Abstract

The Ramanujan sequence {????????}????≥0 , defined as ????0=1/2 , ????????????????/????!=????????/2−∑????−1????=0????????/????! , ????≥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 {????????}????≥0 and {4/135−????⋅(????????−1/3)}????≥0 are completely monotone. In the present paper, we establish that the sequence {(????+1)(????????−1/3)}????≥0 is also completely monotone. Furthermore, we prove that the analytic function (????1−1/3)−1∑∞????=1(????????−1/3)????????/???????? is universally starlike for every ????≥1 in the slit domain ℂ∖[1,∞) . 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 (????1−1/3)−1∑∞????=1(????????−1/3)???????? is universally convex.