Decreasing properties of two ratios defined by three and four polygamma functions

. In the paper, by virtue of convolution theorem for the Laplace transforms, with the aid of three monotonicity rules for the ratios of two functions, of two definite integrals, and of two Laplace transforms, in terms of the majorization, and in the light of other analytic techniques, the author presents decreasing properties of two ratios defined by three and four polygamma functions.

Abstract.In the paper, by virtue of convolution theorem for the Laplace transforms, with the aid of three monotonicity rules for the ratios of two functions, of two definite integrals, and of two Laplace transforms, in terms of the majorization, and in the light of other analytic techniques, the author presents decreasing properties of two ratios defined by three and four polygamma functions.and its logarithmic derivative ψ(z) = [ln Γ(z)] ′ = Γ ′ (z) Γ(z) are called Euler's gamma function and digamma function respectively.Moreover, the functions ψ ′ (z), ψ ′′ (z), ψ ′′′ (z), and ψ (4) (z) are known as the trigamma, tetragamma, pentagamma, and hexagamma functions respectively.As a set, all the derivatives ψ (k) (z) for k ∈ N 0 = {0} ∪ N are known as polygamma functions.
Recall from Chapter XIII in [13], Chapter 1 in [36], and Chapter IV in [40] that, if a function f (x) on an interval I has derivatives of all orders on I and satisfies (−1) n f (n) (x) ≥ 0 for x ∈ I and n ∈ N 0 , then we call f (x) a completely monotonic function on I.
In [18,Theorem 3.2], the author proved the following conclusions: (1) if and only if k+1 , the function F k,ϑ k (x) is increasing on (0, ∞); (5) the following limits are valid: (6) the double inequality is valid on (0, ∞) and sharp in the sense that the lower and upper bounds cannot be replaced by any greater and less numbers respectively.
for c ∈ R and x ∈ (0, ∞).Then (1) for q ≥ 0, if and only if for m, n ∈ N and the double inequality for m, n, p, q ∈ N with p > m ≥ n > q ≥ 1 and m + n = p + q are valid on (0, ∞) and sharp in the sense that the lower and upper bounds cannot be replaced by any larger and smaller scalars respectively.
In this paper, we aim to confirm these two guesses.We also supply an alternative proof of Theorem 1.1.

Lemmas
The following lemmas are necessary in this paper.
By Lemma 2.1, straightforward differentiation gives e t − e 2st = 0, and e t (st + 1) is increasing in s ∈ 0, 1  2 .This means that the partial derivative ∂ ln[g(st)g((1−s)t)] ∂t is increasing in s ∈ 0, 1 2 for fixed t > 0. As a result, the inequality (2.2) is valid.The proof of Lemma 2.5 is complete.□ Lemma 2.6 ([18, Lemma 2.1]).For k ∈ N, we have the limits Lemma 2.7.For m, n, p, q ∈ N such that (p, q) ≻ (m, n), the function and U (x) V (x) are both increasing or both decreasing in x ∈ (a, b), then the ratio and U (x) V (x) are increasing and the other is decreasing in x ∈ (a, b), then the ratio R(t) is decreasing in t.

Decreasing property of a ratio defined by three polygamma functions
In this section, we prove that the function Q m,n (x) defined in (1.3) is decreasing.
Proof.By virtue of the integral representation (2.1), we can rearranged Q m,n (x) as By Lemma 2.3, we obtain By Lemma 2.4, we only need to prove the ratio is decreasing on (0, ∞).Hence, it suffices to show that the ratio g(st) g s (t) for fixed s ∈ (0, 1) is increasing in t ∈ (0, ∞).This increasing property of g (st)  g s (t) has been proved in Lemma 2.5 in this paper.As a result, the function Q m,n (x) defined in (1.3) is decreasing on (0, ∞).
Making use of the limits (2.3) and (2.4) in Lemma 2.6 yields The proof of Theorem 3.1 is complete.□

Decreasing property of a ratio defined by four polygamma functions
In this section, we prove that the function Q m,n;p,q defined in (1.4) is decreasing.
Proof.By the limits (2.3) and (2.4) in Lemma 2.6, we obtain Making use of the integral representation (2.1) yields Utilizing Lemma 2.3 gives Employing Lemma 2.4 tells us that, it suffices to prove the increasing property in t of the ratio where Lemma 2.7 implies that the ratio ϕm,n(s) ϕp,q(s) is increasing in s ∈ 0, 1 2 for (p, q) ≻ (m, n).Further making use of the inequality (2.2) in Lemma 2.5 and utilizing Lemma 2.8 reveal that the function Q m,n;p,q (x) is decreasing in x ∈ (0, ∞).The proof of Theorem 4.1 is complete.□

An alternative proof of Theorem 1.1
In this section, we supply an alternative proof of Theorem 1.1.
For q = 0, we have where we used the integral representation (2.1) and Lemma 2.3.From the second property in Lemma 2.5, it follows that the function is decreasing in t ∈ (0, ∞) and has the limits Consequently, basing on Lemma 2.9, we see that, if and only if the function F p,m,n,0;c (x) is completely monotonic on (0, ∞).

Remarks
Finally, we list several remarks on our main results and their proofs.Remark 6.1.The papers [3,4,7,8,42] are related to Theorem 1.1.Theorems 3.1 and 4.1 in this paper are related to some results reviewed and surveyed in [15,26] and closely related references therein.Remark 6.2.Lemma 2.5 in this paper generalizes the second item in [19,Lemma 2.3], which reads that the function g(2t) g 2 (t) is decreasing from (0, ∞) onto (0, 1 for n ≥ 1 is decreasing from (0, ∞) onto the interval n n+1 , n+1 n+2 .Remark 6.5.Direct differentiation gives The decreasing property of Q m,n (x) in Theorem 3.1 implies that the inequality We guess that, for m, n ∈ N, the function should be completely monotonic in x ∈ (0, ∞).Generally, one can discuss necessary and sufficient conditions on Ω m,n ∈ R such that the function and its opposite are respectively completely monotonic on (0, ∞).
Remark 6.6.It is immediate that The decreasing property of Q m,n;p,q (x) in Theorem 4.1 implies that the inequality We guess that, for (p, q) ≻ (m, n), the function should be completely monotonic in x ∈ (0, ∞).Generally, for (p, q) ≻ (m, n), one can discuss necessary and sufficient conditions on Ω m,n;p,q ∈ R such that the function and its opposite are respectively completely monotonic on (0, ∞).

Declarations
Acknowledgements: The author appreciates anonymous referees for their careful corrections to and valuable comments on the original version of this paper.Availability of data and material: Data sharing is not applicable to this article as no new data were created or analyzed in this study.Statement of conflict of interest: The author states that there is no conflict of interest.Competing interests: The author declares that he has no conflict of competing interests.Funding: Not applicable.
DECREASING PROPERTIES OF TWO RATIOS DEFINED BY THREE AND FOUR POLYGAMMA FUNCTIONS FENG QI Dedicated to my elder brother, Can-Long Qi, and his family

4 . 5 .
property of a ratio defined by three polygamma functions 6 Decreasing property of a ratio defined by four polygamma functions 7 An alternative proof of Theorem 1Motivations In the literature [1, Section 6.4], the function Γ(z) = ∞ 0 t z−1 e −t d t, ℜ(z) > 0