On geometric properties of the generating function for the Ramanujan sequence

Abstract

The Ramanujan sequence $\{\theta _{n}\}_{n \ge 0}$, defined as $\theta _{0}= {1}/{2}$, ${n^{n}} \theta _{n}/{n !} = {e^{n}}/{2} - \sum _{k=0}^{n-1} {n^{k}}/{k !}\,$, $n \ge 1$, has been studied on many occasions and in many different contexts. Adell and Jodrá (Ramanujan J 16:1–5, 2008) and Koumandos (Ramanujan J 30:447–459, 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 $(\theta _{1}- 1/3 )^{-1}\sum _{n=1}^{\infty } (\theta _{n}- 1/3 ) z^{n} / n^{\alpha }$ 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 Ruscheweyh et al. (Israel J Math 171:285–304, 2009), namely that the function $(\theta _{1}- 1/3 )^{-1}\sum _{n=1}^{\infty } (\theta _{n}- 1/3 ) z^{n}$ is universally convex.