Vol. 7, No. 2, 2014

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Convex and subharmonic functions on graphs

Matthew J. Burke and Tony L. Perkins

Vol. 7 (2014), No. 2, 227–237

We explore the relationship between convex and subharmonic functions on discrete sets. Our principal concern is to determine the setting in which a convex function is necessarily subharmonic. We initially consider the primary notions of convexity on graphs and show that more structure is needed to establish the desired result. To that end, we consider a notion of convexity defined on lattice-like graphs generated by normed abelian groups. For this class of graphs, we are able to prove that all convex functions are subharmonic.

convex, subharmonic, discrete, graphs
Mathematical Subject Classification 2010
Primary: 26A51
Secondary: 31C20
Received: 1 April 2013
Revised: 21 June 2013
Accepted: 5 July 2013
Published: 16 November 2013

Communicated by Ronald Gould
Matthew J. Burke
Spring Hill College
4000 Dauphin Street
Mobile, AL 36608-1791
United States
Tony L. Perkins
Department of Mathematics
Spring Hill College
4000 Dauphin Street
Mobile, AL 36608-1791
United States