HIGHER ORDER TURAN INEQUALITIES FOR THE PARTITION FUNCTION

The Turan inequalities and the higher order Turan inequalities arise in the study of the Maclaurin coefficients of real entire functions in the Laguerre-Polya class. A sequence {a(n)}(n >= 0) of real numbers is said to satisfy the Turan inequalities or to be log-concave if for n >= 1, a(n)(2) - a(n-1)a(n+1) >= 0. It is said to satisfy the higher order Turan inequalities if for n >= 1, 4(a(n)(2) - a(n) - 1a(n+1)) (a(n+1)(2) - a(n)a(n+2)) - (a(n)a(n+1) - a(n-1)a(n+2))(2) >= 0. For the partition function p(n), DeSalvo and Pak showed that for n > 25, the sequence {p(n)}(n>25) is log-concave, that is, p(n)(2) - p(n - 1)p(n + 1) > 0 for n > 25. It was conjectured by the first author that p(n) satisfies the higher order Turan inequalities for n >= 95. In this paper, we prove this conjecture by using the Hardy Ramanujan Rademacher formula to derive an upper bound and a lower bound for p(n + 1)p(n - 1)/p(n)(2). Consequently, for n >= 95, the Jensen polynomials p(n 1) + 3p(m)x + 3p(n + 1)x(2) p(n + 2)x(3) have only distinct real zeros. We conjecture that for any positive integer m >= 4 there exists an integer N(m) such that for n >= N(m), the Jensen polynomial associated with the sequence (p(n), p(n + 1),..., p(n m)) has only real zeros. This conjecture was posed independently by Ono.