From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions

Abstract

In this paper, the authors review origins, motivations, and generalizations of a series of inequalities involving finitely many exponential functions and sums. They establish three new inequalities involving finitely many exponential functions and sums by finding convexity of a function related to the generating function of the Bernoulli numbers. They also survey the history, backgrounds, generalizations, logarithmically complete monotonicity, and applications of a series of ratios of finitely many gamma functions, present complete monotonicity of a linear combination of finitely many trigamma functions, construct a new ratio of finitely many gamma functions, derive monotonicity, logarithmic convexity, concavity, complete monotonicity, and the Bernstein function property of the newly constructed ratio of finitely many gamma functions. Finally, they suggest two linear combinations of finitely many trigamma functions and two ratios of finitely many gamma functions to be investigated.

Increasing property and logarithmic convexity of functions involving Riemann zeta function

“…In this paper, we use the notation For more information and recent developments of the gamma function Γ(z) and its logarithmic derivatives ψ (n) (z) for n ≥ 0, please refer to [1,Chapter 6], [25,Chapter 3], or recently published papers [14,18,20,21,31] and closely related references therein.…”

Monotonicity and complete monotonicity of two functions constituted via three derivatives of a function involving trigamma function

“…There are a number of papers and mathematicians dedicated to investigation of complete monotonicity of some functions involving the gamma and polygamma functions. For more information and details, please refer to the papers [2,9,11,15] and closely related references therein.…”

A further generalization of the Catalan numbers and its explicit formula and integral representation

“…form a sequence of integers (see [10,11,53]), can be interpreted combinatorially (see [6,16,50]), date back to the year 1730 (see [17,18,21]), and can be generated (see [24,50,54]) by is the classical Euler gamma function (see [1,Chapter 6], [25,Chapter 5], [55,Chapter 3], and [33]). The Catalan numbers C n have been combinatorially generalized as the Fuss numbers (see [8] and [16, pp.…”