On a conjecture of Clark and Ismail

Abstract

Let Φ m ( x ) = - x m ψ ( 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 Φ m ( 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 m 0 such that for all m ⩾ m 0 the function Φ m ( m ) is not completely monotonic on ( 0 , ∞ ) .