Narrow Results

We provide some conditions for the graph of a Hölder-continuous function on , where is a closed disk in ℂ, to be polynomially convex. Almost all sufficient conditions known to date — provided the function (say F) is smooth — arise from versions of the Weierstrass Approximation Theorem on . These conditions often fail to yield any conclusion if rank_ℝDF is not maximal on a sufficiently large subset of . We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also allows us to make some observations on the polynomial hull of a graph in ℂ² at an isolated complex tangency.