Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 61–79

A CONVERSE DEFECT RELATION FOR QUASIMEROMORPHIC MAPPINGS

Swati Sastry

Institute of Mathematical Sciences, CIT Campus Taramani, Madras -113, India; sastry@imsc.ernet.in

Abstract. Let f: R^{n} → R̄^{n} be a nonconstant K -quasimeromorphic map. We prove first that given C > 1, there exists θ > 1, θ depending only on n, K, C, such that whenever a_{1}, …, a_{q} ∈ R̄^{n} are distinct, we have n(r, a_{j}) ≤ CA(θr) for j = 1, …, q and r ∈ E, where E = E(f, a_{1}, …, a_{q}) has infinite logarithmic measure. This result is then used to obtain the following converse to the defect relation as established by S. Rickman. Let f: R^{n} → R̄^{n} be a nonconstant K -quasimeromorphic map. Then there exist constants C_{1} > 1 and θ_{1} > 1, depending only on n and K such that for a_{1}, …, a_{q} ∈ R̄^{n} any distinct points, we have lim sup_{{r→∞ \atop r∈E}} ∑_{j=1}^{q}((n(r, a_{j}))/(A(θ_{1}r)) − 1)_{+} ≤ C_{1} where E can be taken to be the same set as above. Any improvement or enlargement of the set E for the first result is immediately valid for the second (main) result.

1. Introduction

Quasiregular (and quasimeromorphic) mappings form a natural generalization of analytic (and meromorphic) maps to real n-dimensions. We abbreviate these classes as qr and qm. These functions retain some of the most important topological properties of analytic functions. A study of the value distribution theory of such maps has been a subject of interest for many years. For an overview of results in this area we refer to [R2].

Rickman has shown [R3] that a weak form of Picard’s theorem holds for these mappings. Moreover in [R2], [R6] he proved that for a nonconstant, real n-dimensional, n ≥ 3, K -qm function f, there exists a set E ⊂ [1, ∞) of finite logarithmic measure, and a constant C(n, K) < ∞, depending only on n and K such that

lim sup_{{r→∞ \atop r ∉ E}} ∑_{j=1}^{q}(1 − \frac{n(r, a_{j})}{A(r)})_{+} ≤ C(n, K),

where a_{1}, …, a_{q} are distinct points. For n = 3 this is qualitatively sharp, as can be seen from [R6, Theorem 1.7]. Thus Nevannlinna’s defect relation generalizes in qualitative form to qm maps.

1991 Mathematics Subject Classification: Primary 30C65.

This is a portion of the author’s thesis at Purdue University. Research partially supported by NSF.

In this paper, we consider a converse inequality. For a nonconstant meromorphic function f in the plane, it was shown by J. Miles [Mi] that there exist absolute constants K < ∞ and C ∈ (0, 1) and a set E = E(f) ⊂ [1, ∞) having lower logarithmic density at least C such that if a_{1}, …, a_{q} are distinct elements of the Riemann sphere, then

lim sup_{{r→∞ \atop r∈E}} ∑_{j=1}^{q}(\frac{n(r, a_{j})}{A(r)} − 1)_{+} ≤ K.

Here we extend the above result, for meromorphic functions in the plane, to qm maps and all dimensions.

The proof breaks up into two parts: Sections 3 and Section 4. In Section 3 we show that n(r, a_{j}) ≤ CA(θr) for any given q points a_{1}, …, a_{q} and r taking values in a set E of infinite logarithmic measure. This is an extension of [R1, 5.16], where the case q = 1 is considered. The proof is a slight modification of the proof of the same. In Section 4 we first obtain an estimate which holds for all except possibly one value a_{j}. This estimate holds without the exceptional r-set, but the a_{j} chosen as exception does depend upon r. For such an a_{j} we then use the bound obtained in Theorem 3-1. An important open problem is to get a result such as Theorem 3-1 off an exceptional set which does not depend on a. The main analytic tool is path families, a natural generalization to space of extremal length.

I thank Professor David Drasin for suggesting this problem to me, as part of my thesis, and also for his constant encouragement and guidance.

2. Notation and definitions

We denote by R^{n} the real euclidean n-space, and by R̄^{n} the one-point compactification R̄^{n} = R^{n} ∪ {∞}. Set

\begin{aligned}B_{r}(x) & = \{ y ∈ R^{n} : |x − y| < r \}, S(x, r) = ∂B_{r}(x), \\ B(r) & = B_{r}(0), S(r) = S(0, r), and S = S(1). \end{aligned}

The Lebesgue measure in R^{n} is denoted by ℒ^{n} and the normalized k -dimensional Hausdorff measure in R^{n} by ℋ ^{k}. We set ω_{n−1} = ℋ ^{n−1}(S). The Euclidean metric in R^{n} is d. If γ: Δ → R̄^{n} is a path, we denote its locus γΔ by |γ|.

R̄^{n} is equipped with the spherical metric,

\begin{aligned}d[x, y] & = |x − y|/[(1 + |x|^{2})(1 + |y|^{2})]^{1/2}; x, y ≠ ∞ \\ d[x, ∞] & = 1/(1 + |x|^{2})^{1/2}. \end{aligned}

Definition. Let n ≥ 2, and let G be a domain in R^{n}. A continuous mapping f: G → R^{n} is called quasiregular if (1) f is in the local Sobolev space W_{n,loc}^{1}(G); i.e., f has distributional partial derivatives which are locally L^{n} -integrable, and (2) there exists a constant K, 1 ≤ K ≤ ∞, such that

(2-1)  |f′(x)|^{n} ≤ KJ_{f}(x)

holds for almost every x ∈ G. Here |f′(x)| is the sup norm of the formal derivative f′(x) defined by means of partial derivatives and J_{f}(x) is the Jacobian determinant of f at x. The smallest K in (2-1) is the outer dilatation K_{O}(f), and the smallest K, 1 ≤ K ≤ ∞, for which

J_{f}(x) ≤ K inf_{|h|=1} |f′(x)h|^{n} a.e.

holds is the inner dilatation K_{I}(f) of f. K(f) = max(K_{O}(f), K_{I}(f)) is the maximal dilatation of f. If f is quasiregular and K(f) ≤ K, it is called K - quasiregular.

Let G ⊂ R̄^{n} be a domain. A mapping f: G → R̄^{n} is called quasimeromorphic if either fG = {∞} or the set E = f^{−1}(∞) is discrete and f_{1} = f|G \ (E ∪ {∞}) is quasiregular. We set K(f) = K(f_{1}), K_{O}(f) = K_{O}(f_{1}), and K_{I}(f) = K_{I}(f_{1}).

For a definition of the modulus of a family of curves we refer to [Vu].

If f: R^{n} → R̄^{n} is nonconstant and qm, the counting function n(r, y) is defined for r > 0, y ∈ R̄^{n}, by

n(r, y) = ∑_{x∈f^{−1}(y)∩B̄(r)} i(x, f),

where i(x, f) is the local topological index; see [MRV1].

A(r) is the average of n(r, y) over R̄^{n} with respect to the spherical metric. If r, t > 0, ν(r, S(a, t)) is the average of the counting function over the sphere S(a, t) with respect to ℋ ^{n−1} ,

\begin{aligned}ν(r, S(a, t)) & = \frac{1}{ω_{n−1}} ∫_{S} n(r, a + ty) dℋ ^{n−1}(y), \\ A(r) & = \frac{2^{n}}{ω_{n}} ∫_{R^{n}} \frac{n(r, y)}{(1 + |y|^{2})^{n}} dy. \end{aligned}

In particular, when S(a, t) = S(t), we set ν(r, S(t)) = ν(r, t), and also ν(r, 1) = ν(r).

Let f: G → R̄^{n} be qm. A domain D such that D̄ ⊂ G is called a normal domain if f∂D = ∂fD. If x ∈ G and U is a normal domain such that U ∩ f^{−1}(f(x)) = {x}, then U is called a normal neighbourhood of x. By [MRV1, 2.10], every point in G has arbitrarily small normal neighbourhoods.

We repeatedly use the following result [R4, p. 228, 2.1]. If θ > 1 and r, s, t > 0, then

(2-2)  ν(θr, t) ≥ ν(r, s) − \frac{K_{I}| log(t/s)|^{n−1}}{(log θ)^{n−1}} .

We also need a comparison between averages on non-concentric spheres, S and S(a, t) ⊂ B(1/2), for t small enough, say t < 1/4. This can be obtained by applying the above result to the map ϕ ∘ f, where ϕ is a quasiconformal map of R̄^{n} onto R̄^{n}, which is the translation x → x − a in B_{t}(a) and it is the identity map outside B(1). ϕ can be taken to be 4-bilipschitz. Thus we get,

(2-3)  ν(r, S(a, t)) ≤ ν(2r) + c_{1}(log(1/t))^{n−1},

where we may take c_{1} = 4^{2n−2}K/(log 2)^{n−1} , since ϕ is 4^{2n−2} -quasiconformal.

3. An upper bound on n(r, a)/A(θr)

Theorem 3-1. Let f: R^{n} → R̄^{n} be a nonconstant K -quasimeromorphic map. Then for each C > 1, there exists θ > 1, θ = θ(C, n, K), such that for every a_{1}, …, a_{q} ∈ R̄^{n}, there exists a set E = E(a_{1}, …, a_{q}) ⊂ [1, ∞), with ∫_{E} dλ/λ = ∞, such that

(3-2)  n(r, a_{j}) ≤ CA(θr)  for j = 1, … q, r ∈ E.

Note that here the role of E is different from that in [R4]. We begin with an adaptation of [R1, 5.4] to the case that a ≠ 0. It is a quantification of the fact that a nonconstant qm map is light.

Lemma 3-3. Let f: R^{n} → R̄^{n} be a nonconstant K -quasimeromorphic map. Choose 1 < u < v, t > 0 and r > 0. Let a ∈ R^{n} be given. Set

(3-4)  H_{a,f}(r, t) = { λ ∈ [r, ur] : S(λ) ∩ f^{−1}(B_{t}(a)^{c}) ≠ ∅ },

ϕ_{a,f}(r, t) = ∫_{H_{a,f} (r,t)} \frac{dλ}{λ}.

Then,

(3-5)  ν(vr, S(a, t)) ≥ [1 − \frac{2ω_{n−1}K_{I}K_{O}}{c_{n}ϕ_{a,f}(r, t)(log v/u)^{n−1}} ]n(r, a)

where c_{n} > 0 is the constant in [V1, 10.11] which depends only on n.

Proof. Using (2-2), we may obtain [R1, 5.5] without the constant c′, as

(3-6)  ν(vr, t) ≥ [1 − \frac{2K_{I}K_{O}ω_{n−1}}{c_{n}ϕ(r, t)(log v/u)^{n−1}} ]n(r, 0).

Let g(z) = f(z) − a. Then ν_{g}(vr, t) ≡ ν_{f}(vr, S(a, t)) and n_{g}(r, 0) = n_{f}(r, a). Let ζ = w−a, so that g(z) = ζ∘f(z), and also S(λ)∩g^{−1}(B(t)^{c}) = S(λ)∩f^{−1}(B_{t}(a)^{c}). Hence H_{0,g}(r, t) = H_{a,f}(r, t) and ϕ_{0,g}(r, t) = ϕ_{a,f}(r, t). Now (3-6) applied to g gives (3-5).

Proof of Theorem 3-1. We divide the proof into three steps. The second step proves the theorem under the normalization a_{1}, …, a_{q} ∈ B(1/2). The first and third steps are merely to facilitate this normalization.

Step I: Let C > 1 be given. Let a_{1}, … a_{q} ∈ R̄^{n}. By a rotation of the sphere we may assume that a_{1}, …, a_{q} ∈ B(τ/2) for some τ ≥ 1. Let σ > 0 be such that the balls {B̄_{στ} (a_{j})} are disjoint and {B̄_{στ} (a_{j})} ⊂ B(τ/2) for all j. We claim that for given r_{0} > 0, there exists r_{1} ≥ r_{0} such that for all r ∈ [r_{1}, u^{1/4}r_{1}],

(3-7)  n(r, a_{j}) ≤ CA(θr)  for j = 1, …, q,

where u > 1 is defined in (3-11). By repeating this argument, we obtain our set E = ∪_{i=1}^{∞}[r_{i}, u^{1/4}r_{i}], so that E has infinite logarithmic measure. We may assume that n(r_{0}, a_{j}) ≥ 1 for all j, since the j ’s for which n(r, a_{j}) = 0 for all r satisfy the claim. Let

(3-8)  C′ = C^{1/4} > 1.

By [R1, 4.10] we choose r_{0} so that for r ≥ r_{0} ,

(3-9)  ν(r) < C′A(2r).

We assume ∞ is an essential singularity (i.e. f has no limit in R̄^{n} as we approach ∞), for otherwise f extends to R̄^{n} as a qm map and it has finite degree [MRV2], [MS]. By [R1, 3.1] we then have that A(r) → ∞. So we may choose r_{0} such that for r ≥ r_{0}

(3-10)  C′^{2}K_{I}(\frac{log τ}{log 2} )^{n−1} + C′c_{1}(log \frac{1}{σ} )^{n−1} < (C′^{4} − C′^{3})A(r).

Step II: In this step we replace f by f/τ and a_{1}, …, a_{q} by a_{1}/τ, …, a_{q}/τ. However, for convenience of notation, we still call them f and a_{1}, …, a_{q}. Note that we are now in the situation a_{1}, …, a_{q} ∈ B(1/2), {B̄_{σ}(a_{j})} disjoint and each B̄_{σ}(a_{j}) ⊂ B(1/2). In order to apply Lemma 3-3 we define u > 1 by

(3-11)  \frac{1}{C′} = 1 − \frac{4ω_{n−1}K_{O}K_{I}}{c_{n}(log u)^{n}}

where c_{n} > 0 is as in [V1, 10.11].

For u > 1, as above and t, r > 0, let ϕ_{j}(r, t) ≡ ϕ_{a_{j},f}(r, t) be as in Lemma 3-3, and let

(3-12)  Ψ(t) = sup_{r≥r_{0}} min_{1≤j≤q} ϕ_{j}(r, t).

Then Ψ is decreasing in t.

Case (i): Ψ(σ) ≥ (7/8) log u.

Then, by the definition of Ψ(σ), there exists r_{1} ≥ r_{0} such that min_{j} ϕ_{j}(r_{1}, σ) ≥ (3/4) log u; i.e.

(3-13)  ϕ_{j}(r_{1}, σ) ≥ (3/4) log u,  1 ≤ j ≤ q.

From the definition of ϕ_{j}(r_{1}, σ), we note that

\begin{aligned}ϕ_{j}(r_{1}, σ) & = ∫_{H_{j}(r_{1},σ)} \frac{dλ}{λ} = ∫_{H_{j}(r_{1},σ)∩[r_{1},u^{1/4}r_{1}]} \frac{dλ}{λ} + ∫_{H_{j}(r_{1},σ)∩[u^{1/4}r_{1},ur_{1}]} \frac{dλ}{λ} \\ & ≤ \frac{1}{4} log u + ∫_{H_{j}(r_{1},σ)∩[u^{1/4}r_{1},ur_{1}]} \frac{dλ}{λ}. \end{aligned}

From this and (3-13) we obtain for r ∈ [r_{1}, u^{1/4}r_{1}] and for all j = 1, …, q,

(3-14)  ϕ_{j}(r, σ) ≥ ∫_{H_{j}(r_{1},σ)∩[u^{1/4}r_{1},ur_{1}]} \frac{dλ}{λ} ≥ \frac{1}{2} log u.

We now apply Lemma 3-3 with a = a_{j}, t = σ, r ∈ [r_{1}, u^{1/4}r_{1}], v = u^{2} along with (3-14) and (3-11) to obtain

(3-15)  \begin{aligned}ν(vr, S(a_{j}, σ)) & ≥ [1 − \frac{2ω_{n−1}K_{I}K_{O}}{c_{n}ϕ_{j}(r, σ)(log u)^{n−1}} ]n(r, a_{j}) \\ & ≥ [1 − \frac{4ω_{n−1}K_{I}K_{O}}{c_{n}(log u)^{n}} ]n(r, a_{j}) \\ & = \frac{1}{C′} n(r, a_{j})  j = 1, …, q. \end{aligned}

Now using (2-3) with t = σ and (3-15), we get for r ∈ [r_{1}, u^{1/4}r_{1}] and j = 1, …, q, that

(3-16)  n(r, a_{j}) ≤ C′ν(vr, S(a_{j}, σ)) ≤ C′ν(2vr) + C′c_{1}(log 1/σ)^{n−1}.

Case (ii): Ψ(σ) < (7/8) log u.

Since f is discrete, for each fixed r, ϕ_{j}(r, t) → log u as t → 0. Let t_{0} = inf{ t : t ≤ σ, Ψ(t) ≤ (7/8) log u }. One checks that t_{0} > 0. We may assume t_{0} < σ. Let δ be so small that

(3-17)  0 < δ < min\{\frac{1}{2}t_{0}, σ − t_{0}\},  \frac{4δ}{t_{0}} < (log 2)(\frac{C′ − 1}{K_{I}C′^{2}} )^{1/(n−1)}

and let

(3-18)  t_{1} = t_{0} − δ,  t′_{1} = t_{0} + δ.

Since Ψ(t_{1}) > \frac{7}{8} log u, there exists r_{1} ≥ r_{0} with min_{j} ϕ_{j}(r_{1}, t_{1}) ≥ \frac{3}{4} log u; i.e.

ϕ_{j}(r_{1}, t_{1}) ≥ \frac{3}{4} log u, j = 1, …, q.

From this we may conclude, exactly as in Case (i), that for r ∈ [r_{1}, u^{1/4}r_{1}],

(3-19)  ϕ_{j}(r, t_{1}) ≥ \frac{1}{2} log u,  j = 1, …, q.

Now we apply Lemma 3-3 with r ∈ [r_{1}, u^{1/4}r_{1}], t = t_{1} , a = a_{j}, v = u^{2} , along with (3-19) and (3-11), to obtain

(3-20)  \begin{aligned}ν(vr, S(a_{j}, t_{1})) & ≥ [1 − \frac{2ω_{n−1}K_{I}K_{O}}{c_{n}ϕ_{j}(r, t_{1})(log u)^{n−1}} ]n(r, a_{j}) \\ & ≥ [1 − \frac{4ω_{n−1}K_{I}K_{O}}{c_{n}(log u)^{n}} ]n(r, a_{j}) \\ & ≥ \frac{1}{C′} n(r, a_{j}),  1 ≤ j ≤ q. \end{aligned}

Let t_{0} < t < t′_{1} . By (3-12), Ψ(t) ≡ sup_{r≥r_{0}} min_{j} ϕ_{j}(r, t) ≤ (7/8) log u, and since 2vr ≥ r ≥ r_{0} , we find for an appropriate 1 ≤ l ≤ q, that ϕ_{l}(2vr, t) ≡ min_{j} ϕ_{j}(2vr, t) ≤ (7/8) log u. Then by the definition of ϕ_{l}(2vr, t) there exists ϱ ∈ [2vr, 2vur] such that S(ϱ) ∩ f^{−1}(B_{t}(a_{l})^{c}) = ∅. The analysis of [MRV1, 2.5], which is stated only for qr maps but applies as well to qm maps, shows that every component of f^{−1}(B_{t}(a_{l})^{c}) which meets B̄(ϱ) is a normal domain contained in B(ϱ). Hence

(3-21)  n(ϱ, y) = n(ϱ, z)  for all y, z ∈ B̄_{t}(a_{l})^{c}.

In particular, since t < t′_{1} < σ and the {B̄(a_{j}, σ)} are disjoint, we have for j ≠ l, n(ϱ, y) = n(ϱ, a_{j} + t_{1}y) for all y ∈ S. And so on averaging,

(3-22)  ν(ϱ) = ν(ϱ, S(a_{j}, t_{1}))  j ≠ l.

For j = l, since t < t′_{1} , we note from (3-21) that n(ϱ, y) = n(ϱ, a_{l} + t′_{1}y) for all y ∈ S. So again on averaging,

(3-23)  ν(ϱ) = ν(ϱ, S(a_{l}, t′_{1})).

We now replace ν(ϱ, S(a_{l}, t′_{1})) by ν(ϱ, S(a_{l}, t_{1})) with controllable error. Letting θ = 2, s = t_{1} , t = t′_{1} , r = vr, we obtain from (2-2) that

(3-24)  ν(vr, S(a_{l}, t_{1})) ≤ ν(2vr, S(a_{l}, t′_{1})) + \frac{K_{I}(log(t′_{1}/t_{1}))^{n−1}}{(log 2)^{n−1}} .

Now we find, using (3-18) and (3-17), that

log \frac{t′_{1}}{t_{1}} = log(1 + \frac{2δ}{t_{0} − δ} ) < \frac{2δ}{t_{0} − δ} < \frac{4δ}{t_{0}} < (log 2)(\frac{C′ − 1}{K_{I}C′^{2}} )^{1/(n−1)}.

Hence, from (3-24),

(3-25)  ν(vr, S(a_{l}, t_{1})) ≤ ν(2vr, S(a_{l}, t′_{1})) + (C′ − 1)/C′^{2}.

Since n(r, a_{l}) ≥ n(r_{0}, a_{l}) ≥ 1 as stated in Step I, we have from (3-20) that ν(vr, S(a_{l}, t_{1})) ≥ 1/C′. Substituting this inequality on the right hand side of (3-25) and unraveling, we obtain,

ν(vr, S(a_{l}, t_{1})) ≤ C′ν(2vr, S(a_{l}, t′_{1})).

But since 2vr ≤ ϱ ≤ 2vur, the last inequality, together with (3-20) and (3-23) gives for r ∈ [r_{1}, u^{1/4}r_{1}],

(3-26)  n(r, a_{l}) ≤ C′^{2}ν(ϱ, S(a_{l}, t′_{1})) = C′^{2}ν(ϱ).

And again using the fact that 2vr ≤ ϱ along with (3-20) and (3-22), we find for j ≠ l, r ∈ [r_{1}, u^{1/4}r_{1}]

(3-27)  n(r, a_{j}) ≤ C′ν(ϱ, S(a_{j}, t_{1})) = C′ν(ϱ).

Using the inequality 2vr ≤ ϱ, we conclude in both cases, from (3-26), (3-27) and (3-16) that, for j = 1, …, q, r ∈ [r_{1}, u^{1/4}r_{1}],

(3-28)  n(r, a_{j}) ≤ C′^{2}ν(ϱ) + C′c_{1}(log 1/σ)^{n−1}.

Finally, we recall the change of scale we made in the beginning of Step II, and conclude from (3-28) that for r ∈ [r_{1}, u^{1/4}r_{1}],

(3-29)  n(r, a_{j}) ≤ C′^{2}ν(ϱ, τ) + C′c_{1}(log 1/σ)^{n−1}

for the original f and a_{1}, …, a_{q}.

Step III: First we use (2-2) to replace ν(ϱ, τ) by ν(2ϱ) in (3-29) and get

n(r, a_{j}) ≤ C′^{2}ν(2ϱ) + C′^{2}K_{I}(\frac{log τ}{log 2} )^{n−1} + C′c_{1}(log 1/σ)^{n−1}.

Using (3-9), (3-10) (3-8) and ϱ ≤ 2uvr we now get for r ∈ [r_{1}, u^{1/4}r_{1}] and j = 1, …, q,

n(r, a_{j}) ≤ C′^{3}A(4ϱ) + (C′^{4} − C′^{3})A(4ϱ) ≤ CA(θr),

where θ = 8uv = 8u^{3} . This proves the theorem.

4. The main result

We first prove an intermediate result, i.e., the estimate (4-2). This is an essential fact needed for the main theorem.

Theorem 4-1. Let n ≥ 2 and K ≥ 1. There exist positive constants θ_{0} = θ_{0}(n, K), b = b(n, K) such that if f: R^{n} → R̄^{n} is a nonconstant K -qm map and a_{1}, …, a_{q} ∈ R̄^{n}, are any distinct points, with q > 1, then there exist r_{0} = r_{0}(a_{1}, …, a_{q}, f) > 0 such that for each r ≥ r_{0} , we have

(4-2)  ∑_{{j=1 \atop j≠J(r)}}^{q} n(r, a_{j}) ≤ [q + \frac{4K_{I}b}{(log 2)^{n−1}} + 2]A(16θ_{0}r),

for some J(r) ∈ {1, …, q}. The constants θ_{0} and b are given by

(4-3)  log θ_{0} = \frac{ω_{n−1}K_{O}c_{1}}{2^{n−4}c_{n}n} ,  b = \frac{2K_{O}ω_{n−1}}{c_{n} log θ_{0}}

with c_{1} and c_{n} as in (2-3) and (3-5) respectively.

Observe that there is no exceptional set for the r-values here. However, the estimate obtained is close to what we want, save for one a_{J(r)} . For this a_{J(r)} we use Theorem 3-1. We thus obtain our main result, Theorem 4-26, on the same exceptional set of r-values as that obtained in Theorem 3-1. It is worth noting that any enlargement or improvement of the set E of Theorem 3-1, is also valid for Theorem 4-26.

Proof of Theorem 4-1. Again we divide the proof into three steps with main body of the proof being in the second step.

Step I: We may assume, as in the proof of Theorem 3-1, that ∞ is an essential singularity, so that A(r) → ∞ as r → ∞. By a rotation we assume that a_{1}, …, a_{q} ∈ R^{n}. Let τ ≥ 1 and σ > 0 be such that B_{στ} (a_{j}) ⊂ B(τ/2), and the {B̄_{στ} (a_{j})} are disjoint. We set r_{0} = max(r_{1}, r_{2}), where r_{1} and r_{2} are obtained below. Choose r_{1} = r_{1}(τ, q, f) > 0 such that for r ≥ r_{1} ,

(4-4)

(i) [q + \frac{K_{I}b}{(log 2)^{n−1}} ]K_{I}(\frac{log τ}{log 2} )^{n−1} ≤ \frac{K_{I}b}{(log 2)^{n−1}} ν(r)

(ii) ν(r) < \frac{q}{q − 1}A(2r) by [R1, 4.10].

Step II: Again by replacing f by f/τ we reduce to the case τ = 1. Since ν(r) → ∞ as r → ∞, we can choose r_{2} = r_{2}(σ, q, f) > 0 such that for r ≥ r_{2} ,

(4-5)

(i) [bν(2θ_{0}r)]^{1/n} + 1 < [2bν(2θ_{0}r)]^{1/n},

(ii) log 2 < (bν(2θ_{0}r))^{1/(n−1)} − (bν(2θ_{0}r))^{1/n},

(iii) \frac{1}{1 + (log(σ/2))/(bν(2θ_{0}r))^{1/(n−1)}} < 2^{1/n},

(iv) 2 exp (−\frac{1}{2}(bν(2θ_{0}r))^{1/n}) < σ,

(v) c_{1}qb < (bν(2θ_{0}r))^{1/n}.

Fix r ≥ r_{2} . Since f is qm , ℋ ^{n}(∂B(θ_{0}r)) = 0 implies ℋ ^{n}(f(∂B(θ_{0}r))) = 0, by [Vu, 10.5(3)]. From this and Fubini’s theorem it follows that ℋ ^{n−1}(f(∂B(θ_{0}r))∩ S(a_{j}, σ_{1})) = 0 for a.e.

σ_{1} ∈ [exp {−(bν(2θ_{0}r))^{1/(n−1)}}, 2 exp {−(bν(2θ_{0}r))^{1/(n−1)}}],

for each j = 1, …, q. Hence there exists ε_{1} ∈ [1, 2] such that for

(4-6)  σ_{1} = ε_{1} exp {−(bν(2θ_{0}r))^{1/(n−1)}}

(4-7)  ℋ ^{n−1}(f(∂B(θ_{0}r)) ∩ S(a_{j}, σ_{1})) = 0  for all j = 1, …, q.

Then by (4-6) and (4-5) (ii) we have

(4-8)  σ_{1} ≤ 2 exp {−(bν(2θ_{0}r))^{1/(n−1)}} < exp {−(bν(2θ_{0}r))^{1/n}} = σ_{2}

and by (4-5) (iv),

σ_{2} = exp {−(bν(2θ_{0}r))^{1/n}} < σ.

Let α_{j} and β_{j} be the maps of S onto S(a_{j}, σ_{1}) and S(a_{j}, σ_{2}) respectively given by α_{j}(y) = a_{j} + σ_{1}y, β_{j}(y) = a_{j} + σ_{2}y.

For y ∈ S, let γ_{y}^{j}: [0, 1] → R^{n} be the line segment joining a_{j} to β_{j}(y), parametrized so that γ_{y}^{j}: [0, 1/2] joins a_{j} to α_{j}(y) ∈ S(a_{j}, σ_{1}), gyj: [1/2, 1] joins α_{j}(y) to β_{j}(y) ∈ S(a_{j}, σ_{2}).

Comparison of n(r, a_{j}) with n(θ_{0}r, α_{j}(y)): Let f | X denote f restricted to X and let Λ_{y}^{j} = {λ_{1}, …, λ_{h}} be a maximal sequence of f | B(4θ_{0}r + 1)-liftings of γ_{y}^{j} | [0, 1/2] starting at points of f^{−1}(a_{j}) ∩ B̄(r), as defined in [R1]. Then necessarily h = n(r, a_{j}). The following crucial lemma has been inspired by the proof of [R2, 3.2].

Lemma 4-9. The family of curves

ℱ_{j} = ⋃_{y∈S}Λ_{y}^{j}

lies completely in B(θ_{0}r), except perhaps for one j = J(r) ∈ {1, …, q}.

Proof. Note that by definition, all paths in ℱ_{j} start at preimages of a_{j} in B̄(r). We prove the lemma by contradiction. Suppose there exist j ≠ k and η_{j} ∈ ℱ_{j}, η_{k} ∈ ℱ_{k}, such that η_{j}, η_{k} ⊈ B(θ_{0}r). Let Γ be the family of paths in B(θ_{0}r) \ B̄(r) joining the loci |η_{j}| and |η_{k}|. Note that |f(η_{j})| and |f(η_{k})| are line segments starting at a_{j} and a_{k} and contained in B̄(a_{j}, σ_{1}) and B̄(a_{k}, σ_{1}) respectively. Hence each path in fΓ contains sub-paths which join S(a_{j}, σ_{1}) to S(a_{j}, σ) and S(a_{k}, σ) to S(a_{k}, σ_{1}). Set

ϱ(z) = \{ \begin{matrix}(2 log(σ/σ_{1})|z − a_{j}|)^{−1}, & σ_{1} < |z − a_{j}| < σ \\ (2 log(σ/σ_{1})|z − a_{k}|)^{−1}, & σ_{1} < |z − a_{k}| < σ \\ 0, & otherwise. \end{matrix}

Then ϱ is well-defined by the choice of σ. Also, ϱ is admissible for the family fΓ, and by [MRV1, 3.2] we obtain

M(Γ) ≤ K_{O} ∫_{R^{n}} ϱ(z)^{n}n(θ_{0}r, z) dℒ^{n}(z)

= \frac{K_{O}}{(2 log(σ/σ_{1}))^{n}} ∫_{\{σ_{1}<|z−a_{j}|<σ\}} n(θ_{0}r, z)|z − a_{j}|^{−n} dℒ^{n}(z)

+ \frac{K_{O}}{(2 log(σ/σ_{1}))^{n}} ∫_{\{σ_{1}<|z−a_{k}|<σ\}} n(θ_{0}r, z)|z − a_{k}|^{−n} dℒ^{n}(z)

= I + II. (4 − 10)

We obtain an estimate for I. Exactly the same estimate holds for II as well. By transfering the integral of (4-10) into polar coordinates, we find that,

\begin{aligned}I & = K_{O}(2 log(σ/σ_{1}))^{−n} ∫_{σ_{1}}^{σ} ∫_{S} n(θ_{0}r, a_{j} + τy) dℋ ^{n−1}(y)τ^{−1} dτ \\ & ≡ K_{O}ω_{n−1}(2 log(σ/σ_{1}))^{−n} ∫_{σ_{1}}^{σ} ν(θ_{0}r, S(a_{j}, τ))τ^{−1} dτ. \end{aligned}

Using (2-3), with θ = θ_{0} ,

(4-11)

I ≤ \frac{K_{O}ω_{n−1}}{(2 log(σ/σ_{1}))^{n}} ∫_{σ_{1}}^{σ}\{ν(2θ_{0}r) + c_{1}(log(1/τ))^{n−1}\}τ^{−1} dτ

≤ \frac{K_{O}ω_{n−1}}{(2 log(σ/σ_{1}))^{n}} [ν(2θ_{0}r) log(σ/σ_{1}) + c_{1} \frac{(log(1/σ_{1}))^{n}}{n} ]

≤ \frac{K_{O}ω_{n−1}}{2^{n}} [\frac{ν(2θ_{0}r)}{(log(σ/σ_{1}))^{n−1}} + \frac{c_{1}}{n} (\frac{log 1/σ_{1}}{log σ/σ_{1}} )^{n}]

Now using (4-6), the fact that ε_{1} ∈ [1, 2], and (4-5) (iii), we find that

(4-12)

\frac{log(1/σ_{1})}{log(σ/σ_{1})} = \frac{log(1/ε_{1}) + (bν(2θ_{0}r))^{1/(n−1)}}{log σ + log(1/ε_{1}) + (bν(2θ_{0}r))^{1/(n−1)}}

≤ \frac{(bν(2θ_{0}r))^{1/(n−1)}}{log σ + log(1/2) + (bν(2θ_{0}r))^{1/(n−1)}}

= \frac{1}{1 + (log σ/2)/(bν(2θ_{0}r))^{1/(n−1)}} ≤ 2^{1/n}.

Also, since ε_{1} < 2, (4-6) and (4-5) (iv) yield that

\frac{σ}{σ_{1}} > \frac{2 exp\{−\frac{1}{2}(bν(2θ_{0}r))^{1/n}\}}{ε_{1} exp\{−(bν(2θ_{0}r))^{1/(n−1)}\}} > exp\{\frac{1}{2}(bν(2θ_{0}r))^{1/(n−1)}\},

and hence

(4-13)  (log \frac{σ}{σ_{1}} )^{n−1} > \frac{bν(2θ_{0}r)}{2^{n−1}}.

Substituting (4-12) and (4-13) into (4-11) we get

I ≤ K_{O}ω_{n−1}2^{−n}[\frac{2^{n−1}}{b} + \frac{2c_{1}}{n} ] ≤ \frac{K_{O}ω_{n−1}}{2b} + \frac{ω_{n−1}K_{O}c_{1}}{2^{n−1}n} .

The same estimate holds for II. Substituting these and the value of b from (4-3) into (4-10) we obtain

M(Γ) ≤ \frac{K_{O}ω_{n−1}}{b} + \frac{ω_{n−1}K_{O}c_{1}}{2^{n−2}n} = \frac{c_{n} log θ_{0}}{2} + \frac{ω_{n−1}K_{O}c_{1}}{2^{n−2}n} .

Further by [V1, (10.12)], M(Γ) ≥ c_{n} log θ_{0} so that

\frac{c_{n} log θ_{0}}{2} ≤ \frac{ω_{n−1}K_{O}c_{1}}{2^{n−2}n} .

But this contradicts our choice of θ_{0} in (4-3). This proves the lemma.

From this lemma, we find that for j ≠ J(r), ℱ_{j} ⊂ B(θ_{0}r). If J(r) does not exist, so that ℱ_{j} ⊂ B(θ_{0}r) for all j, we then set J(r) = q. Fix j ≠ J, and y ∈ S. Then Λ_{y}^{j} = {λ_{1}, …, λ_{h}} ⊂ B(θ_{0}r), and since Λ_{y}^{j} is a maximal sequence of f | B(4θ_{0}r + 1) lifts of γ_{y}^{j} | [0, 1/2] we have , for all j ≠ J, y ∈ S,

(4-14)  h = n(r, a_{j}) ≤ n(θ_{0}r, α_{j}(y)).

Now set

(4-15)  A_{j} = S(a_{j}, σ_{1}) ∩ {f(B_{f} ∩ B̄(8θ_{0}r)) ∪ f(∂B(θ_{0}r))}

where B_{f} is the branch set, i.e. the set of points where f is not a local homeomorphism. From [MR, 3.1] we note that for all j = 1, …, q,

ℋ ^{n−1}(S(a_{j}, σ_{1}) ∩ f(B_{f} ∩ B̄(8θ_{0}r))) = 0.

This along with (4-7) implies that ℋ ^{n−1}(A_{j}) = 0 for all j. Further, we have that ℋ ^{n−1}(α_{j}^{−1}(A_{j})) = 0 for all j.

Set

(4-16)  S′ = S \ [⋃_{j=1}^{q}α_{j}^{−1}(A_{j})].

Comparison of n(θ_{0}r, α_{j}(y)) with n(2θ_{0}r, β_{j}(y)). For any y ∈ S, we redefine Λ_{y}^{j} = {λ_{1}, …, λ_{g}} to be a maximal sequence of f | B(4θ_{0}r + 1)-liftings of γ_{y}^{j} | [1/2, 1], starting at points of f^{−1}(α_{j}(y)) ∩ B̄(θ_{0}r), where g = n(θ_{0}r, α_{j}(y)). Let the set of such sequences be Ω_{y}^{j}. For Λ_{y}^{j} ∈ Ω_{y}^{j} we set

N(Λ_{y}^{j}) = card { ν : |λ_{ν}| ⊂ B̄(2θ_{0}r) }

and define

(4-17)  p_{j}(y) = sup_{Λ_{y}^{j}∈Ω_{y}^{j}} N(Λ_{y}^{j}).

Fix an extremal sequence Λ̂_{y}^{j} ∈ Ω_{y}^{j}; i.e. N(Λ̂_{y}^{j}) = p_{j}(y). Then by the definition of a maximal sequence of f -liftings, we have,

(4-18)  p_{j}(y) ≤ n(2θ_{0}r, β_{j}(y)).

We shall integrate n(θ_{0}r, α_{j}(y)) − p_{j}(y) on S and for this we need the following lemma, which is almost entirely an imitation of [R4, 4.1].

Lemma 4-19. Let S′ and p_{j} be as in (4-16) and (4-17), then p_{j} is upper semi-continuous on S′.

Proof. Let y_{0} ∈ S′, then by (4-16) and (4-15), α_{j}(y_{0}) ∉ f(B_{f} ∩ B̄(8θ_{0}r)) ∪ f(∂B(θ_{0}r)). So if f^{−1}(α_{j}(y_{0})) ∩ B̄(θ_{0}r) = {x_{1}, …, x_{g}}, with g = n(θ_{0}r, α_{j}(y_{0})), then {x_{1}, …, x_{g}} ⊂ B(θ_{0}r). Let y_{1}, y_{2}, … be a sequence in S′ such that y_{h} → y_{0} . The lemma asserts that

lim sup_{h→∞} p_{j}(y_{h}) ≤ p_{j}(y_{0}).

By choosing a subsequence we may assume that for some integer m, p_{j}(y_{h}) ≡ m holds for all h ≥ 1. Also n(θ_{0}r, α_{j}(y)) is upper semi-continuous in y because n(r, y) is. Hence if g_{h} = n(θ_{0}r, α_{j}(y_{h})), then lim sup_{h→∞} g_{h} ≤ g. We choose and fix the following:

(i) Normal neighbourhoods V_{1}, …, V_{g} ⊂ B(θ_{0}r) of the points x_{1}, …, x_{g}, respectively, such that α_{j}(y_{h}) ∈ ⋂_{ν=1}^{g} f(V_{ν}), h ≥ 1. (This then implies f^{−1}(α_{j}(y_{h})) ∩V_{ν} ≠ ∅ for all ν, so that g_{h} ≥ g; i.e. g_{h} = g.)

(ii) For each h ≥ 1 a maximal sequence Λ̂_{y_{h}}^{j} = {λ_{h,1}, …, λ_{h,g}} ∈ Ω_{y_{h}}^{j} such that λ_{h,ν} starts at a point ζ_{h,ν} in f^{−1}(α_{j}(y_{h}))∩V_{ν} for ν = 1, …, g, and |λ_{h,ν}| ⊂ B̄(2θ_{0}r) for ν = 1, …, m (since p_{j}(y_{h}) ≡ m).

We divide the ν ’s, 1 ≤ ν ≤ g, into two groups. First let ν ∈ {1, …, m} be fixed. We claim that the family { λ_{h,ν} : h = 1, 2, …} is equicontinuous on 1/2 ≤ t ≤ 1. Indeed, choose ε > 0. For t ∈ [1/2, 1] there exists δ_{t} > 0 such that U(ξ, f, ϱ) is a normal neighbourhood of ξ with d(U(ξ, f, ϱ)) < ε for each ξ ∈ f^{−1}(γ_{y_{0}}^{j}(t)) ∩ B̄(2θ_{0}r), and

(4-20) B̄(2θ_{0}r)∩f^{−1}(B(γ_{y_{0}}^{j}(t), ϱ)) ⊂ ⋃_{ξ}{U(ξ, f, ϱ) : ξ ∈ f^{−1}(γ_{y_{0}}^{j}(t))∩B̄(2θ_{0}r)}

whenever 0 < ϱ < δ_{t}. We cover γ_{y_{0}}^{j}([1/2, 1]) with a finite number of balls B(γ_{y_{0}}^{j}(t), δ_{t}/2), say B(η_{u}, ϱ_{u}), u = 1, …, v. Again by taking a subsequence of the {y_{h}} we have γ_{y_{h}}^{j}([1/2, 1]) ⊂ ⋃_{u=1}^{v} B(η_{u}, ϱ_{u}), and |α_{j}(y_{h}) − α_{j}(y_{0})| ≤ δ = min_{1≤u≤v}{ϱ_{u}/8}, |β_{j}(y_{h}) − β_{j}(y_{0})| ≤ δ for all h ≥ 1. Fix t ∈ [1/2, 1]. Since γ is continuous there exists u such that for any h ≥ 1

γ_{y_{h}}^{j}(t′) ∈ B(η_{u}, 2ϱ_{u}) for |t′ − t| < δ.

For each such h there exists then ξ ∈ f^{−1}(η_{u}) ∩ B̄(2θ_{0}r) such that, by (4-20)

|λ_{h,ν}(t′)| ⊂ U(ξ, f, 2ϱ_{u}) for |t′ − t| < δ.

And since d(U(ξ, f, 2ϱ_{u})) < ε for all h ≥ 1, the family {λ_{h,ν}}_{h≥1} is equicontinuous. By Ascoli’s theorem we may conclude that {λ_{h,ν}}_{h≥1} converges uniformly to a path λ_{ν}: [1/2, 1] → B̄(2θ_{0}r). The path λ_{ν} is a maximal f | B(4θ_{0}r + 1)-lift of γ_{y_{0}}^{j} | [1/2, 1].

Next fix ν ∈ {m+1, …, g}. Let the end-point of λ_{h,ν}, in B(4θ_{0}r+1), occur at t = t_{h} < 1 and set t_{0} = lim sup_{h→∞} t_{h}. We shall construct a maximal f | B(4θ_{0}r+ 1)-lift λ_{ν} of γ_{y_{0}}^{j} | [1/2, 1] with end-point t_{0} as follows. By taking subsequences of {t_{h}} again, we may assume t_{0} = lim_{h→∞} t_{h}. As above we conclude that the paths λ_{h,ν} ∘ G_{h}, where G_{h} maps [1/2, t_{0}) affinely onto [1/2, t_{h}), converges uniformly on compact subsets of [1/2, t_{0}) to a path λ̃_{ν}: [1/2, t_{0}) → B̄(4θ_{0}r + 1) which is then a lift of γ_{y_{0}}^{j} | [1/2, t_{0}). The path has an extension to a path λ̄_{ν}: [1/2, t_{0}] → B̄(4θ_{0}r + 1), by [MRV3, 3.12]. If Δ ⊂ [1/2, t_{0}] is the largest interval such that 1/2 ∈ Δ and λ̄_{ν}Δ ⊂ B̄(4θ_{0}r + 1), then λ_{ν} = λ̄_{ν} | Δ is maximal f | B(4θ_{0}r + 1)- lift of γ_{y_{0}}^{j} | [1/2, 1], and we have constructed paths λ_{1}, …, λ_{g}, each of which is a maximal lift of γ_{y_{0}}^{j} | [1/2, 1]. Next we will show that Λ_{y_{0}} = {λ_{1}, …, λ_{g}} ∈ Ω_{y_{0}} ; i.e. Λ_{y_{0}} is a maximal sequence of f | B(4θ_{0}r +1)-liftings of γ_{y_{0}}^{j} | [1/2, 1], as defined in [R1]. We need only check that

card { ν : λ_{ν}(t) = x } ≤ i(x, f) for all t and x.

Let A = {ν : λ_{ν}(t) = x} ≠ ∅, and let U(x, f, ϱ) be normal neighbourhood of x. There exists h_{0} such that |λ_{h,ν}| ∩ U ≠ ∅ for all h ≥ h_{0} , ν ∈ A. Let h ≥ h_{0} . We may easily find a point η = γ_{y_{h}}^{j}(t′) in ⋂_{ν∈A}{f(|λ_{h,ν}| ∩ U)}. Let ξ_{1}, …, ξ_{w} be the points in { λ_{h,ν}(t′) : ν ∈ A } ⊂ f^{−1}(η) ∩ U. Since {λ_{h,1}, …, λ_{h,g}} is a maximal sequence, we have for u = 1, …, w,

θ_{u} = card { u : λ_{h,ν}(t′) = ξ_{u} } ≤ i(ξ_{u}, f).

Further, by the choice of η and since U is a normal neighbourhood of x,

card A = ∑_{u=1}^{w} θ_{u} ≤ ∑_{u=1}^{w} i(ξ_{u}, f) ≤ n(U, η) = n(U, x) = i(x, f),

where the last inequality is true because f^{−1}(f(x)) ∩ U = {x}. This proves that Λ_{y_{0}} = {λ_{1}, …, λ_{g}} obtained above is a maximal sequence of f | B(4θ_{0}r + 1)- liftings of γ_{y_{0}}^{j} | [1/2, 1], such that |λ_{ν}| ⊂ B̄(2θ_{0}r) for 1 ≤ ν ≤ m. Thus p_{j}(y_{0}) ≥ N(Λ_{y_{0}}) = m. This proves the lemma.

Set

(4-21)  q_{j}(y) = n(θ_{0}r, α_{j}(y)) − p_{j}(y).

q_{j}, being the difference of two measurable functions, is measurable relative to S′. With Λ̂_{y}^{j} such that p_{j}(y) = N(Λ̂_{y}^{j}), for k = 1, 2, …, let

\begin{aligned}E_{k}^{j} & = \{ y ∈ S′ : q_{j}(y) = k \}, E_{k}^{j}′ = \{ y + a_{j} : y ∈ E_{k}^{j} \} \\ Γ_{k}^{j} & = \{ γ_{y}^{j} | [1/2, 1] : y ∈ E_{k}^{j} \} \\ Δ_{k}^{j} & = \{ λ_{ν} : λ_{ν} ∈ Λ̂_{y}^{j}, y ∈ E_{k}^{j}, |λ_{ν}| ⊈ B̄(2θ_{0}r) \}. \end{aligned}

Then ℋ ^{n−1}(E_{k}^{j}) = ℋ ^{n−1}(E_{k}^{j}′) and by the definition of E_{k}^{j} and the fact that ℋ ^{n−1}(S \ S′) = 0, we have

(4-22)  \begin{aligned}\frac{1}{ω_{n−1}} ∫_{S} q_{j}(y) dℋ ^{n−1}(y) & = \frac{1}{ω_{n−1}} ∑_{k=1}^{∞} kℋ ^{n−1}(E_{k}^{j}) \\ & = \frac{1}{ω_{n−1}} ∑_{k=1}^{∞} kℋ ^{n−1}(E_{k}^{j}′). \end{aligned}

We get ℋ ^{n−1}(E_{k}^{j}′) = (log(σ_{2}/σ_{1}))^{n−1}M(Γ_{k}^{j}) using a standard estimate, [V1, 7.7]. Thus (4-22) becomes

\frac{1}{ω_{n−1}} ∫_{S} q_{j}(y) dℋ ^{n−1}(y) = \frac{1}{ω_{n−1}} ∑_{k=1}^{∞} kM(Γ_{k}^{j})(log(σ_{2}/σ_{1}))^{n−1}

= \frac{1}{ω_{n−1}} (log(σ_{2}/σ_{1}))^{n−1} ∑_{k=1}^{∞} kM(Γ_{k}^{j}).

Further, Väisälä’s inequality [V2, 3.1] gives us kM(Γ_{k}^{j}) ≤ K_{I}M(Δ_{k}^{j}). Also note that since the {Γ_{k}^{j}}_{j,k} are disjoint, so are the {Δ_{k}^{j}}_{j,k}, and by [V1, 6.7],

∑_{{bj=1 \atop j≠J}}^{q} ∑_{k=1}^{∞} M(Δ_{k}^{j}) ≤ M(⋃q_{{j=1 \atop j≠J}} ⋃_{k=1}^{∞}Δ_{k}^{j}).

Using these two estimates, summing over j ≠ J and recalling σ_{1} from (4-6) we get

∑_{j≠J} \frac{1}{ω_{n−1}} ∫_{S} q_{j}(y) dℋ ^{n−1}(y) ≤ \frac{1}{ω_{n−1}} (log(σ_{2}/σ_{1}))^{n−1}K_{I}M(⋃q_{{j=1 \atop j≠J}} ⋃_{k=1}^{∞}Δ_{k}^{j})

≤ \frac{1}{ω_{n−1}} (log(σ_{2}/σ_{1}))^{n−1}K_{I} \frac{ω_{n−1}}{(log 2)^{n−1}} (4-23)

≤ \frac{K_{I}}{(log 2)^{n−1}} bν(2θ_{0}r).

If y ∈ S, then by (4-14), (4-21) and (4-18) we have

(4-24)  n(r, a_{j}) ≤ q_{j}(y) + n(2θ_{0}r, β_{j}(y)).

On integrating over S, and summing over j ≠ J, we obtain using (4-23),

(4-25)  ∑_{j≠J} n(r, a_{j}) ≤ \frac{K_{I}bν(2θ_{0}r)}{(log 2)^{n−1}} + ∑_{j≠J} ν(2θ_{0}r, S(a_{j}, σ_{2})).

But from (2-3) and (4-8)

ν(2θ_{0}r, S(a_{j}, σ_{2})) ≤ ν(4θ_{0}r) + c_{1}(bν(2θ_{0}r))^{1−1/n}.

Using this (4-25) becomes

∑_{j≠J} n(r, a_{j}) ≤ \frac{K_{I}b}{(log 2)^{n−1}} ν(2θ_{0}r) + (q − 1)ν(4θ_{0}r) + (q − 1)c_{1}(bν(2θ_{0}r))^{1−1/n}.

Finally we use (4-5) (v) in the above inequality to get

\begin{aligned}∑_{j≠J} n(r, a_{j}) & ≤ \frac{K_{I}b}{(log 2)^{n−1}} ν(2θ_{0}r) + (q − 1)ν(4θ_{0}r) + ν(2θ_{0}r) \\ & ≤ [q + \frac{K_{I}b}{(log 2)^{n−1}} ]ν(4θ_{0}r). \end{aligned}

In the situation when τ > 1 this gives us

∑_{j≠J} n(r, a_{j}) ≤ [q + \frac{K_{I}b}{(log 2)^{n−1}} ]ν(4θ_{0}r, τ).

Step III: Recall r_{0} = max(r_{1}, r_{2}). Fix r ≥ r_{0} , and use (2-2) to replace ν(4θ_{0}r, τ) by ν(8θ_{0}r) to get,

∑_{j≠J} n(r, a_{j}) ≤ [q + \frac{K_{I}b}{(log 2)^{n−1}} ](ν(8θ_{0}r) + K_{I}(\frac{log τ}{log 2} )^{n−1}).

The inequality (4-4) (i) reduces this to

∑_{j≠J} n(r, a_{j}) ≤ [q + \frac{2K_{I}b}{(log 2)^{n−1}} ]ν(8θ_{0}r),

and by (4-4) (ii) we obtain,

∑_{j≠J} n(r, a_{j}) ≤ [q + \frac{4K_{I}b}{(log 2)^{n−1}} + 2]A(16θ_{0}r).

This proves Theorem 4-1.

Theorem 4-26. For n ≥ 2, and K ≥ 1, let f: R^{n} → R̄^{n} be a nonconstant K -qm function. Then there exist constants C_{1} = C_{1}(n, K) > 1, θ_{1} = θ_{1}(n, K) > 1 such that for every a_{1}, …, a_{q} ∈ R̄^{n}, q > 1, there exists a set E ⊂ [1, ∞) with ∫_{E} dλ/λ = ∞ such that

(4-27)  lim sup_{{r→∞ \atop r∈E}} ∑_{j=1}^{q}[\frac{n(r, a_{j})}{A(θ_{1}r)} − 1]_{+} ≤ C_{1}.

Proof of Theorem 4-26. We first use Theorem 3-1 with some fixed value of C, say C = 2, and obtain a corresponding θ and a set E ⊂ [1, ∞) with ∫_{E} dλ/λ = ∞, such that for j = 1, …, q, r ∈ E,

(4-28)  n(r, a_{j}) ≤ 2A(θr).

We will then show that (4-27) holds with

θ_{1} = max(16θ_{0}, θ), C_{1} = 4 + \frac{4K_{I}b}{(log 2)^{n−1}}

where b has been defined in (4-3). As in Theorem 4-1, we assume that a_{1}, …, a_{q} ∈ B(τ/2) and σ > 0 such that B_{στ} (a_{j}) ⊂ B(τ/2) and B̄_{στ} (a_{j}) are disjoint.

Now apply Theorem 4-1 and obtain r_{0} = r_{0}(σ, τ, q, f) > 0. Fix r ∈ E such that r ≥ r_{0} . If ((n(r, a_{j})/A(θ_{1}r)) − 1) ≤ 0 for (q − 1) values of j, then by (4-28) there is nothing to prove. So let Q = {1 ≤ j ≤ q : ((n(r, a_{j})/A(θ_{1}r)) − 1) > 0} for all j ∈ Q. We assume cardQ = q′ ≥ 2.

Again we apply Theorem 4-1, to the same function f, but using the set {a_{j} : j ∈ Q} = {a′_{j}}. Note that the same σ and τ, as for the {a_{j}}, work for {a′_{j}}. Theorem 4-1 yields r′_{0} = r′_{0}(σ, τ, q′, f). From (4-4) and (4-5) (v) we see that we may choose r′_{0}(σ, τ, q′, f) = r_{0}(σ, τ, q, f); i.e. r′_{0} = r_{0} . So we have for r ∈ E, r ≥ r_{0} = r′_{0} , by (4-2),

∑_{{j∈Q \atop j≠J}} n(r, a_{j}) ≤ [q′ + \frac{4K_{I}b}{(log 2)^{n−1}} + 2]A(16θ_{0}r) ≤ [q′ + \frac{4K_{I}b}{(log 2)^{n−1}} + 2]A(θ_{1}r);

i.e.,

∑_{{j∈Q \atop j≠J}}[\frac{n(r, a_{j})}{A(θ_{1}r)} − 1] ≤ [3 + \frac{4K_{I}b}{(log 2)^{n−1}} ].

For j = J, since r ∈ E, we have from (4-28) that

n(r, a_{J}) ≤ 2A(θr) ≤ 2A(θ_{1}r).

Hence

∑_{j∈Q}[\frac{n(r, a_{j})}{A(θ_{1}r)} − 1] ≤ [4 + \frac{4K_{I}b}{(log 2)^{n−1}} ] = C_{1}.

And by the definition of Q,

∑_{j=1}^{q}[\frac{n(r, a_{j})}{A(θ_{1}r)} − 1]_{+} ≤ C_{1},

where r ∈ E, r ≥ r_{0} . The theorem is proved.

A converse defect relation for quasimeromorphic mappings 79

References

[Mi] Miles, J.: Bounds on the ratio n(r, a)/S(r) for meromorphic functions. - Trans. Amer. Math. Soc. 162, 1971 , 383–393.

[MR] Martio, O., and S. Rickman: Measure properties of the branch set and its image of quasiregular mappings. - Ann. Acad. Sci. Fenn. Ser. A I Math. 541, 1973, 1–16.

[MRV1] Martio,O., S. Rickman, and J. Väisälä: Definitions for quasiregular mappings. - Ann. Acad. Sci. Fenn. Ser. A I Math. 448, 1969, 1–40.

[MRV2] Martio, O., S. Rickman, and J. Väisälä: Distortion and singularities of quasiregular mappings. - Ann. Acad. Sci. Fenn. Ser. A I Math, 465, 1970, 1–13.

[MRV3] Martio, O., S. Rickman, and J. Väisälä: Topological and metric properties of quasiregular mappings. - Ann. Acad. Sci. Fenn. Ser. A I Math. 488, 1971, 1–31.

[MS] Martio, O., and U. Srebro: Periodic quasimeromorphic mappings in R^{n}. - J. Analyse Math. 28, 1975, 20–40.

[R1] Rickman, S.: On the value distribution of quasimeromorphic maps. - Ann. Acad. Sci. Fenn. Ser. A I Math. 2, 1976, 447–466.

[R2] Rickman, S.: On the number of omitted values of entire quasiregular mappings. - J. Analyse Math. 37, 1980, 100–117.

[R3] Rickman, S.: Value distribution of quasiregular mappings. - In Proceedings of the Conference on Value Distribution Theory, Joensuu. Lecture Notes in Math. 981, Springer– Verlag, 1981, 220–245.

[R4] Rickman, S.: A defect relation for quasimeromorphic mappings. - Ann. of Math. 114, 1981, 165–191.

[R5] Rickman, S.: Quasiregular mappings. - Ergeb. Math. Grenzgeb. (to appear).

[R6] Rickman, S.: Defect relation and its realization for quasiregular mappings. - Preprint, year?

[V1] Väisälä, J.: Lectures on n-dimensional quasiconformal mappings. - Lecture Notes in Math. 229, Springer Verlag, 1971.

[V2] Väisälä, J.: Modulus and capacity inequalities for quasiregular mappings. - Ann. Acad. Sci. Fenn. Ser. A I Math. 509, 1972, 1–14.

[Vu] Vuorinen, M.: Conformal geometry and quasiregular mappings. - Lecture Notes in Math. 1319, Springer–Verlag, 1988.

Received 28 April 1993