Higher order Turán inequalities for Boros-Moll sequences

Abstract

We prove that for the Boros-Moll sequences { d i ( m ) } i = 0 m \{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 ≥ 0 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 ≥ 3 m\geq 3 and 1 ≤ i ≤ m − 2 1\leq i\leq m-2 . As a consequence, the 3rd associated Jensen polynomials d i ( m ) + 3 d i + 1 ( m ) x + 3 d i + 2 ( m ) x 2 + d i + 3 ( m ) x 3 d_i(m)+3d_{i+1}(m)x+3d_{i+2}(m)x^2+d_{i+3}(m)x^3 have only real zeros.