LAUSR

In this note, we point out several gaps in the paper “On the lower bound for a class of harmonic functions in the half space” by Zhang, Deng and Kou (Acta Math. Sci. Ser. B Engl. Ed., 32(4), 2012) and give the main results under weaker conditions. The origin of our work lies in Zhang, Deng and Kou [5]. In [5] Lemmas 1 and 2 and therefore also Theorem 1 are erroneous. We give now the correction of these statements. The present notation and terminology in the same as used in [5]. To this end, we start with an auxiliary proposition. Actually, this proposition is a direct corollary of [2, p. 3296], in which harmonic majorization Theorems with respect to a half-space and their applications were introduced. But it plays an important role in our discussions. Proposition 1. Let H be an admissible domain with boundary ∂H in Rn. If u and v are two harmonic functions in H, then we have ∫ ∂H ( u(x) ∂v(x) ∂n − v(x) ∂n ) dσ(x) = 0, where dσ(x) is the surface element of sphere in H and ∂/∂n denotes differentiation along the inward normal into H. We now return to [5, Lemma 1] and give a corrected proof of it. This result does not seem easy to be proved, hence we refer to utilize a slightly different approach. For more details about this procedure we refer to [1], where a different problem is studied by a similar argument. Lemma 1. Let u(x) be a harmonic function in the upper half space R+ and continuous on ∂R+ . Then ∫ {x∈R+:|x|=R} u(x) nxn Rn+1 dσ(x) + ∫ {x∈R+:r<|x |