More properties of the Ramanujan sequence

Abstract

The Ramanujan sequence $\{{\theta }_n{\}}_{n\ge 0}$, defined as $${\theta }_0=\frac{1}{2},{\theta }_n=\left(\frac{e^n}{2}-\sum _{k=0}^{n-1}\frac{n^k}{k!}\right)\cdot \frac{n!}{n^n},n\ge 1,$$ 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 $\{{\theta }_n{\}}_{n\ge 0}$ and $\{4/135-n\cdot ({\theta }_n-1/3){\}}_{n\ge 0}$ are completely monotone. In the present paper we establish that the sequence $\{(n+1)({\theta }_n-1/3){\}}_{n\ge 0}$ 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 $\mathbb{C}\setminus [1,\infty )$. 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.

12 p