Completely monotonic degrees for a difference between the logarithmic and psi functions

Abstract

Completely monotonic degrees for a difference between the logarithmic and psi functions: A difference between logarithmic and psi functions. Journal of Computational and Applied Mathematics, Elsevier, 2019, 361, pp.Abstract. In the paper, the authors firstly present a concise proof for complete monotonicity of a function involving a difference between the logarithmic and psi functions, secondly compute completely monotonic degree of the above-mentioned function, and finally pose several conjectures on completely monotonic degrees of remainders and their derivatives for the asymptotic formula of the logarithm of the classical Euler gamma function.2010 Mathematics Subject Classification. Primary 26A48; Secondary 33B15, 44A10.

Qi’s conjectures on completely monotonic degrees of remainders of asymptotic formulas of di- and trigamma functions

“…This implies that the completely monotonic degree of the function ψ(x) -log(x) + 1 2x + 1 12x 2 is 2 at least. This is also a part of the proof of Theorem 2 in reference [22] by Qi and Liu. Our method, with the help of the Koumandos-Pedersen theorem, is the same as that of Qi and Liu, a method of integration-by-part, essentially.…”

“…Then the result was improved by [3], and it was proved that the function Φ(x) = x 2 [ψ(x) -log(x)] + x/2 + 1/12 is completely monotonic on (0, ∞). A concise proof of the complete monotonicity of the function Φ(x) was presented by Qi and Liu [22]. Meanwhile, they proved that…”

“…In [11], the completely monotonic degree of the function R n (x) for n ≥ 0 on (0, ∞) was proved to be at least n. This means that the functions (-1) m [R n (x)] (m) for m, n ≥ 0 are completely monotonic on (0, ∞). Qi and Liu [22] asked a question: What are the completely monotonic degrees of the completely monotonic functions (-1) m [R n (x)] (m) for m, n ≥ 0 on (0, ∞)? Several conjectures posed by Qi can be recited as follows:…”

Monotonicity properties for a ratio of finite many gamma functions

“…(1) If (-1) k f (k) (x) ≥ 0 for all k ≥ 0 and x ∈ (0, ∞), then we call f (x) a completely monotonic function on (0, ∞). See the review papers [22,31,36] and [35, Chapter IV]. f (x) is a completely monotonic function on (0, ∞), then we call f (x) a logarithmically completely monotonic function on (0, ∞).…”

Monotonicity, convexity and inequalities related to complete $(p,q,r)$-elliptic integrals and generalized trigonometric functions

“…x > 0 converges. For more information on (logarithmically) completely monotonic functions, please refer to [16], [24]- [26], [36], [39], [40], [45], and closely related references therein. Now we state and prove monotonicity, logarithmic convexity and complete monotonicity of complete (p, q, r)-elliptic integrals and, consequently, derive several Turán-type inequalities for complete (p, q, r)-elliptic integrals.…”