COMPLETE MONOTONICITY OF A DIFFERENCE BETWEEN THE EXPONENTIAL AND TRIGAMMA FUNCTIONS

Abstract

In the paper, by directly verifying an inequality which gives a lower bound for the first order modified Bessel function of the first kind, the authors supply a new proof for the complete monotonicity of a difference between the exponential function $e^{1/t}$ and the trigamma function ${\psi}^{\prime}(t)$ on (0, ${\infty}$).

1. INTRODUCTION

In [3, Lemma 2], the inequality

on (0, ∞) was discovered and employed, where 𝜓(t) denotes the digamma function

and Γ is the classical Euler gamma function which may be defined for ℜ(z) > 0 by

The functions 𝜓′(z) and 𝜓′′(z) are respectively called the trigamma function and the tetragamma function. As a whole, the derivatives 𝜓(k)(z) for k ∈ {0} ∪ are called polygamma functions.

An infinitely differentiable function f defined on an interval I is said to be a completely monotonic function on I if it satisfies

for all k ∈ {0} ∪ on I. Some properties of the completely monotonic functions can be found in, for example, [2, 8].

In [5, Theorem 3.1] and [6, Theorem 1.1], the following theorem was proved by three methods totally.

Theorem 1.1. The function

is completely monotonic on (0, ∞) and

The second main result of the paper [6] is [6, Theorem 1.2] which has been referenced in [4, Section 1.2] and [5, Lemma 2.1] as follows.

Theorem 1.2. For k ∈ {0} ∪ and z ≠ 0, let

For ℜ(z) > 0, the function Hk(z) has the integral representations

and

where the hypergeometric series

for bi ∉ {0, −1, −2, ... }, the shifted factorial (a)0 = 1 and

for n > 0 and any real or complex number a, and the modified Bessel function of the first kind

for ν ∈ and z ∈ .

When k = 0, the integral representations (1.6) and (1.7) may be written as

and

for ℜ(z) > 0. Hence, by the well known formula

for ℜ(z) > 0 and n ∈ , see [1, p. 260, 6.4.1], the function h(t) defined by (1.3) has the following integral representation

Proposition 1.3 (Hausdorff-Bernstein-Widder Theorem [8, p. 161, Theorem 12b]).

A necessary and sufficient condition that f(x) should be completely monotonic for 0 < x < ∞ is that

where α(t) is non-decreasing and the integral converges for 0 < x < ∞.

Combining the complete monotonicity in Theorem 1.1 and the integral representation (1.14) with the necessary and sufficient condition in Proposition 1.3, it was revealed in [6] that

Replacing by t in (1.16) yields [6, Theorem 1.3] below.

Theorem 1.4. For t > 0, we have

We note that the complete monotonicity in Theorem 1.1 is the basis of the inequality (1.17) and some results in the subsequent papers [4, 5].

The aim of this paper is, with the help of the integral representation (1.14) but without using Proposition 1.3, to supply a new proof of Theorems 1.1 and 1.4 in a converse direction with that in [4, 5, 6]. In other words, Theorem 1.4 will be firstly and straightforwardly proved, and then Theorem 1.1 will be done.

 

2. A NEW PROOF OF THEOREMS 1.1 AND 1.4

By the definition of the modified Bessel function Iν(z) in (1.10), it is easy to see that

Hence, in order to prove (1.16), it suffices to show

which is equivalent to

Consequently, the proof of the inequality (1.16), that is, Theorem 1.4, is thus complete.

Substituting the inequality (1.16) into the integral representation (1.14) leads to h(t) > 0 and for k ∈

on (0, ∞). As a result, the function h(t) is completely monotonic on (0, ∞).

The limit (1.4) follows immediately from taking t → ∞ on both sides of the integral representation (1.14). Theorem 1.1 is thus proved.

Remark 2.1. The inequality (2.1) is equivalent to

An immediate differentiation yields

Q′ (u) = eu (u2 − 4u + 6) − 2 (u + 3), Q′′ (u) = eu (u2 − 2u + 2) − 2, Q′′′ (u) = u2eu.

Since Q′′′ (u) and Q′′ (0) = 0, it follows that Q′′ (u) > 0 on (0, ∞). Owing to Q′(0) = 0 and Q′′ (u) > 0, it is derived that Q′ (u) > 0. Finally, since Q(0) = 0, the function Q(u) is positive on (0, ∞). This gives an alternative proof of the inequality (2.1).

Remark 2.2. This is a slightly modified version of the preprint [7].