On the positive definiteness of $n\mapsto e^{pn^\alpha}$

The proof of Theorem 2 in [1] contains two errors which, however, do not make the theorem false.Firstly , in (6) the factor 20 should have been 5/4, so (6) should have read as follows: $$\frac{5}{4}\alpha^2(\alpha-1)(log 2)\biggr[(\frac{3}{2})^{\alpha-2}+1- 2(\frac{5}{4})^\alpha-2\biggl]\geq,$$ which makes the condition harder to satisfy. Secondly , the sentence following (6) is nonsense.However, the factor $[\cdots]$ has a positive derivative (with respect to $\alpha$) as soon as $$\alpha>2+\frac{\log\frac{2\log\frac{5}{4}}{\log\frac{3}{2}}}{\log\frac{6}{5}}=2.52614…$$ Moreover, the corrected inequality (6) certainly holds for $ \alpha\geq4$, so the proof is saved.