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Let D be a classical Cartan domain and let B be the unit ball. We find the exact value of the supremum of the set of positive numbers p satisfying the condition: pB C f( D) for a certain holomorphic imbeddingf: D B. 1. This paper is concerned with the following problem: Let D be a bounded homogeneous domain in C". By S(D) we denote the family of holomorphic imbeddingsf of D into the unit ball Bn in C' such that f(D) 3 0. For each f E S (D) we define pf sup{ p > 0: pB C f(D)} where pB. = (pz: z E B"J. Further we define p(D) = sup{pf: f E S(D)}. Obviously p(D) is invariant under biholomorphic mappings. It is required to find the exact value of p(D). Alexander [1] proved that for the unit polydisc U in C', p(U) = n-',". In this paper we note that by appealing to the method of Alexander we are able to find the values p(D) for the classical Cartan domains. By a classical Cartan domain we understand a domain of one of the following four types: RI(r, s) = {Z = (Zjk): IZZ' > 0, where Z is an r X s matrix) (r 0, where Z is a symmetric matrix of order p), RII,(q) = {Z = (Zk): I -ZZ' > 0, where Z is a skew-symmetric matrix of order Rlv(n) = {z = (zl,...,z ): 1 + I zz, 12-2zi' >0 1Izz' > 0. Here I is the identity matrix and Z denotes the conjugate matrix of Z and Z' the transposed matrix of Z. The complex dimensions of these four domains are rs, p(p + 1)/2, q(q 1)/2, n, respectively. We shall prove the following theorems. Received by the editors April 16, 1980. 1980 Mathematics Subject Classification. Primary 32H99. Kei' words and phrases. Classical Cartan domain, holomorphic imbedding, bounded homogeneous domain, biholomorphic mapping. ?j 1982 American Mathematical Society 0002-9939/82/00001096/$02.00

A note on holomorphic imbeddings of the classical Cartan domains into the unit ball