Short proofs of three theorems on harmonic functions

We present elementary proofs-shorter than any others that we know for three related theorems. THEOREM I. A function u that is harmonic and positive in the upper half-space {x E R': x?, > O} and zero on the boundary hyperplane must be of the form ax,. When n = 2, for instance, this implies that an entire function which maps the upper half-plane into itself and is real on the real axis is an affine function az + b. Many proofs have been given, for instance [1-7]. Our proof is similar to the one given in [4] for the planar case. THEOREM II. A function u that is harmonic in Rn and bounded from one side by a polynomial must be a polynomial, and of no higher degree. This is a strong form of Liouville's theorem. THEOREM III. A function f, meromorphic in the whole complex plane, real on the real axis, with Im f (z) > 0 when Im z > 0, has the form f(z) = bo + b1z + C +E(Ak Ak) z z -ak ak where bo E R, b1 > 0, c < 0, ak E R \ {0}, Ak < 0, and the series converges uniformly on every compact set that avoids the poles. This is a theorem of Chebotarev and Mefman [1, p. 197]. PROOF OF THEOREM I. By the reflection principle we may assume that u is harmonic in the whole space R', and that u is an odd function of x,. Write 00 00 (1) u(x) =E u (x) = L r'u. (w) ^X=1 ,7=1 where x = rw, r = Ixi, and each u7 is a homogeneous polynomial of degree j. The u, inherit from u the properties of harmonicity and anti-symmetry in the variable x,. In particular ul(x) = axn, and the function lu3(x)/xnl extends continuously to the unit sphere and has some upper bound c3 there. Multiply (1) by g(w) = Ckwn Uk (w) and integrate over the unit sphere. Since spherical harmonics of Received by the editors June 8, 1987. 1980 Mathematics Subject (Cassificatzon (1985 Revzssion). Primary 31B05, Secondary 30D30.