ABSTRACTIn a recent article, Bhatia and Jain have considered the family of functions hr(x) = (1 − x1/2)/(1 − xr/2), x > 0, depending on the positive parameter r. Let H denote the set of parameter values r > 0 such that hr is completely monotonic. We give a new proof of their result that [1, 4]⊆H and improve their result that 9∉H by showing that H⊂[1, 8[. For 4   0. Using this it is established that 5 ∈ H, 6∉H. Using Maple it seems plausible that there exists a number r0 such that H = [1, r0], and we estimate that r0 is approximately 5.62.
['Christian Berg']
On a Family of Functions Studied by Bhatia and Jain