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

Abstract

Several conjectures posed by Qi on completely monotonic degrees of remainders for the asymptotic formulas of the digamma and trigamma functions are proved. MSC: 26A48; 33B15; 44A10

Increasing property and logarithmic convexity of functions involving Riemann zeta function

“…In this paper, we use the notation For more information and recent developments of the gamma function Γ(z) and its logarithmic derivatives ψ (n) (z) for n ≥ 0, please refer to [1,Chapter 6], [25,Chapter 3], or recently published papers [14,18,20,21,31] and closely related references therein.…”

Several Functions Originating from Fisher–Rao Geometry of Dirichlet Distributions and Involving Polygamma Functions

“…In the paper [32,33], the author noticed that the completely monotonic function G(t) defined in (28) and expressed by (34) had been studied in [34] (Theorem 1), which reads that the function u α G(u) is of complete monotonicity on (0, ∞) if and only if α ≤ 0. In other words, the completely monotonic degree of the function ψ ′ (u) − 1 u − 1 2u 2 with respect to u on (0, ∞) is 2.…”

Decreasing properties of two ratios defined by three and four polygamma functions

“…As a generalization of decreasing property of real functions of one variable, one can consider (logarithmically) complete monotonicity and completely monotonic degrees. For details, please refer to [14,19,31,39,41,44,49] and the review article [26].…”