International Journal of Mathematics and Mathematical Sciences

1. Introduction

Notation. Our notation is rather standard; see, for example, [1–3] 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.

The function is quasinearly subharmonic if is -quasinearly subharmonic for some . For the definition and properties of quasinearly subharmonic functions, see, for example, [1, 4–7] 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 Radó-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 Koebe’s one-quarter and distortion theorems, Kojić 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 13–15] and [8, pages 171-172].

Below we give a partial generalization to Kojić’s 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 Radó-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 Koebe’s results, Kojić 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 Kojić’s 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.