[Paper Review] Some properties of the psi and polygamma functions

Feng Qi, Bai‐Ni Guo|arXiv (Cornell University)|Mar 5, 2009

Mathematical Inequalities and Applications13 references3 citations

TL;DR

This paper establishes new monotonicity and concavity properties of functions involving the psi and polygamma functions, generalizing known inequalities. It proves that ϕθ(x) = ψ(x) + ln(e^{θ/x} - 1) is strictly increasing and concave for 0 < θ ≤ 1, with precise asymptotic limits, and derives sharp inequalities for ψ′(x) and ψ′′(x) that extend and reverse known bounds depending on θ.

ABSTRACT

In this paper, some monotonicity and concavity results of several functions involving the psi and polygamma functions are proved, and then some known inequalities are extended and generalized.

Motivation & Objective

• To extend and generalize known inequalities involving the psi and polygamma functions by analyzing the monotonicity and concavity of composite functions.
• To establish necessary and sufficient conditions for the monotonicity and convexity of ϕθ(x) = ψ(x) + ln(e^{θ/x} - 1) on (0, ∞).
• To correct and refine a previously incorrect inequality (1.8) by proving a corrected and extended version (1.9) valid for x > -1.
• To derive new sharp inequalities for the trigamma and tetragamma functions based on the monotonicity analysis of ϕθ(x).
• To investigate the monotonicity and asymptotic behavior of related functions such as f(x), g(x), and h(x), and to conjecture their convexity/concavity properties.

Proposed method

• Analyzes the function ϕθ(x) = ψ(x) + ln(e^{θ/x} - 1) for θ > 0, computing its first and second derivatives to determine monotonicity and concavity/convexity.
• Uses the integral representation and series expansion of the psi and polygamma functions, particularly ψ(x) = -γ + ∑_{k=0}^∞ [1/(k+1) - 1/(k+x)] and ψ^{(m)}(x) = (-1)^{m+1} ∫_0^∞ t^m e^{-xt}/(1 - e^{-t}) dt.
• Applies the mean value theorem to compare derivatives of composite functions, such as in f′(x) = (x/2)[ψ''(1+ξ(x)) - ψ''(1+x/2)].
• Employs differentiation of series expressions to prove positivity of higher-order derivatives, such as [ (u²−1)ψ′(u) − ψ(u²) ]′ > 0 for u > 0.
• Uses asymptotic expansions and known limits of ψ(x) and ψ^{(m)}(x) to evaluate lim_{x→0⁺} ϕθ(x) and lim_{x→∞} ϕθ(x).
• Transforms functions like f(x) and g(x) using the reflection and recurrence identities of the psi function to extend their domain to (−1, ∞).

Experimental results

Research questions

• RQ1 For which values of θ > 0 is the function ϕθ(x) = ψ(x) + ln(e^{θ/x} - 1) strictly increasing or concave on (0, ∞)?
• RQ2 What are the necessary and sufficient conditions on θ for ϕθ(x) to be strictly increasing or strictly concave?
• RQ3 How can the incorrect inequality (1.8) be corrected and generalized to hold for x > -1?
• RQ4 What are the sharp inequalities for ψ′(x) and ψ′′(x) that follow from the monotonicity of ϕθ(x) for θ ∈ (0,1] and θ ≥ 2?
• RQ5 Is the function h(x) = (x²−1)ψ′(x) − ψ(x²) strictly concave on (−1,1) and strictly convex on (1, ∞), as conjectured?

Key findings

• The function ϕθ(x) = ψ(x) + ln(e^{θ/x} - 1) is strictly increasing and strictly concave on (0, ∞) if and only if 0 < θ ≤ 1.
• For 0 < θ ≤ 1, the inequality ψ′(x) > (θ e^{θ/x}) / [x²(e^{θ/x} - 1)] holds for all x > 0, and for θ ≥ 2, the inequality reverses.
• The function ϕθ(x) satisfies lim_{x→0⁺} ϕθ(x) = −γ if θ = 1, lim_{x→0⁺} ϕθ(x) = ∞ if θ > 1, and lim_{x→0⁺} ϕθ(x) = −∞ if 0 < θ < 1.
• The corrected inequality −γ + xψ′(1 + x/2) < ψ(x+1) < −γ + xψ′(√(x+1)) holds for all x > 0, and reverses for −1 < x < 0.
• The functions f(x) = ψ(x+1) − xψ′(1 + x/2) and g(x) = xψ′(√(x+1)) − ψ(x+1) are both strictly increasing on (−1, ∞), with lim_{x→−1⁺} f(x) = −∞ and lim_{x→∞} f(x) = ∞.
• The function h(x) = (x²−1)ψ′(x) − ψ(x²) is strictly increasing on (−1, ∞), with lim_{x→−1⁺} h(x) = −∞ and lim_{x→∞} h(x) = ∞, and is conjectured to be strictly concave on (−1,1) and strictly convex on (1, ∞).