Wermer originally
introduced his subharmonicity theorem to find analytic structure in certain
polynomial hulls in C2 and later, with Aupetit, applied it to improve a well known
result of Bishop providing analytic structure in the spectra of certain uniform
algebras. We make a simpler application to yield analytic structure in the presence of
an element of constant modulus on certain fibers provided by another element, and to
extend Wermer’s maximality theorem.