A sufficient condition for a polynomial to be stable

A real polynomial is called Hurwitz (stable) if all its zeros have negative real parts. For a given n ∈ N we find the smallest possible constant dn > 0 such that if the coefficients of F (z) = a0 + a1z + ··· +an zn are positive and satisfy the inequalities akak+1 > dnak−1ak+2 for k = 1, 2, . . . ,n −2, then F (z) is Hurwitz