On a conjecture of Clark and Ismail

Abstract

Let Φm (x) = -xmψ(m) (x), where ψ denotes the logarithmic derivative of Euler's gamma function. Clark and Ismail prove in a recently published article that if m ∈ {1,2,..., 16}, then Φm(m) is completely monotonic on (0, ∞), and they conjecture that this is true for all natural numbers m. We disprove this conjecture by showing that there exists an integer m0 such that for all m ≥ m0 the function Φm(m) is not completely monotonic on (0, ∞). © 2005 Elsevier Inc. All rights reserved.