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+⋯+anzn are positive and satisfy the inequalities akak+1>dnak−1ak+2 for k=1,2,…,n−2, then F(z) is Hurwitz.