Vol. 26, No. 1, 1968

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A maximum principle and geometric properties of level sets

Thomas Allan Cootz

Vol. 26 (1968), No. 1, 39–46

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.

Mathematical Subject Classification 2000
Primary: 31A05
Secondary: 30A44
Received: 21 March 1967
Published: 1 July 1968
Thomas Allan Cootz