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 \(\rho(g) <\rho\) or \(\rho(g)=\rho\), \(\tau (g)\leq\tau\). The values of \(\rho(g)\) and \(\tau(g)\) are unaffected if \(M_\infty (g,r)\) is replaced by \(M_2(g,r)\) in their definitions. Proposition 1: If \(h\) is of growth \((\rho,\tau)\), then so also is \(f\). For certain class of functions of order l the following result is proved. Proposition 2: If \(M_2(h,r)=O(r^p e^{\lambda r})\), \((r\to+ \infty)\), \(p\in\mathbb{R}\), \(\lambda>0\), 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 \(t\).
scientific article; zbMATH DE number 1346406
scientific article