Abstract
We will give the growth properties of harmonic functions of order greater than one in a half space, which generalize the result obtained by B. Levin in a half plane.
1. Introduction and Main Theorem
Let and be the sets of all real numbers and of all positive real numbers, respectively. Let denote the -dimensional Euclidean space with points , where and . The boundary and closure of an open set of are denoted by and , respectively. The upper half space is the set , whose boundary is .
For a set , , we denote and by and , respectively. We identify with and with , writing typical points as , , where , and putting
Let denote the open ball with center at the origin and radius in . We use the standard notations and . In the sense of Lebesgue measure and . Let denote -dimensional surface area measure and let denote differentiation along the inward normal into .
The estimate we deal with has a long history which can be traced back to Levin’s estimate of harmonic functions from below (see, e.g., [1, page 209]).
Theorem A. Let be a constant and let, be harmonic in the upper half space and continuous on . Suppose that Then where is a constant independent of , , , and the function .
Further versions and refinements of Theorem 1 may be found in [2, Chapter 1], [3, 4] and in the paper of Krasichkov-Ternovskiǐ [5].
In this paper, we will consider functions harmonic in and continuous on . In what follows we shall denote by various values which do not depend on , , , and the function .
We prove in this note analogous estimates for in .
Theorem 1. Suppose that Then where and is nondecreasing on .
Remark 2. If and , Theorem 1 is just the result of Theorem A.
Theorem 3. If (4) and (5) hold, then where , is a sufficiently large number, and is defined in Theorem 1.
2. Main Lemmas
Carleman’s formula [6] connects the modulus and the zeros of a function analytic in (see, e.g., [7, page 224]). Nevanlinna’s formula (see [1, page 193]) refers to a harmonic function in a half disk. Ren obtained a generalized Nevanlinna-type formula in a half space and Poisson integral forumla for half balls, resepctively, which play important roles in our discussions.
Lemma 4 (see [8]). If , then one has where
Lemma 5 (see [8]). Let and let be a function in and continuous in . Then where , , , and is the volume of the unit -ball in .
3. Proof of Theorem 1
By applying Lemma 4 to , we have
It immediately follows from (4) that
Hence from (11) and (12) we have
And (14) gives
Since , by applying Lemma 5 to , we have where
We remark that
If we put and in (16), then we finally have from (13) and (18) where
We obtain that from (15) and (5), respectively.
From (16), (19), and (21), we have for
For , we have from (5)
Thus the conclusion immediately follows from (22) and (23).
4. Proof of Theorem 3
By modifying (15), we have
Then (21), (22), and (23) are replaced accordingly by the following estimates:
All (16), (19), (25), and (21) give from which the conclusion immediately follows.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors wish to express their appreciation to Dr. Xuna Miao for some very useful conversations related to this problem. This work was supported by the National Natural Science Foundation of China under Grants nos. 11301140 and U1304102.