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We give an example of a separately subharmonic function which is not subharmonic. It has been known since Hartogs, cf. [3], that a function / defined on a domain in Cn which is separately holomorphic, i.e. / is holomorphic in each variable separately while the others remain fixed, is in fact a holomorphic function of several variables. Subsequently, by work of Lelong, see [5], and Avanissian, [2] a similar result was obtained for separately subharmonic functions which in addition are locally bounded from above. Lelong also showed that separately harmonic functions are harmonic and gave conditions for separately real-analytic functions to be real-analytic, see [6]. Siciak improved on these results, see [7]. Arsove showed in [1] that in Lelong's theorem local boundedness may be replaced by possessing an integrable majorant. The question whether separate subharmonicity in itself is enough for subharmonicity, has been around at least since 1966, cf. [1, 4]. In this note we give an example showing that the answer is no. The author thanks Professor S. Krantz for bringing this question to his attention. THEOREM. There exists a nonnegative function f on C2 with the following properties. 1. For every zq, u>o G C, f(zo, •) and f(-,wo) are continuous subharmonic functions on C. 2. The function f is not subharmonic on C2. PROOF. Let an = (l/n)e¿/("+1), n = 1,2,.... Put K'n = {zGC:\z\< n, 1/n < arg z < 2ir} U {0}, Kn=K'nU{an}, n = l,2,.... By Runge's theorem the holomorphic function fn(z) := 0 on a small neighborhood of K'n, fn(z) := n + 1 on a small neighborhood of an, can be uniformly approximated by polynomials on Kn. Hence there exist polynomials Pn such that |Pn| < 1/2 on K'n, Pn(an) = n + 1. Now let hn(z) — max{|P„(z)| — 1,0}, a continuous subharmonic function on C, which equals 0 on K-_ Received by the editors October 15, 1987 and, in revised form, March 29, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 31C05. Research supported by a fellowship of the Royal Netherlands Academy of Arts and Sciences. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

Separately subharmonic functions need not be subharmonic