Notes on judgment criteria of convex functions of several variables

Huannan Shi̇,Jian Zhang,Pei Wang,Wei-Shih Du

doi:10.53006/rna.986088

Abstract

By transferring the judgment of convex functions of several variables into the judgment of convex functionsof one variable, the authors discuss the convexity of some convex functions of several variables.

Highlights

The theory of convex analysis plays an important role in almost all branches of mathematics, physics, dynamic systems theory, optimization, and so forth
Convex function theory is an important part of the general topic of convexity with a long history and full of application value
In [12], Wu and Zhu proved the following judgment theorem for convex functions of several variables

Summary

Introduction

The theory of convex analysis plays an important role in almost all branches of mathematics, physics, dynamic systems theory, optimization, and so forth. As for the judgment criteria of convex functions of several variables, we have the following theorems. In [12], Wu and Zhu proved the following judgment theorem for convex functions of several variables. 1.3.1]) Let φ be a continuously dierentiable function on the convex set Ω ⊂ Rn. φ is a convex function on Ω if and only if φ(x) ≥ φ(y) + [∇φ(y)]T (x − y), x, y ∈ Ω, where ∇φ(y) =. The following judgment theorem allows us to convert the convexity of functions of several variables into the convexity of functions of one variable to judge. Theorem 1 is usually applied to judge the convexity of functions of several variables. Such applications often increase the diculty and complexity of proofs. We will give some important examples to demonstrate the conciseness

Applications of Theorem 3

Convexity of arithmetic mean of integral form