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].
참고문헌
- M. Abramowitz & I.A. Stegun (Eds): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Applied Mathematics Series 55, 9th printing, Washington, 1970.
- B.-N. Guo & F. Qi: A completely monotonic function involving the tri-gamma function and with degree one. Appl. Math. Comput. 218 (2012), no. 19, 9890-9897; Available online at http://dx.doi.org/10.1016/j.amc.2012.03.075.
- B.-N. Guo & F. Qi: Refinements of lower bounds for polygamma functions. Proc. Amer. Math. Soc. 141 (2013), no. 3, 1007-1015; Available online at http://dx.doi.org/10.1090/S0002-9939-2012-11387-5.
- F. Qi: Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions. arXiv preprint, available online at http://arxiv.org/abs/1302.6731.
- F. Qi & C. Berg: Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function. Mediterr. J. Math. 10 (2013), no. 4, 1685-1696; Available online at http://dx.doi.org/10.1007/s00009-013-0272-2.
- F. Qi & S.-H. Wang: Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions. arXiv preprint, available online at http://arxiv.org/abs/1210.2012.
- F. Qi & X.-J. Zhang: Complete monotonicity of a difference between the exponential and trigamma functions. arXiv preprint, available online at http://arxiv.org/abs/1303.1582.
- D.V. Widder: The Laplace Transform. Princeton University Press, Princeton, 1946.
피인용 문헌
- On complete monotonicity for several classes of functions related to ratios of gamma functions vol.2019, pp.1, 2019, https://doi.org/10.1186/s13660-019-1976-z
- A LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTION INVOLVING THE RATIO OF GAMMA FUNCTIONS vol.5, pp.4, 2014, https://doi.org/10.11948/2015049
- Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers vol.8, pp.2, 2014, https://doi.org/10.1515/ausm-2016-0019