Wiman and Arima Theorems for Quasiregular Mappings

Abstract

Wiman's theorem says that an entire holomorphic function of order less than 1/2 has a minimum modulus converging to along a sequence. Arima's theorem is a refinement of Wiman's theorem. Here we generalize both results to quasiregular mappings in the manifold setup. The so called fundamental frequency has an important role in this study.

1. Main Results

It follows from the Ahlfors theorem that an entire holomorphic function of order has no more than distinct asymptotic curves where stands for the largest integer . This theorem does not give any information if , This case is covered by two theorems: if an entire holomorphic functionhas orderthen (Wiman [1]) and ifis an entire holomorphic function of orderandis a number satisfying the conditionsthen there exists a sequence of circular arcsalong whichtends touniformly with respect to (Arima [2]).

Below we prove generalizations of these theorems for quasiregular mappings for . The next two theorems are generalizations of the theorems of Wiman and of Arima for quasiregular mappings on manifolds.

Theorem 1.1.

Let be -dimensional noncompact Riemannian manifolds without boundary. Assume that is a special exhaustion function of the manifold and is a nonnegative growth function on the manifold , which is a subsolution of (3.4) with the structure conditions (3.2), (3.3) and the structure constants , , .

Let be a nonconstant quasiregular mapping. Suppose that the manifold is such that

(1.1)

If now

(1.2)

then

(1.3)

Here

(1.4)

is a constant, is the maximal dilatation of , is an -sphere in the manifold , is a fundamental frequency of an open subset , and , where the infimum is taken over all open sets with . (See Sections 4 and 6.)

Theorem 1.2.

Let be -dimensional noncompact Riemannian manifolds without boundary. Assume that is a special exhaustion function of the manifold and is a nonnegative growth function on the manifold , which is a subsolution of (3.4) with the structure conditions (3.2), (3.3) and the structure constants , , .

Let be a quasiregular mapping and . If for some the mapping satisfies the condition

(1.5)

then for each there exists an -sphere and an open set , for which

(1.6)

The proofs of these results are based upon Phragmén-Lindelöf's and Ahlfors' theorems for differential forms of -classes obtained in [3].

For -harmonic functions on abstract cones, similar theorems were obtained in [4].

Our notation is as in [3, 5]. We assume that the results of [3] are known to the reader and we only recall some results on qr-mappings.

2. Quasiregular Mappings

Let and be Riemannian manifolds of dimension . A continuous mapping of the class is called a quasiregular mapping if satisfies

(2.1)

almost everywhere on . Here is the formal derivative of , further, . We denote by the Jacobian of at the point , that is, the determinant of .

The best constant in the inequality (2.1) is called the outer dilatation of and denoted by . If is quasiregular, then the least constant for which we have

(2.2)

almost everywhere on is called the inner dilatation and denoted by . Here

(2.3)

The quantity

(2.4)

is called the maximal dilatation of and if , then the mapping is called -quasiregular.

If is a quasiregular homeomorphism, then the mapping is called quasiconformal. In this case, the inverse mapping is also quasiconformal in the domain and .

Let and be Riemannian manifolds of dimensions and , , and with scalar products , , respectively. The Cartesian product has the natural structure of a Riemannian manifold with the scalar product

(2.5)

We denote by and the natural projections of the manifold onto submanifolds.

If and are volume forms on and , respectively, then the differential form is a volume form on .

Theorem 2.1 (see [5]).

Let be a quasiregular mapping and let . Then the differential form is of the class on with the structure constants , , and .

Remark 2.2.

The structure constants can be chosen to be

(2.6)

where and are, respectively, the greatest and smallest positive roots of the equation

(2.7)

3. Domains of Growth

Let be an unbounded domain and let be a holomorphic function continuous on the closure . The Phragmén-Lindelöf principle [6] traditionally refers to the alternatives of the following type:

( ) if everywhere on , then either grows with a certain rate as or for all ;

( ) if on , then either grows with a certain rate as or for all .

Here the rate of growth of the quantities and depends on the "width" of the domain near infinity.

It is not difficult to prove that these conditions are equivalent with the following conditions:

( ) if on and in , then either grows with a certain rate as or ;

( ) if on and in , then either grows with a certain rate as or .

Let be an unbounded domain in and let be a quasiregular mapping. We assume that . It is natural to consider the Phragmén-Lindelöf alternative under the following assumptions:

(a) and everywhere in ;

(b) and on , ;

(c) on and on .

Several formulations of the Phragmén-Lindelöf theorem under various assumptions can be found in [7–11]. However, these results are mainly of qualitative character. Here we give a new approach to Phragmén-Lindelöf type theorems for quasiregular mappings, based on isoperimetry, that leads to almost sharp results. Our approach can be used to prove Phragmén-Lindelöf type results for quasiregular mappings of Riemannian manifolds.

Let be an -dimensional noncompact Riemannian -manifold with piecewise smooth boundary (possibly empty). A function is called a growth function with as a domain of growth if (i) (ii) if and

We consider a quasiregular mapping , , where is a noncompact Riemannian -manifold, , and . We assume that . In what follows, we mean by the Phragmén-Lindelöf principle an alternative of the form: either the function has a certain rate of growth in or .

By choosing the domain of growth and the growth function in a special way, we can obtain several formulations of Phragmén-Lindelöf theorems for quasiregular mappings. In view of the examples in [7], the best results are obtained if an -harmonic function is chosen as a growth function. In the case (a), the domain of growth is and as the function of growth, it is natural to choose ; in the case (b), the domain is the set , , and ; in the case (c), the domain of growth is and .

In the general case, we shall consider growth functions which are -solutions of elliptic equations [12]. Namely, let be a Riemannian manifold and let

(3.1)

be a mapping defined a.e. on the tangent bundle Suppose that for a.e. the mapping is continuous on the fiber that is, for a.e. , the function is defined and continuous; the mapping is measurable for all measurable vector fields (see [12]).

Suppose that for a.e. and for all , the inequalities

(3.2)

(3.3)

hold with and for some constants . It is clear that we have .

We consider the equation

(3.4)

Solutions to (3.4) are understood in the weak sense, that is, -solutions are -functions satisfying the integral identity

(3.5)

for all with compact support in .

A function in is an -subsolution of (3.4) in if

(3.6)

weakly in , that is,

(3.7)

whenever , is nonnegative with compact support in .

A basic example of such an equation is the -Laplace equation

(3.8)

4. Exhaustion Functions

Below we introduce exhaustion and special exhaustion functions on Riemannian manifolds and give illustrating examples.

4.1. Exhaustion Functions of Boundary Sets

Let , , be a locally Lipschitz function such that

(4.1)

For arbitrary , we denote by

(4.2)

the -balls and -spheres, respectively.

Let be a locally Lipschitz function such that there exists a compact with for a.e. . We say that the function is an exhaustion function for a boundary set of if for an arbitrary sequence of points , the function if and only if .

It is easy to see that this requirement is satisfied if and only if for an arbitrary increasing sequence , the sequence of the open sets is a chain, defining a boundary set . Thus the function exhausts the boundary set in the traditional sense of the word.

The function is called the exhaustion function of the manifold if the following two conditions are satisfied:

(i)for all , the -ball is compact;

(ii)for every sequence with , the sequence of -balls generates an exhaustion of , that is,

(4.3)

Example 4.1.

Let be a Riemannian manifold. We set where is a fixed point. Because almost everywhere on , the function defines an exhaustion function of the manifold .

4.2. Special Exhaustion Functions

Let be a noncompact Riemannian manifold with the boundary (possibly empty). Let satisfy (3.2) and (3.3) and let be an exhaustion function, satisfying the following additional conditions:

( ) there is such that is compact and is a solution of (3.4) in the open set

( ) for a.e. , ,

(4.4)

Here is the element of the dimensional Hausdorff measure on Exhaustion functions with these properties will be called the special exhaustion functions ofwith respect to. In most cases, the mapping will be the Laplace operator (3.8) and, unless otherwise stated, is the -Laplace operator.

Since the unit vector is orthogonal to the -sphere , the condition means that the flux of the vector field through -spheres is constant.

In the following, we consider domains in as manifolds . However, the boundaries of are allowed to be rather irregular. To handle this situation, we introduce -transversality property for .

Let be a -exhaustion function. We say that satisfies the -transversality property if for a.e. , and for every , there exists an open set

(4.5)

with piecewise regular boundary such that

(4.6)

(4.7)

(4.8)

where is the unit inner normal to .

We say that satisfies the -transversality condition if satisfies the -transversality condition for the -Laplace operator . In this case, (4.8) reduces to

(4.9)

Example 4.2.

Let be a bounded domain in and let

(4.10)

be a cylinder with base . The function , , is an exhaustion function for . Since every domain in can be approximated by smooth domains from inside, it is easy to see that for the domain can be used as an approximating domain . Note that the transversality condition (4.8) is automatically satisfied for the -Laplace operator .

Lemma 4.3.

Suppose that an exhaustion function satisfies (3.4) in and that the function is continuously differentiable. If satisfies the -transversality condition, then is a special exhaustion function on the manifold .

Proof.

It suffices to show . Let and . Choose an open set as in the definition of the -transversality condition. for every , and (4.6)–(4.8) together with the Gauss formula imply for a.e.

(4.11)

Since is arbitrary, follows.

Example 4.4.

Fix . Let be an orthonormal system of coordinates in . Let be an unbounded domain with piecewise smooth boundary and let be a -dimensional compact Riemannian manifold with or without boundary. We consider the manifold .

We denote by , , and the points of the corresponding manifolds. Let and be the natural projections of the manifold .

Assume now that the function is a function on the domain satisfying the conditions , , and (3.8). We consider the function .

We have

(4.12)

Because is a special exhaustion function of , we have

(4.13)

Let be an arbitrary point where the boundary has a tangent hyperplane and let be a unit normal vector to .

If , then where the vector is orthogonal to and is a vector from . Thus

(4.14)

because is a special exhaustion function on and satisfies the property on . If , then the vector is orthogonal to and

(4.15)

because the vector is parallel to .

The other requirements for a special exhaustion function for the manifold are easy to verify.

Therefore, the function

(4.16)

is a special exhaustion function on the manifold.

Example 4.5.

We fix an integer , and set

(4.17)

It is easy to see that everywhere in , where . We shall call the set

(4.18)

a -ball and the set

(4.19)

a -sphere in .

We shall say that an unbounded domain is -admissible if for each , the set has compact closure.

It is clear that every unbounded domain is -admissible. In the general case, the domain is -admissible if and only if the function is an exhaustion function of . It is not difficult to see that if a domain is -admissible, then it is -admissible for all .

Fix . Let be a bounded domain in the -plane and let

(4.20)

The domain is -admissible. The -spheres are orthogonal to the boundary and therefore everywhere on the boundary. The function

(4.21)

satisfies (3.4). By Lemma 4.3, the function is a special exhaustion function of the domain . Therefore, the domain has -parabolic type for and -hyperbolic type for .

Example 4.6.

Fix . Let be a bounded domain in the plane with a (piecewise) smooth boundary and let

(4.22)

be the cylinder domain with base

The domain is -admissible. The -spheres are orthogonal to the boundary and therefore everywhere on the boundary, where is as in Example 4.5.

Let where is a function with . We have and since , we obtain

(4.23)

From the equation

(4.24)

we conclude that the function

(4.25)

satisfies(3.8) inand thus it is a special exhaustion function of the domain

Example 4.7.

Let , where , , be the spherical coordinates in . Let , be an arbitrary domain with a piecewise smooth boundary on the unit sphere . We fix and consider the domain

(4.26)

As above it is easy to verify that the given domain is -admissible and the function

(4.27)

is a special exhaustion function of the domain for .

Example 4.8.

Let be a compact Riemannian manifold, with piecewise smooth boundary or without boundary. We consider the Cartesian product , . We denote by , , and the points of the corresponding spaces. It is easy to see that the function

(4.28)

is a special exhaustion function for the manifold . Therefore, for , the given manifold has -parabolic type and for -hyperbolic type.

Example 4.9.

Let , where , , be the spherical coordinates in . Let be an arbitrary domain on the unit sphere . We fix and consider the domain

(4.29)

with the metric

(4.30)

where are -functions on and is an element of length on .

The manifold is a warped Riemannian product. In the cases, , , and the manifold is isometric to a cylinder in . In the cases, , , the manifold is a spherical annulus in .

The volume element in the metric (4.30) is given by the expression

(4.31)

If , then the length of the gradient in takes the form

(4.32)

where is the gradient in the metric of the unit sphere .

For the special exhaustion function , (3.8) reduces to the following form:

(4.33)

Solutions of this equation are the functions

(4.34)

where and are constants.

Because the function satisfies obviously the boundary condition as well as the other conditions of special exhaustion functions listed in (4.2), we see that under the assumption

(4.35)

the function

(4.36)

is a special exhaustion function on the manifold .

Theorem 4.10.

Let be a special exhaustion function of a boundary set of the manifold . Then

(i)if , the set has -parabolic type,

(ii)if , the set has -hyperbolic type.

Proof.

Choose such that . We need to estimate the -capacity of the condenser . We have

(4.37)

where

(4.38)

is a quantity independent of . Indeed, for the variational problem [3, ()], we choose the function , for ,

(4.39)

and for . Using the Kronrod-Federer formula [13, Theorem ], we get

(4.40)

Because the special exhaustion function satisfies (3.8) and the boundary condition , one obtains for arbitrary ,

(4.41)

Thus we have established the inequality

(4.42)

By the conditions, imposed on the special exhaustion function, the function is an extremal in the variational problem [3, ()]. Such an extremal is unique and therefore the preceding inequality holds as an equality. This conclusion proves (4.37).

If , then letting in (4.37) we conclude the parabolicity of the type of . Let . Consider an exhaustion and choose such that the -ball contains the compact set .

Set . Then for , we have

(4.43)

and hence

(4.44)

and the boundary set has -hyperbolic type.

5. Wiman Theorem

Now we will prove Theorem 1.1.

5.1. Fundamental Frequency

Let be an open set. We need further the following quantity:

(5.1)

where the infimum is taken over all functions (U) with (By the definition, is a -function on an open set , if belongs to this class on every component of .). Here is the gradient of on the surface .

In the case , this quantity is well-known and can be interpreted, in particular, as the best constant in the Poincaré inequality. Following [14], we shall call this quantity the fundamental frequency of the rigidly supported membrane .

Observe a useful property of the fundamental frequency.

Lemma 5.1.

Let be an open set and let be the components of , . Then

(5.2)

Proof.

To prove this property, we fix arbitrary functions with . Set for and for . Hence

(5.3)

Summation yields

(5.4)

and we obtain

(5.5)

This gives

(5.6)

The reverse inequality is evident. Indeed, if is a component of , then evidently

(5.7)

and hence

(5.8)

We also need the following statement.

Lemma 5.2.

Under the above assumptions for a.e. , we have

(5.9)

where is the fundamental frequency of the membrane defined by formula (5.1) and

(5.10)

where

(5.11)

For the proof, see Lemma in [10].

We now use these estimates for proving Phragmén-Lindelöf type theorems for the solutions of quasilinear equations on manifolds.

Theorem 5.3.

Let be an exhaustion function. Suppose that the manifold satisfies the condition

(5.12)

Let be a continuous solution of (3.4) with (3.2) and (3.3) on such that

(5.13)

Then either everywhere on or

(5.14)

(5.15)

In particular, if is a special exhaustion function on , then

(5.16)

Here

(5.17)

and where is the constant of Lemma 5.2.

Proof.

We assume that at some point we have . We consider the set

(5.18)

By, [3, Corollary ] the set is noncompact.

The function is an exhaustion function on . Using the relation [3, 6.74] for the function on , we have

(5.19)

where .

By Lemma 5.2, the following inequality holds

(5.20)

Because , it follows that and hence

(5.21)

Thus using the requirement (3.3) for (3.4), we arrive at the estimate

(5.22)

Further we observe that from the condition on , it follows that

(5.23)

From this relation, we arrive at (5.14).

The proof of (5.15) is carried out exactly in the same way by means of the inequality [3, 5.75].

In order to convince ourselves of the validity of (5.16), we observe that by the maximum principle we have

(5.24)

But is a special exhaustion function and therefore by (4.37) we can write

(5.25)

where is a number independent of .

The relation (5.15) implies then that (5.16) holds.

Example 5.4.

Let be a compact Riemannian manifold with nonempty piecewise smooth boundary, , and let , . Choosing as a special exhaustion function of the function , defined in Example 4.8, we have

(5.26)

Then using the fact that , we find

(5.27)

Therefore, on the basis of (5.1) we get

(5.28)

Computation yields

(5.29)

where is an element of -dimensional area on . Therefore,

(5.30)

and we obtain

(5.31)

where the infimum is taken over all functions with

(5.32)

In the particular case , Theorem 5.3 has a particularly simple content. Here is a function of one variable, and is isometric to . Therefore, and by (5.31) we have

(5.33)

In the same way, (5.16) can be written in the form

(5.34)

Let . We do not know of examples where the quantity (5.31) had been exactly computed. Some idea about the rate of growth of the quantity in the Phragmén-Lindelöf alternative can be obtained from the following arguments. Simplifying the numerator of (5.31) by ignoring the second summand, we get the estimate

(5.35)

For each fixed , the function is finite on , because from the definition of the fundamental frequency it follows that

(5.36)

From this we get

(5.37)

Thus

(5.38)

Here is the inverse function of . Because

(5.39)

the relation (5.16) can be written in the form (5.34).

Example 5.5.

Let be an arbitrary domain with nonempty boundary. We consider a warped Riemannian product equipped with the metric (4.30) of the domain . We now analyze Theorem 5.3 in this case.

The function , given by (4.36) under the requirement (4.35), is a special exhaustion function on . We compute the quantity as follows:

(5.40)

Therefore, observing that

(5.41)

we have

(5.42)

Thus

(5.43)

Further we get

(5.44)

Thus the relation (5.16) attains the form

(5.45)

5.2. Proof of Theorem 1.1

We assume that

(5.46)

Consider the set

(5.47)

It is clear that for a suitable choice of , the set is not empty.

By assumptions, the function satisfies (3.4) with (3.2), (3.3) and structure constants , , . Since is quasiregular, by Lemma of [12] the function is a subsolution of another equation of the form (3.4) with structure constants , , where and are outer and inner dilatations of . In view of the maximum principle for subsolutions, the set does not have relatively compact components. Without restricting generality, we may assume that is connected. Because for sufficiently large , the condition

(5.48)

holds; we see that

(5.49)

Therefore, the condition (1.1) on the manifold implies the following property:

(5.50)

Observing that

(5.51)

we see that by (1.2)

(5.52)

with the constant of Theorem 1.1.

It is easy to see that . Using (5.16) with for the function in the domain , we see that on . This contradicts with the definition of the domain .

Example 5.6.

As the first corollary, we shall now prove a generalization of Wiman's theorem for the case of quasiregular mappings where is a warped Riemannian product.

For , let

(5.53)

be a ring domain in and let be an -dimensional Riemannian manifold on with the metric

(5.54)

where are continuously differentiable on and is an element of length on .

As we have proved in Example 4.9, under condition (4.35), the function

(5.55)

is a special exhaustion function on .

Let be a quasiregular mapping. We set . This function is a subsolution of (3.4) with and also satisfies all the other requirements imposed on a growth function.

We find

(5.56)

and further

(5.57)

Therefore, the requirement (1.1) on the manifold will be fulfilled if

(5.58)

holds.

Because

(5.59)

we see that, in view of (1.2), it suffices that

(5.60)

In this way, we get the following corollary.

Corollary 5.7.

Let be a nonconstant quasiregular mapping from the warped Riemannian product and a special exhaustion function of . If the manifold has property (5.58) and the mapping has property (5.60), then

(5.61)

Example 5.8.

Suppose that under the assumptions of Example 5.6, we have (in addition) , , and the functions , that is, with the metric is an -dimensional half-cylinder. As the special exhaustion function of the manifold , we can take . The condition (5.58) is obviously fulfilled for the manifold.

The condition (5.60) for the mapping attains the form

(5.62)

Corollary 5.9.

If is a half-cylinder and is a nonconstant quasiregular mapping satisfying (5.62), then

(5.63)

We assume that in Example 5.8 the quantities , , and the functions , , that is, the manifold is . As the special exhaustion function, we choose . This function satisfies (3.6) with and . The condition (5.58) for the manifold is obviously fulfilled.

The condition (5.62) attains the form

(5.64)

where

(5.65)

We have the following corollary.

Corollary 5.10.

Let be a nonconstant quasiregular mapping satisfying (5.64). Then

(5.66)

6. Asymptotic Tracts and Their Sizes

Wiman's theorem for the quasiregular mappings asserts the existence of a sequence of spheres , , along which the mapping tends to . It is possible to further strengthen the theorem and to specify the sizes of the sets along which such a convergence takes place. For the formulation of this result it is convenient to use the language of asymptotic tracts discussed by MacLane [15].

6.1. Tracts

Let be a domain in the complex plane and let be a holomorphic function on . A collection of domains is called an asymptotic tract of if

( )each of the sets is a component of the set

(6.1)

( )for all , we have and .

Two asymptotic tracts and are considered to be different if for some we have .

Below we shall extend this notion to quasiregular mappings of Riemannian manifolds. We study the existence of an asymptotic tract and its size.

Let be -dimensional connected noncompact Riemannian manifolds and let be a growth function on , which is a positive subsolution of (3.4) with structure constants , , .

A family is called an asymptotic tract of a quasiregular mapping if

( )each of the sets is a component of the set

(6.2)

( )for all , we have and .

Let be a quasiregular mapping having a point as a Picard exceptional value, that is, and attains on all values of for some .

The set has -capacity zero in and there is a solution in of (3.4) such that as or (cf. [12, Chapter 10, polar sets] ). As the growth function on , we choose the function . It is clear that this function is a subsolution of (3.4) in .

The function also is a subsolution of an equation of the form (3.4) on . Because the mapping attains all values in the punctured ball , then among the components of the set

(6.3)

there exists at least one having a nonempty intersection with . Then by the maximum principle for subsolutions, such a component cannot be relatively compact.

Letting , we find an asymptotic tract , along which a quasiregular mapping tends to a Picard exceptional value .

Because one can find in every asymptotic tract a curve along which , we obtain the following generalization of Iversen's theorem [16].

Theorem 6.1.

Every Picard exceptional value of a quasiregular mapping is an asymptotic value.

The classical form of Iversen's theorem asserts that if is an entire holomorphic function of the plane, then there exists a curve tending to infinity such that

(6.4)

We prove a generalization of this theorem for quasiregular mappings of Riemannian manifolds.

The following result holds.

Theorem 6.2.

Let be a nonconstant quasiregular mapping between -dimensional noncompact Riemannian manifolds without boundaries. If there exists a growth function on which is a positive subsolution of (3.4) with and on a special exhaustion function, then the mapping has at least one asymptotic tract and, in particular, at least one curve on along which .

Proof.

Let be a special exhaustion function of the manifold . Set

(6.5)

If , then tends uniformly on to for . The asymptotic tract generates mutual inclusion of the components of the set .

Let . For an arbitrary , we consider the set

(6.6)

Because is a subsolution, the nonempty set does not have relatively compact components. By a standard argument, we choose for each , as , a component of the set having property (b) of the definition of an asymptotic tract. We now easily complete the proof for the theorem.

6.2. Proof of Theorem 1.2

We fix a growth function and a special exhaustion function as in Section 4. Let be a nonconstant quasiregular mapping. We set

(6.7)

Let be the quantity defined in (6.5). The case is degenerate and has no interest in the present case.

Suppose now that . For , we consider the set , defined in the proof of the preceding theorem. Define

(6.8)

Because is a subsolution of an equation of the form (3.4) on by [3, Theorem ], we have for an arbitrary

(6.9)

Using the inequality () of [10] for the quantity , we get

(6.10)

where

(6.11)

By [3, 5.71], we have

(6.12)

where

(6.13)

But is a special exhaustion function and as in the proof of (4.37) we get

(6.14)

for all sufficiently large . Hence

(6.15)

and further

(6.16)

where and is defined in Lemma 5.2.

Under these circumstances, from the condition (1.5) for the growth of , it follows that for all and for all sufficiently large , we have

(6.17)

If we assume that for all

(6.18)

then because was arbitrary, it would follow from (6.17) that on which is impossible.

Hence there exists for which

(6.19)

Letting , we see that . Using each time the relation (6.17), we get Theorem 1.2.

In the formulation of the theorem, we used only a part of the information about the sizes of the sets which is contained in (6.17). In particular, the relation (6.17) to some extent characterizes also the linear measure of those for which the intersection of the sets with the -spheres is not too narrow.

We consider the case of warped Riemannian product with the metric described in Example 5.6. Let be a special exhaustion function of the manifold of the type (4.36) with , satisfying condition (4.35).

Here, as in Example 5.6,

(6.20)

where and is the image of the set under the similarity mapping

(6.21)

of .

Let be a nonconstant quasiregular mapping. We choose as a growth function the function . This function satisfies (3.6) with and The condition (1.5) can be written as follows:

(6.22)

Hence we obtain

Corollary 6.3.

If a quasiregular mapping has the property (6.22) for some , then for each there are spheres , , , and open sets for which

(6.23)

where as above

(6.24)

Corresponding estimates of the quantities and were given in [7] in terms of the -dimensional surface area and in terms of the best constant in the embedding theorem of the Sobolev space into the space on open subsets of the sphere. This last constant can be estimated without difficulties in terms of the maximal radius of balls contained in the given subset.