First and second order characterizations of pseudolinear functions

This work is concerned with first and second order characterizations of pseudolinear functions. It is shown that for continuously differentiable function f(x), with ∇f(x)ℝ0, pseudolinearity is completely characterized by a special property of the normalized gradient ∇f(x)/||∇f(x)||. Applying some results on the quasi-Hessian matrices the eigenvalues and the eigenvectors of the Hessian ∇²f(x) of a twice continuously differentiable pseudolinear function f(x) is characterized.