## Involve: A Journal of Mathematics

• Involve
• Volume 7, Number 2 (2014), 227-237.

### Convex and subharmonic functions on graphs

#### Abstract

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.

#### Article information

Source
Involve, Volume 7, Number 2 (2014), 227-237.

Dates
Revised: 21 June 2013
Accepted: 5 July 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.involve/1513733659

Digital Object Identifier
doi:10.2140/involve.2014.7.227

Mathematical Reviews number (MathSciNet)
MR3133721

Zentralblatt MATH identifier
1282.05059

Subjects
Primary: 26A51: Convexity, generalizations
Secondary: 31C20: Discrete potential theory and numerical methods

Keywords
convex subharmonic discrete graphs

#### Citation

Burke, Matthew; Perkins, Tony. Convex and subharmonic functions on graphs. Involve 7 (2014), no. 2, 227--237. doi:10.2140/involve.2014.7.227. https://projecteuclid.org/euclid.involve/1513733659

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