Some properties of several functions involving polygamma functions and originating from the sectional curvature of the beta manifold

Abstract

In the paper, the author investigates some properties, including analyticity, limits, monotonicity, complete monotonicity, and inequalities, of several functions involving the tri-, tetra-, and penta-gamma functions and originating from computation of the sectional curvature of the beta manifold.

Necessary and sufficient conditions for a ratio involving trigamma and tetragamma functions to be monotonic

“…In [8,Theorem 4], the author turned out that, (1) is valid and sharp in the sense that the constants −2 and −1 cannot be replaced by any bigger and smaller ones respectively. In this paper, we mainly generalize the double inequality (1.2) by finding necessary and sufficient conditions in the following theorem.…”

“…Consequently, when α ≥ 2, the function H α (x) is completely monotonic on (0, ∞); when α ≤ 1, the function −H α (x) is completely monotonic on (0, ∞). By the way, the proof of necessary conditions on α for ±H α (x) to be completely monotonic on (0, ∞) is the same as in the proof of [8,Theorem 4]. The required proof is complete.…”

Lower Bound of Sectional Curvature of Manifold of Beta Distributions and Complete Monotonicity of Functions Involving Polygamma Functions

“…Remark 6.7. This paper is a slightly revised version of the preprint [13] and the fifth one in a series of articles including [10,14,15,16].…”

“…In [7,Proposition 3] and [8,Proposition 13], the sectional curvature K(x, y) of the manifold of the beta distributions was given by In [7,Proposition 4] and [8,Proposition 14], the following limits were computed: In [14,Theorem 4.1] and [16,Theorem 4], the author presented that…”

Complete monotonicity of a difference constituted by four derivatives of a function involving trigamma function

“…Remark 4.3. This paper is the tenth one in a series of articles including [6,7,9,10,11,12,13,14,16].…”