# Order version of the Hahn-Banach theorem, and envelopes. II. Applications to the function theory

## TL;DR

This paper extends the Hahn-Banach theorem to order structures, providing envelope existence results and applying them to complex function theory, including subharmonic and holomorphic functions.

## Contribution

It introduces an order-based version of the Hahn-Banach theorem and applies it to envelopes and complex function theory, offering new existence criteria and applications.

## Key findings

- Existence conditions for envelopes in convex sets and cones.
- Applications to subharmonic and harmonic minorants in complex analysis.
- Insights into zero sets and representation of meromorphic functions.

## Abstract

Chapter 1 deals with the problem of the existence of an upper/lower envelope from a convex cone or, more generally, a convex set for functions on the projective limit of vector lattices with values in the completion of the Kantorovich space or on the extended real line. Vector, ordinal and topological dual interpretations of the existence conditions of such envelope and the method of its construction are given. Chapter 2 presents applications to the existence of a nontrivial (pluri)subharmonic and/or (pluri) harmonic minorant for functions in domains from finite-dimensional real or complex space. General approaches to the problems of nontriviality of weight classes of holomorphic functions, to the description of zero (sub)sets for such classes of holomorphic functions, to the problem of representation of a meromorphic function as a ratio holomorphic functions from a given weight class are indicated.