More properties of the Ramanujan sequence

The Ramanujan sequence $\{{\theta }_n{\}}_{n\ge 0}$, defined as $${\theta }_0=\frac{1}{2},{\theta }_n=(\frac{e^n}{2}-\sum _{k=0}^{n-1}\frac{n^k}{k!})\cdot \frac{n!}{n^n},n\ge 1,$$ has been studied on many occasions and in many different contexts. J.Adell and P.Jodra (2008) and S. Koumandos (2013) showed, respectively, that the sequences ${\theta_{n}}{n \geq 0}$and${4/135 - n \cdot (\theta{n}- 1/3 )}{n \geq 0}$ are completely monotone. In the present paper we establish that the sequence $\{(n+1)({\theta }_n-1\mathrm{/}3){\}}_{n\ge 0}$ is also completely monotone. Furthermore, we prove that the analytic function $(\theta{1}- 1/3 )^{-1} \sum_{n=1}^{\infty} (\theta_{n}- 1/3 ) \cdot z^{n} / n^{\alpha} $ is universally starlike for every $\alpha \ge 1$ in the slit domain $\mathbb{C}\setminus [1,\mathrm{\infty })$. This seems to be the first result putting the Ramanujan sequence into the context of analytic univalent functions and is a step towards a previous stronger conjecture, proposed by S.Ruscheweyh, L.Salinas and T.Sugawa in 2009, namely that the function $(\theta_{1}- 1/3 )^{-1} \sum_{n=1}^{\infty} (\theta_{n}- 1/3 ) \cdot z^{n} $ is universally convex.