Explicit proof of an inequality related to the Omega-matrix

Abstract

The authors recently published a paper on some properties of the solution curves for the last n-1 equations of F(x)+Ax=b where x=[x₁,x₂,...,x_n]^T is a variable, F(x):Rⁿ → Rⁿ is a nonlinear function of which the first and second derivatives are strictly positive, A ∈ R^{n × n} is an Ω-matrix, and b ∈ Rⁿ is a constant vector. In that paper, the authors showed that any solution curve possesses neither maximal points nor inflection points with respect to x₁, by making use of a fundamental property of Ω-matrices, which is expressed in the form of inequality. However, the proof was a little unclear as shown in Introduction. The objective of this paper is to give an explicit proof for the property of Ω-matrices, which makes the author's previous result more rigorous.