Some classes of completely monotonic functions, II

Abstract

A function f : (0, ∞) → R is said to be completely monotonic if (−1) n f (n) (x) ≥ 0 for all x > 0 and n = 0, 1, 2, . . .. In this paper we present several new classes of completely monotonic functions. Our functions have in common that they are defined in terms of the classical gamma, digamma, and polygamma functions. Moreover, we apply one of our monotonicity theorems to prove a new inequality for prime numbers. Some of the given results extend and complement theorems due to Bustoz & Ismail, Clark & Ismail, and other researchers.

Some Logarithmically Completely Monotonic Functions Related to the Gamma Function

“…It is remarked that many complete monotonicity results in [2,3,4,11,21] and closely-related references therein can be restated indeed as the logarithmically complete monotonicity.…”

“…For example, they play a role in complex analysis, number theory, potential theory, probability theory [9], physics [20], numerical and asymptotic analysis, integral transforms [41], and combinatorics. Some related references are listed in [2,3,4,5,6,15,23,41]. In recent years, inequalities and (logarithmically) completely monotonic functions involving the gamma, psi, or polygamma functions are established by some mathematicians (see [2,3,4,11,15,17,21,31] and related references therein).…”

Completely monotonic functions involving the gamma and $q$-gamma functions

“…In the 1980s Ismail et al, [21], [9], realized that some of the inequalities are consequences of the complete monotonicity of functions of similar structure. We continue to witness progress in this area as can be seen from the more recent works [1,2,3,4,6,22] (see also [5]). This paper is a further contribution to the subject.…”

“…The authors are grateful to the referee for many helpful comments and for informing us of several references including [6] and [8].…”

Time response analysis of fractional-order control systems: A survey on recent results

“…In [60] also by benefiting from the properties of completely and absolutely monotonic functions [38], [2], different classes of irrational transfer functions with monotonic step responses have been introduced. Two forms of these classes have been presented in the following theorems.…”