Abstract:   Let p(n) denote the partition function. Desalvo and Pak proved the log-concavity of p(n) for n>25 and the inequality p(n-1)p(n)(1+1/n)>p(n)/p(n+1) for n>1. Let r(n)= and Δ be the difference operator respect to n. Desalvo and Pak pointed out that their approach to proving the log-concavity of p(n) may be employed to prove a conjecture of Sun on the log-convexity of {r(n)}_n≥61, as long as one finds an appropriate estimate of Δ²log r(n-1). In this paper, we obtain a lower bound for Δ²log r(n-1), leading to a proof of this conjecture. From the log-convexity of {r(n)}_n≥61} and , we are led to a proof of another conjecture of Sun on the log-convexity of . Furthermore, we show that . Finally, by finding an upper bound of Δ²log, we prove an inequality on the ratio analogous to the above inequality on the ratio p(n-1)/p(n).

  AMS Classification:  05A20

  Download:   pdf