On the geometric means of entire functions of several complex variables

Abstract

Let f(z, z.,) be an entire function of the n (? 2) complex variables Z1 ... . Zn, holomorphic for Izt I rt, t= 1 ... , n. We have considered the case of only two complex variables for simplicity. Recently many authors have defined the arithmetic means of the function lf(zl, z2)I and have investigated their properties. In the present paper, the geometric means of the function lf(zl, Z2)t have been defined and the asymptotic behavior of certain growth indicators for entire functions of several complex variables have been studied and the results are given in the form of theorems. 1. Let f(zlg .. * Zn) = ki.. knZll* *nt kl ,...,kn> be an entire function of the n (> 2) complex variables z1, ..., ,Zn, holomorphic for $ztl rt, t= 1, .. ., n. Let us denote the maximum modulus of the function f(ZlX* .. X Zn) as M(rl, ... rn) = max if(z1, ..., zZn) (t= 1, .. , n). lztl <rt Here we consider the case of only two complex variables for simplicity. The results can easily be extended to several complex variables. The geometric mean of If(z1, z2)1 for Iztj ? rt (t= 1, 2) has been defined as [4] 1 rzz rzz (1.1) G(r1, r2) = exp {(2_)2 J J log lf(rlei0i, r2ei02)1 dO1 d02}. Further, let us define 4 (k+ 1) 2 r1 r2 loG(, dx xi (1.2) gk(rl, r2) = exp ( )k J (xx2)k log G(xl 1 2 where k is any positive number. Received by the editors September 20, 1968 and, in revised form, March 25, 1970. AMS 1969 Subject Classifications. Primary 3205, 3210; Secondary 3217.