A Question concerning Positive Type Polynomials

1. The problem. In [l], the following problem is posed: Let 0>0, a„ = l. Define 0°+ = (Po+ = U"s,1 (Pj). Further, let (Pi+ (respectively, (Pt; (Pt) denote the class of polynomials, all of whose roots are real and negative (respectively, are equal and negative; have negative real part). What is the smallest integer Nj = Nj(e) such that (x2 — x+e)QNj(x)E(P+ where e>l/4 and Qn5(x) is a polynomial of degree Nj which belongs to (P/? Here, j takes any of the values 0, 1,2, 3. If j = 0, the answer is contained in the statement and proof of Theorem 1 of [2]. For 7 = 1, the smallest integer n (namely Nx) for which (x2 — x+e)Y[l=i (x+bi)E(?+ is exactly the integer N mentioned earlier, as the correspondence y—>2^4x, e—>(4^42)-1, at-^Abi shows. Clearly, bi^O, all i and when n = Ni, bt>0, i = l, 2, • • • , Ni For j = 2, the above question is intimately related to the second part of the research problem while for j = 3, the connection is more remote. All of these questions are rendered easier by an inversion, that is, the fixing of re and the quest for the smallest e=en such that