On the positive definiteness of \(n\mapsto e^{pn^{\alpha}}\) (Q1585378)

For \(\alpha> 2\), let \(Q^+(\alpha)\) be the infimum of those \(q> 0\) such that the sequence \(n\mapsto e^{pn^\alpha}\) is positive definite on \(\mathbb{N}_0\) (i.e., a Hamburger moment sequence) whenever \(p\geq q\), and let \(Q^-(\alpha)\) be the supremum of those \(q> 0\) such that this sequence is not positive definite if \(0< p< q\). It is known that \(0< Q^-(\alpha)\leq Q^+(\alpha)< \infty\) for all \(\alpha> 2\). We show that \[ {\alpha\over 4^\alpha}\log{9\over 8}\leq Q^-(\alpha)= Q^+(\alpha)\leq {\alpha\over 4^\alpha}\log 2 \] for \(\alpha\geq 11.7\), and denoting by \(Q(\alpha)\) the common value of \(Q^+(\alpha)\) and \(Q^-(\alpha)\) whenever these are equal, \[ \lim_{n\to\infty} {Q(\alpha)4^\alpha\over \alpha}= \log{9\over 8}. \]