Abstract
The Turán inequalities and the higher order Turán inequalities arise in the study of the Maclaurin coefficients of real entire functions in the Laguerre–Pólya class. A sequence { a n } n ≥ 0 \{a_{n}\}_{n\geq 0} of real numbers is said to satisfy the Turán inequalities or to be log-concave if for n ≥ 1 n\geq 1 , a n 2 − a n − 1 a n + 1 ≥ 0 a_n^2-a_{n-1}a_{n+1}\geq 0 . It is said to satisfy the higher order Turán inequalities if for n ≥ 1 n\geq 1 , 4 ( a n 2 − a n − 1 a n + 1 ) ( a n + 1 2 − a n a n + 2 ) − ( a n a n + 1 − a n − 1 a n + 2 ) 2 ≥ 0 4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\geq 0 . For the partition function p ( n ) p(n) , DeSalvo and Pak showed that for n > 25 n>25 , the sequence { p ( n ) } n > 25 \{ p(n)\}_{n> 25} is log-concave, that is, p ( n ) 2 − p ( n − 1 ) p ( n + 1 ) > 0 p(n)^2-p(n-1)p(n+1)>0 for n > 25 n> 25 . It was conjectured by the first author that p ( n ) p(n) satisfies the higher order Turán inequalities for n ≥ 95 n\geq 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 p(n+1)p(n-1)/p(n)^2 . Consequently, for n ≥ 95 n\geq 95 , the Jensen polynomials p ( n − 1 ) + 3 p ( n ) x + 3 p ( n + 1 ) x 2 + p ( n + 2 ) x 3 p(n-1)+3p(n)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 m\geq 4 there exists an integer N ( m ) N(m) such that for n ≥ N ( m ) n\geq N(m) , the Jensen polynomial associated with the sequence ( p ( n ) , p ( n + 1 ) , … , p ( n + m ) ) (p(n),p(n+1),\ldots ,p(n+m)) has only real zeros. This conjecture was posed independently by Ono.
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