Treffer: An Entire Holomorphic Function Associated to an Entire Harmonic Function: An entire holomorphic function associated to an entire harmonic function

Given a harmonic function \(h\) on \(\mathbb{R}^N\), \(N\geq 2\), \((h\in {\mathcal H}_N)\) there is a unique holomorphic function \(f\) on \(\mathbb{C}\), \((f\in{\mathcal E})\) such that \(f(t)=h(t,0,\dots,0)\) for \(t\in\mathbb{R}\). This paper has two purposes. Firstly to obtain theorems on the connection between the growth rates of \(h\) and \(f\) which are more comprehensive and precise than those obtained formerly, and secondly to apply these results to deduce uniqueness theorems for harmonic functions. One of them is given below as theorem 1. Let \(M_2(g,r)= (\int_S |g(rx) |^2 d\sigma(x))^{1/2}\), where \(S\) is the unit sphere in \(R^N\) if \(g=h\in{\mathcal H}\), or the unit circle in \(\mathbb{C}\) if \(g=f\in{\mathcal E}\). The maximum value of \(|g|\) on \(S\) is denoted by \(M_\infty(g,r)\). Let \(\rho(g)\) and \(\tau(g)\) be the order and type of \(g\) respectively. It is said that \(g\) is of growth \((\rho,\tau)\) if \(\rho(g) 0\), then \(M_\infty (f, r)=O(r^{p+(2N-3)/4} e^{\lambda r})\), \((r\to+\infty\). Theorem 1. Suppose that \(h\in{\mathcal H}_N\) and \(h(m,0, \dots,0)=0\) for all \(m\in\mathbb{Z}\). (i) If \(M_2 (h,r)=O(r^p e^{\pi r})\), \((r\to+ \infty)\), for some \(p\geq(3-2N)/4\), then \(h(t,0,\dots, 0)=P(t)\sin (\pi t)\) for all real \(t\), where \(P\) is a polynomial of degree at most \(p+(2N-3)/4\). (ii) If \(M_2(h,r)=o(r^{3-2N)/4}e^{\pi r})\), \((r\to +\infty)\), then \(h(t,0, \dots,0)=0\) for all real \(t\).