There are many results in
function theory which relate the behavior of a function in the interior of a
domain to its behavior on the boundary. A well known result of this sort is the
theorem of study: if the map of the unit disc under a univalent analytic
function f(z) is convex, then the map of every concentric disc contained
therein is also convex. This theorem has been generalized in many different
directions including more general properties of univalent functions, and the
convex and star-shaped properties for level surfaces of harmonic functions in
E3. The results for univalent functions depend basically upon Schwarz’s
lemma, while the results for level surfaces of harmonic functions have been
shown previously by means of rather complicated forms of the maximum
principle.
In §1, we give a simple and direct proof of a very general theorem, depending
upon a form of the maximum principle, which is then shown in §2 to easily give the
known results as well as several new ones. Some related new problems are discussed
in §3.
