More properties of the Ramanujan sequence

Abstract

The Ramanujan sequence $ \{\theta_{n}\}_{n \geq 0}$, defined as $$ \theta_{0}= \frac{1}{2} \ , \ \ \ \theta_{n} = \left(\ \ \frac{e^{n}}{2} - \sum_{k=0}^{n-1} \frac{n^{k}}{k !} \ \ \right) \cdot \frac{n !}{n^{n}} \ , \ \ n \geq 1 \ ,$$ 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 $\{(n+1)(\theta_{n}- 1/3 )\}_{n \geq 0}$ 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 \geq 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.