Higher order Turán inequalities for Boros-Moll sequences

Abstract:

We prove that for the Boros-Moll sequences $\{d_i(m)\}_{i=0}^m$, the higher order Turán inequalities $4(d_i(m)^2 -d_{i-1}(m)d_{i+1}(m))(d_{i+1}(m)^2-d_i(m)d_{i+2}(m)) -(d_i(m)d_{i+1}(m)-d_{i-1}(m)d_{i+1}(m))^2\geq 0$ hold for $m\geq 3$ and $1\leq i\leq m-2$. As a consequence, the 3rd associated Jensen polynomials $d_i(m)+3d_{i+1}(m)x+3d_{i+2}(m)x^2+d_{i+3}(m)x^3$ have only real zeros.