Qi\u2019s conjectures on completely monotonic degrees of remainders of asymptotic formulas of di- and trigamma functions

Ai-Min Xu,Zhong-Di Cen

doi:10.1186/s13660-020-02345-5

Abstract

Several conjectures posed by Qi on completely monotonic degrees of remainders for the asymptotic formulas of the digamma and trigamma functions are proved.

Highlights

A function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I which alternate successively in sign, that is,(–1)nf (n)(x) ≥ 0 (1.1)for all x ∈ I and for all n ≥ 0
In this paper, following the method due to Koumandos and Pedersen [11], we prove that degxcm (–1)2R0(x) = 2, degxcm (–1)2R1(x) = 3, degxcm –R0(x) = 1, (1.7) (1.8) (1.9)
3 Proofs of the conjectures we present our theorems and proofs

Summary

Introduction

A function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I which alternate successively in sign, that is,. When the function xα[f (x) – f (∞)] is completely monotonic on (0, ∞) if and only if 0 ≤ α ≤ r, the number r, denoted by degxcm[f (x)], is called the completely monotonic degree of f (x) with respect to x ∈ (0, ∞). A concise proof of the complete monotonicity of the function Φ(x) was presented by Qi and Liu [22] They proved that degxcm Φ(x)/x2 = 2. Motivated by (1.7) and (1.8), which can be verified as in [15], Qi and Mahmoud [23, 24] corrected and modified two conjectures stated in (3) as follows: The completely monotonic degrees of (–1)mR(nm)(x) for m ≥ 2 with respect to x ∈ (0, ∞) satisfy degxcm (–1)m R0(x) (m) = m, degxcm (–1)m R1(x) (m) = m + 1.

It follows from

Since et