LOCAL HOLOMORPHIC CONVEXITY OF A TWO-MANIFOLD IN C,2 175
on B and takes the constant value β on ∂B. Now Re/ takes maximum
and minimum values on the compact set B. If both of these are assumed on
∂B, then they are the same and Re/ = Re β is constant on B. Otherwise
Re/ takes an extreme value on B. In either case, B contains a critical
point of Re/, which proves that OeB. But 0 cannot be a relative minimum
of Re/ because Q has a negative eigenvalue and R(z) = o([z∣2). Similarly,
0 cannot be a relative maximum because Q has a positive eigenvalue.
This contradiction shows that Lfl has no bounded complementary com-
ponents and completes the proof.
This argument shows that M is really convex with respect to rational
functions of the form p(z)q(w), where p is a polynomial and q is rational.
The second proof shows this directly as a first step, and then appeals to a
technique of Wermer [9] to obtain the conclusion. Since this avoids
Mergelyan’s Theorem 1.1, it may have some interest as a means for obtaining
further results of this type, which may be regarded as differential versions
of Theorem 1.1 in which / is permitted to take complex values.
In general, a compact set M is rationally convex exactly when it coincides
with its rational convex hull Mr, which is the intersection of all rationally
convex sets containing M. The rational hull is clearly rationally convex
and the smallest set with this property which contains M. Another description
of Mr is as the set of all points p in C" such that each rational function r
which is holomorphic near M is also holomorphic near p. For such p, each
such function r must also satisfy
(2.1) ∣r(p)∣ ≤ sup∣r∣,
M
because if it did not, then l∕(r — r(p)) would be holomorphic near M but
with a singularity at p.
Proof 2. It will be shown that each point (α, β) in Mr is in fact contained
in M. Let R(M) denote the uniform closure on M of the rational functions
holomorphic on M. Then R(M) is a Banach algebra, and because of (2.1)
the map r→r(a,β) extends by continuity to an algebra homomorphism
λ of R(M) onto C. It is a standard result [2, pp. 31-32] that there is a proba-
bility measure μ on M representing λ in the sense that
(2.2) λ(g) = ʃ gdμ
for each g in R(M). It will be proved that (α,/) is in M by examining the
support of μ.