Higher Order Turán Inequalities And Double Turán Inequalities For The Partition Function

Recently,the partition function has been studied by many mathematicians and a lot of properties of it have been found.Using analytic method,Hardy and Ramanujan provided a remarkable asymptotic formula for the partition function p?n?.Later,some revision and modification of the Hardy-Ramanujan method made by Rademacher led him to a convergent series for the partition function p?n?.Based on Rademacher's work,Lehmer gave an estimate for the remainder term of the convergent series for the partition function p?n?.The Turán inequal-ity,the higher order Turán inequality and the double Turán inequality arise in the study the Maclaurin coefficients of entire functions in the Laguerre-Pólya class.A real sequence{a_n}_n?0is said to satisfy the Turán inequalities or to be log-concave if for n?1,a_n²-a_n-1a_n+1?0.It is said to satisfy the higher order Turán inequalities if for n?1,4(a_n²-a_n-1a_n+1)(a²_n+1-a_na_n+2)-(a_na_n+1-a_n-1a_n+2)²?0.And we say that it satisfies the double Turán inequalities if for n?2,(a_n²-a_n-1a_n+1)²-(a²_n-1-a_n-2a_n)(a²_n+1-a_na_n+1)?0.Exploring Lehmer's estimate,DeSalvo and Pak proved that the partition function satisfies the Turán inequalities.They also conjectured that the partition function satis-fies a sharper inequality.This conjecture was proved by Chen,Wang and Xie based on the upper bound and the lower bound for the logarithm of the parti-tion function.In this thesis,we provided an upper bound and a lower bound for the partition function and obtained some inequalities on the partition function.Using these bounds,we proved that the partition function satisfies the higher order Turán inequalities and the double Turán inequalities.In Chapter 1,we first introduce the Turán inequality,the higher order Turán inequality and the double Turán inequality and Laguerre-Pólya class.It is known that the sequence of the Maclaurin coefficients of any entire function in the Laguerre-Pólya class satisfies the Turán inequalities and the higher order Turán inequalities.Moreover,if all the coefficients are positive,then it also satisfies the double Turán inequalities.We also introduce two kinds of multiplier sequences,which can be used to sketch the character of Laguerre-Pólya class completely.After that,we introduce the definition of the partition function p?n?and some recent results of it.In Chapter 2,we further study the Hardy-Ramanujan-Rademacher formula and Lehmer's estimate to derive an upper bound and a lower bound for the partition function p?n?,which leads us to bound u_n=p?n-1?p?n+1?/p?n?²and s?n?=u_n-1+u_n+1-u_n-1u_n+1.We also explore the relationship about u_n,u_n+1and s?n?and obtain an upper bound for u_n+1,which is represented by u_nand an upper bound for u_n,which is a function in s?n?.In Chapter 3,we make use of the inequalities in Chapter 2 to give a proof that{p?n?}_n?95satisfies the higher order Turán inequalities and{p?n?}_n?222satisfies the double Turán inequalities.