On a conjecture of Clark and Ismail

Abstract

Let m (x) = −x m (m) (x), where 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 (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 (m) m is not completely monotonic on (0, ∞).

Monotonicity of some functions involving the gamma and psi functions

“…Completely monotonic functions attracted the attention of many authors and a plethora of results on completely monotonic functions have been obtained, most of them concerning functions defined in terms of the gamma and polygamma functions. We refer the reader to [5], [3], [17] [18], [20], [23] for some interesting results, applications, bibliography and helpful information regarding such functions.…”

Completely Monotonic Functions Related to Logarithmic Derivatives of Entire Functions

“…In [4], it was furthermore conjectured that (−x m ψ (m) (x)) (m) would be completely monotonic for all m ≥ 1 but it has turned out not to be the case; see [1]. In this direction we show that the function…”

Asymptotic approximations to the Hardy–Littlewood function

“…Therefore, we will have to bring in some other ideas in order to reach the value of Q(x) which is less than −π/2. In order to find these new ideas, we have looked at the proof of the estimate Q(x) = Ω ± ( ln ln(x)) given in [1] and [10]. This proof is based on a rather interesting argument, which is both constructive and non-constructive.…”

“…Of course, it still takes a lot of work to deduce (19), as we have to show that the sum over all n which do not divide K is not too large. See [1] and [10] for all the details.…”