Lower Estimates for Certain Harmonic Functions in the Half Space

and Applied Analysis 3 where I 1 (x) = ∫ {y∈H: |Y|=R} R2 − |x| 2


Introduction and Main Theorem
Let R and R + be the sets of all real numbers and of all positive real numbers, respectively.Let R  ( ≥ 3) denote the dimensional Euclidean space with points  = (  ,   ), where   = ( 1 ,  2 , . . .,  −1 ) ∈ R −1 and   ∈ R. The boundary and closure of an open set  of R  are denoted by  and , respectively.The upper half space is the set  = {(  ,   ) ∈ R  :   > 0}, whose boundary is .
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  1 be a constant and let, () be harmonic in the upper half space C + and continuous on C + .Suppose that Then where  2 is a constant independent of  1 , , , and the function ().
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 .

Main Lemmas
Carleman's formula [6] connects the modulus and the zeros of a function analytic in C + (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 Nevanlinnatype formula in a half space and Poisson integral forumla for half balls, resepctively, which play important roles in our discussions.

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 where We obtain that from which the conclusion immediately follows.