#### Abstract

After considering a variant of the generalized mean value inequality of quasinearly subharmonic functions, we consider certain invariance properties of quasinearly subharmonic functions. KojiΔ has shown that in the plane case both the class of quasinearly subharmonic functions and the class of regularly oscillating functions are invariant under conformal mappings. We give partial generalizations to her results by showing that in , , these both classes are invariant under bi-Lipschitz mappings.

#### 1. Introduction

*Notation*. Our notation is rather standard; see, for example, [1β3] and the references therein. We recall here only the following. The Lebesgue measure in , , is denoted by . We write for the ball in , with center and radius . Recall that , where . If is an open set in , and , then we write for the distance between the point and the boundary of . Our constants are nonnegative, mostly , and may vary from line to line.

##### 1.1. Subharmonic Functions and Generalizations

Let be an open set in , . Let be a Lebesgue measurable function. We adopt the following definitions.

(i) is*subharmonic*if is upper semicontinuous and if for all balls . A subharmonic function may be on any component of ; see [3, page 9] and [4, page 60]. (ii) is

*nearly subharmonic*if and for all balls . Observe that this definition, see [5, page 51], is slightly more general than the standard one [3, page 14]. (iii)Let . Then is

*-quasinearly subharmonic*if and for all and for all balls . Here .

The function is *quasinearly subharmonic* if is -quasinearly subharmonic for some . For the definition and properties of quasinearly subharmonic functions, see, for example, [1, 4β7] and the references therein.

Proposition 1.1 (cf. [5, Proposition , pages 54-55]). *The following holds. *(i)*A subharmonic function is nearly subharmonic but not conversely. *(ii)*A function is nearly subharmonic if and only if it is -quasinearly subharmonic. *(iii)*A nearly subharmonic function is quasinearly subharmonic but not conversely. *(iv)*If is Lebesgue measurable, then is -quasinearly subharmonic if and only if and
for all balls . *

##### 1.2. Bi-Lipschitz Mappings

Let be an open set in , . Let be arbitrary. A function is -*bi-Lipschitz* if

for all . A function is *bi-Lipschitz* if it is -bi-Lipschitz for some . It is easy to see that if is -bi-Lipschitz, then also is -bi-Lipschitz, where .

Let be an open subset of . Let and . We write

#### 2. On the Generalized Mean Value Inequality

Lemma 2.1. *Let be a bounded open set in , . Fix a point . Let be a domain in . Let be a -quasinearly subharmonic function. Then there is such that
**
for every point and all -BiLip, .*

*Proof. *Take and -BiLip, , arbitrarily. (Observe that the set of bi-Lipschitz mappings is (in general) nonempty.) Write
Using the fact that is a homeomorphism, one sees easily that . Since is -bi-Lipschitz, it follows from a result of RadΓ³-Reichelderfer, see, for example, [8, Theorem , page 99], that . (Observe that bi-Lipschitz mappings satisfy the property and are differentiable almost everywhere, see, for example, [9, Theorem , page 112, Theorem , page 109].) Therefore,
Thus (2.1) holds with .

Theorem 2.2. *Let be an open set in , , with . Fix a point . Let be an open set in . Let be a -quasinearly subharmonic function. Then there is a constant such that (2.1) holds for every point and all -BiLip, .*

*Proof. *Let be arbitrary. It is easy to see that for some . Write and . One sees easily that satisfies the assumptions of Lemma 2.1; that is, is a bounded domain, and . Hence there is a constant such that
for every point and all -BiLip. Since and are -bi-Lipschitz, it follows that and ; see again [8, Theorem , page 99]. Thus for ,
Proceed then as follows:
Therefore
concluding the proof.

#### 3. An Invariance of the Class of Quasinearly Subharmonic Functions

Suppose that and are open sets in the complex plane . If is analytic and is subharmonic, then is subharmonic; see, for example, [3, page 37] and [4, Corollary , page 70]. Using Koebeβs one-quarter and distortion theorems, KojiΔ proved the following partial generalization.

Theorem 3.1 (see [6, Theorem , page 245]). * Let and be open sets in . Let be a -quasinearly subharmonic function. If is conformal, then the composition mapping is -quasinearly subharmonic for some .*

For the definition and properties of conformal mappings, see, for example, [9, pages 13β15] and [8, pages 171-172].

Below we give a partial generalization to KojiΔβs result. Our result gives also a partial generalization to the standard result according to which in , , the class of subharmonic functions is invariant under orthogonal transformations; see [10, page 55].

Theorem 3.2. *Let and be open sets in . Let be a -quasinearly subharmonic function. If is -bi-Lipschitz, then the composition mapping is -quasinearly subharmonic for some . *

*Proof. *It is sufficient to show that there exists a constant such that
for all . To see this, observe first that
where .

Then
Above we have used the routineous fact that for -bi-Lipschitz mappings,
and the already cited change of variable result of RadΓ³-Reichelderfer; see [8, Theorem , page 99]. (Recall again that bi-Lipschitz mappings satisfy the property and are differentiable almost everywhere.)

#### 4. An Invariance of Regularly Oscillating Functions

Let be an open set in , . Let be continuous. Write

The function is a Borel function in . If is differentiable at , then ; see [9, page 11], [11, page 19], and [12, page 93].

A function is *regularly oscillating*, if there is such that

The class of such functions is denoted by OC. The class of all regularly oscillating functions is denoted by RO; see [11, page 19], [13, page 17], [14], [6, page 245], and [12, page 96].

Using again Koebeβs results, KojiΔ proved also the following result.

Theorem 4.1 (see [6, Theorem , page 245]). *Let and be open sets in . Let OC. If is conformal, then OC, where depends only on .*

Below we give a partial generalization to KojiΔβs above result.

Theorem 4.2. *Let and be open sets in . Let OC. If is -bi-Lipschitz, , then OC.*

*Proof. *Let be -bi-Lipschitz. Take and arbitrarily such that . Write and for . Then
Using (3.2) (for ), we get

On the other hand, since is -bi-Lipschitz,
Therefore,
Thus OC.

In addition of regularly oscillating functions, one sometimes considers so-called HC functions, too; see [11, page 19], [13, page 16], and [12, page 93]. Their definition reads as follows. Let be an open set in , . Let . A function is in HC if

The class HC is the union of all HC, . Clearly, HCOC.

Proceeding as above in the proof of Theorem 4.2 one gets the following result.

Theorem 4.3. *Let and be open sets in . Let HC. If is -bi-Lipschitz, , then HC.*

#### Acknowledgment

The first author was partially supported by the Academy of Finland.