Monotonicity properties for a ratio of finite many gamma functions

Feng Qi,Dongkyu Lim

doi:10.1186/s13662-020-02655-4

Abstract

In the paper, the authors consider a ratio of finite many gamma functions and find its monotonicity properties such as complete monotonicity, the Bernstein function property, and logarithmically complete monotonicity.

Highlights

In the paper, the authors consider a ratio of finite many gamma functions and find its monotonicity properties such as complete monotonicity, the Bernstein function property, and logarithmically complete monotonicity
As remarked in [27, Remark 4.1], setting n = m and λ1k = λk1 = λk > 0 for 1 ≤ k ≤ m and letting λij → 0+ for 2 ≤ i, j ≤ m in inequality (3.2) result in
Author details 1Institute of Mathematics, Henan Polytechnic University, Jiaozuo, China

Summary

Introduction

The authors consider a ratio of finite many gamma functions and find its monotonicity properties such as complete monotonicity, the Bernstein function property, and logarithmically complete monotonicity. (1) If (–1)kf (k)(x) ≥ 0 for all k ≥ 0 and x ∈ (0, ∞), we call f (x) a completely monotonic function on (0, ∞). (3) If f (x) is a completely monotonic function on (0, ∞), we call f (x) a Bernstein function on (0, ∞). 5], the paper [18], and [34, Chap.

Results

Conclusion