On a conjecture of Clark and Ismail

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,∞).