A Solution to Qi’s Conjecture on a Double Inequality for a Function Involving the Tri- and Tetra-Gamma Functions

Ladislav Matejíčka

doi:10.3390/math7111098

Abstract

In the paper, the author gives a solution to a conjecture on a double inequality for a function involving the tri- and tetra-gamma functions, which was first posed in Remark 6 of the paper “Complete monotonicity of a function involving the tri- and tetragamma functions” (2015) and repeated in the seventh open problem of the paper “On complete monotonicity for several classes of functions related to ratios of gamma functions” (2019).

Highlights

It is common knowledge that the classical Euler’s gamma function [1,2] is defined by Γ( x ) = Z∞t x−1 e−t dt for x > 0 and the digamma function [3] is defined as the logarithmic derivative of the gamma function ψ( x ) = Γ0 ( x ) Γ( x )The functions ψ, ψ0, ψ00, ψ000, ... are known as polygamma functions [4].Very recently, in the paper [5], F
T x−1 e−t dt for x > 0 and the digamma function [3] is defined as the logarithmic derivative of the gamma function ψ( x ) =
Agarwal surveyed some results related to the function ψ02 + ψ00

Summary

Introduction

It is common knowledge that the classical Euler’s gamma function [1,2] is defined by. T x−1 e−t dt for x > 0 and the digamma function [3] is defined as the logarithmic derivative of the gamma function ψ( x ) =. The functions ψ, ψ0 , ψ00 , ψ000 , ... Are known as polygamma functions [4]. P. Agarwal surveyed some results related to the function ψ02 + ψ00. Agarwal surveyed some results related to the function ψ02 + ψ00 The goal of the paper is to find a solution of the seventh open problem which was first posed as a conjecture in Remark 6 of the paper [6]. + ψ (x) < 4 x β holds on (0, ∞) if and only if α ≥ 6/5 and β ≤ 1

The Key Lemmas

Proof of the Main Result

Findings

Conclusions