The Log-Behavior of √[n]p(n) and √[n]p(n)/n

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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)=√[n]p(n)/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 Δ² r(n-1). In this paper, we obtain a lower bound for Δ² r(n-1), leading to a proof of this conjecture. From the log-convexity of {r(n)}ₙ≥₆₁ and {√[n]n}ₙ≥₄, we are led to a proof of another conjecture of Sun on the log-convexity of {√[n]p(n)}ₙ≥₂₇. Furthermore, we show that ₙ → ₊∞n⁵/²Δ²√[n]p(n)=3π/√24. Finally, by finding an upper bound of Δ² √[n-1]p(n-1), we prove an inequality on the ratio √[n-1]p(n-1)√[n]p(n) analogous to the above inequality on the ratio p(n-1)/p(n).