Some generalizations of the problem of positive definiteness of a piecewise linear function

In this paper, we consider the following problem concerning functions that are positive definite on a real linear space E (the set Φ(E)). Let f,g∈Φ(E) be such that f(0)=g(0) and f≢g. We are looking for the set Pf,g:={λ∈C:fλ∈Φ(E)}, where fλ(x):=λf(x)+(1−λ)g(x), λ∈C.