and (5.7) holds. Then for each a ∈ ℂ^{∗} we have δ(a, f) = δ(a, G).

Proof. Given any a ∈ ℂ and t > 0, write t = ϕ(z), |z| = r, with, say, r_{n} ≤ r < r_{n+1}. Then (5.13) shows that if (1 + o(1))ω(r)ϵ > ω(r), then

{|z| < (1 − ϵ)r} ⊂ ϕ^{−1}{|Z| < t} ⊂ {|z| < (1 + ϵ)r},

so that

(5.17)  n((1 − ϵ)r, a, G) < n(t, a, f) < n((1 + ϵ)r, a, G).

Suppose now that a ∈ {a_{j}}. By using (2.7) and (5.14), the last expression in (5.17) can be simplified: if r > r_{n} then

n((1 + ϵ)r, a, G) < 2(1 − δ(a) + o(1))λ((1 + ϵ)r)S((1 + ϵ)r)
< 2(1 − δ(a) + o(1))(1 + o(1))λ(r)(1 + ϵ)^{λ(r)}S(r)
∼ (1 + o(1))2(1 − δ(a) + o(1))λ(r)(1 + λ(r)ϵ)S(r)
∼ (1 + o(1))2(1 − δ(a) + o(1))λ(r)(1 + O(nω(r_{n}))S(r)
∼ 2(1 − δ(a) + o(1))λ(r)S(r),

and similarly, the left side of (5.17) is asymptotic to 2(1 − δ(a) + o(1))λ(r)S(r). If a ∉ {a_{j}} then n(t, a, f) ∼ λ(r)S(r). Thus each N(t, a, f) can be computed asymptotically. This proves Lemma 5.16.

It remains to arrange (5.14) (or, equivalently, (5.15)). Since the k_{n} decrease, it is clear that

(5.18)  ∥μ_{G}∥_{∞} < k_{0}.

Lemma 5.19. Given numbers η_{n} ↓ 0 we can choose {r′_{n}} so that (5.15) holds.

Proof. We will have r′_{n} < r_{n} < r′_{n+1}, and r_{n} = C_{n}r′_{n}^{M}. Let us suppose that r′_{0} = 0 and that r_{n} has been defined; i. e., we are at the stage (5.5). Let r_{n} < r and |z| = r. First, suppose z ∈ ∪_{0}^{n−1}Ω_{j}, say z ∈ Ω_{p}. While the elementary bound |μ_{G}(z)| < k_{p} for any fixed p does not improve as r → ∞, we observe from the form of ω in

20 DAVID DRASIN

Lemma 5.1 and the expression (2.8) of U that μ_{G} ≡ 0 at points at which | log |g|| is large; i.e., except for neighborhoods of the set Z of (3.11), whose angular measure is o(1) as r → ∞. Hence, by taking r′_{n+1} sufficiently large we can arrange that

∫_{∪_{0}^{n}Ω_{j}∩\{|z|=r\}} μ_{G}(re^{iθ}) dθ < \frac{1}{2}η_{n+1} (r > r′_{n+1}).

Thus take k_{n} = (4π)^{−1}η_{n+1}; this gives (5.15) with n+1 in place of n, we obtain the {r′_{n}}. Note that this k_{n} determines ε = ε_{n} in Lemma 5.1, and so that G in (5.5) be continuous, we must now choose r_{n+1} large enough so that G remains continuous on Γ_{n+1}. In this way we obtain the {r′_{n}} and the Lemma.

Our Theorem is proved.

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e–mail:drasin@math.purdue.edu Purdue University