On a Family of Functions Studied by Bhatia and Jain

ABSTRACT

In a recent article, Bhatia and Jain have considered the family of functions h_r(x) = (1 − x^1/2)/(1 − x^r/2), x > 0, depending on the positive parameter r. Let H denote the set of parameter values r > 0 such that h_r 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 < r < 8, we determine the function ϕ_r having h_r as Laplace transform, so r ∈ H if and only if ϕ_r(t) ⩾ 0 for t > 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.

KEYWORDS:

• completely monotonic function
• complex analysis
• Stieltjes function

2000 AMS SUBJECT CLASSIFICATION:

• Primary 26A48
• Secondary 30E20

Acknowledgment

The author would like to thank Henrik L. Pedersen for helpful discussions.