key: cord-0180748-4cdsenbl
authors: Qi, Feng; Guo, Bai-Ni
title: From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions
date: 2020-01-02
journal: nan
DOI: nan
sha: 9590839485386b20aad9c5d1f94140afe0ba110e
doc_id: 180748
cord_uid: 4cdsenbl

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

was proved to be true.

In [33, Lemma 1.4] , inequality (1.1) was generalized as one which can be reformulated as 1 y − 1 − n k=1 1 y 1/λ k − 1 > 0, (1.2) where y > 1 and λ 1 , λ 2 , . . . , λ n ∈ (0, 1) such that 1 e x/λij − 1 (1.6) was proved to be true for x > 0 and λ ij > 0, where ν i = n j=1 λ ij and τ j = m i=1 λ ij . We observe that (1) inequality (1.1) can be rearranged as

where x = ln y > 0 and λ ∈ (0, 1); (2) inequality (1.2) can be rewritten as

where x = ln y > 0 and λ 1 , λ 2 , . . . , λ n ∈ (0, 1) such that n k=1 λ k = 1; (3) inequality (1.3) can be reformulated as (1.8) without restrictions n k=1 λ k = 1 and λ 1 , λ 2 , . . . , λ n < 1; (4) inequality (1.4) can be rewritten as

where λ k > 0 and x > 0. (5) inequality (1.5) can be reformulated as (9) when taking m = n and λ 1i = λ i1 > 0 for 1 ≤ i ≤ n and letting λ ij → 0 + for 2 ≤ i, j ≤ n, inequality (1.6) becomes 1 e x/ n j=1 λ1j − 1

this inequality is equivalent to (1.8) without restrictions n k=1 λ k = 1 and λ 1 , λ 2 , . . . , λ n < 1. In a word, inequality (1.6) 

validates for x > 0 and λ k > 0? Motivated by inequalities (1.6) and (1.12), we would like to ask a question: what are the largest ranges of α and ρ such that

is valid for x > 0 and λ ij > 0? where ν i = n j=1 λ ij and τ j = m i=1 λ ij . Motivated by proofs of inequalities (1.3), (1.4), and (1.6) in the papers [37, 54, 59] , we would like to ask a question: what is the largest range of α such that the function t α H 1 t is convex on (0, ∞)?

The following lemmas are useful in this paper.

are both decreasing on (a, b).

A function ϕ : [0, ∞) → R is said to be star-shaped if ϕ(νt) ≤ νϕ(t) for ν ∈ [0, 1] and t ≥ 0. A real function ϕ defined on a set S ⊂ R n is said to be super-additive if s, t ∈ S implies s + t ∈ S and ϕ(s + t) ≥ ϕ(s) + ϕ(t). See [27, Chapter 16] and [30, Section 3.4] . (1) if ϕ is convex on [0, ∞) with ϕ(0) ≤ 0, then ϕ is star-shaped;

(2) if ϕ : [0, ∞) → R is star-shaped, then ϕ is super-additive.

Now we give an answer to the third question above and find something more.

(1) if α ≥ 1, the function H α (t) is convex on (0, ∞);

(2) if 0 ≤ α < 1, the function H α (t) has a unique inflection point on (0, ∞);

(3) if α < 0, the function H α (t) has only two inflection points on (0, ∞);

(4) the function H α (t) has the limits

Proof. By direct computation, we have

where

By straightforward calculation, we have

Since 2te t + (4t + 1)e 2t − 2(2t 2 + 2t + 1)e t + 1 ′ (e t − 1) ′ = (4t + 3)e t − (2t 2 + 6t + 3) (4t + 1)e 2t − 2(2t 2 + 2t + 1)e t + 1 + 2t + 2,

and d dt 2t + 3 e t = − 2t + 1 e t , making use of Lemma 2.1 twice, we can deduce that the function H 1 (t) is increasing on (0, ∞).

Since The proof of the existence of inflection points of the function H α (t) on (0, ∞) is straightforward.

It is easy to see

and

the limits in (3.2) follow immediately. The proof of Theorem 3.1 is complete.

Proof. By standard computation and by virtue of (3.3), we have

where

By standard calculation, we have

Employing Lemma 2.1 twice arrives at that the function H (t) is decreasing on (0, ∞). Then the function H 1 t is increasing on (0, ∞), with the limits

Accordingly, the function H 1 t is negative on (0, ∞) and, if α ≥ 0, the second derivative d 2 ln Hα(t) dt 2 is negative on (0, ∞). Hence, if α ≥ 0, the function H α (t) is logarithmically concave on (0, ∞) and the first derivative d ln Hα(t) dt is decreasing on (0, ∞). Combining this with the limits in (3.4) 

It is not difficult to see that, if α < 0, by the limits in (3.4) and (3.5), the second derivative d 2 ln Hα(t) dt 2 has a zero, the first derivative d ln Hα(t) dt has only one zero, and the function ln H α (t) has only one inflection point and has only one maximum point on (0, ∞). The proof of Theorem 3.2 is complete.

Remark 3.1. It is well known [30, Section 1.3] that a logarithmically convex function must be convex, but not conversely. It is also well known [30, Section 1.3] that a concave function must be logarithmically concave, but not conversely. The function H α (t) is an example that a logarithmically concave function may not be concave, that a convex function may not be logarithmically convex, and so on.

Making use of some conclusions in Theorems 3.1 and 3.2, we now start out to derive several inequalities involving exponential functions and sums and to answer the first and second questions above.

Proof. Combining the first conclusion and the limit (3.1) in Theorem 3.1 with Lemma 2.2 yields that, if α ≥ 1, the function H α (t) with redefining H α (0) = 0 is convex, then star-shaped, and then supper-additive on [0, ∞). Consequently, it follows that

Accordingly, we obtain

which can be rearranged as

The proof of Theorem 4.1 is complete.

Remark 4.1. As the deduction of (1.11), setting m = n and λ 1k = λ k1 = λ k > 0 for 1 ≤ k ≤ n and letting λ ij → 0 + for 2 ≤ i, j ≤ n in inequality (4.1) result in inequality (1.12) for α ≥ 0. The inequality (4.1) is equivalent to (1.13) for α ≥ 0 and ρ ≤ 2.

Proof. As did in the proof of Theorem 4.1, we can obtain

Consequently, it follows that

Rearranging and simplifying the above two inequalities lead to (4.2) and (4.3). The proof of Theorem 4.2 is complete.

A new ratio of many gamma functions and its properties Its logarithmic derivative ψ(z) = [ln Γ(x)] ′ = Γ ′ (z) Γ(z) and ψ (k) (z) for k ∈ N are called in sequence digamma function, trigamma function, tetragamma function, and, totally, polygamma functions.

The q-gamma function Γ q (x) for q > 0 and x > 0, the q-analogue of the gamma function Γ(x), can be defined ([4, pp. 493-496] and [9, Section 1.10]) by (1) A logarithmically completely monotonic function is completely monotonic, but not conversely. See [6, 11, 47, 50] 

Let p ∈ (0, 1) and k, n be nonnegative integers such that 0 ≤ k ≤ n. In [2, Theorem] , motivated by inequalities related to binomial probability studied in [23, 24] , with the help of inequality (1.1), Alzer proved [2] that the function

is completely monotonic on (0, ∞). Indeed, Alzer implicitly proved logarithmically complete monotonicity of G(x) on (0, ∞). In [33, Theorem 2.1] and [59] , with the aid of inequalities (1.2) and (1.3), the function G(x) defined in (5.1) and its logarithmically complete monotonicity were generalized as follows. Let m ∈ N, λ i > 0 for 1 ≤ i ≤ m, p i ∈ (0, 1) for 1 ≤ i ≤ m, and m i=1 p i = 1. Then the function

is logarithmically completely monotonic on (0, ∞ 

p xλi i for q ∈ (0, 1) and m ∈ N, which is the q-analogue of the function Q(x) in (5.2), was proved in [37] to be logarithmically completely monotonic on (0, ∞), where λ i > 0 for 1 ≤ i ≤ m and p i ∈ (0, 1) for 1 ≤ i ≤ m with m i=1 p i = 1. By virtue of inequality (1.5), Ouimet considered in [32, Theorem 2.1] the function

and its logarithmically complete monotonicity on (0, ∞). But, the conclusion and its proof in [32, Theorem 2.1] are both wrong. Let λ ij > 0 for 1 ≤ i ≤ m and 1 ≤ j ≤ n, let ν i = n j=1 λ ij and τ j = m i=1 λ ij for 1 ≤ i ≤ m and 1 ≤ j ≤ n, and let

for ρ ∈ R. This function f (t) is a generalization of the function g(t) in (5.3).

In [54, Theorem 4.1] , with the help of inequality (1.6), the following conclusions were obtained:

(1) when ρ ≤ 2, the second derivative [ln f (t)] ′′ is a completely monotonic function of t ∈ (0, ∞) and maps from (0, ∞) onto the open interval

(2) when ρ = 2, the logarithmic derivative [ln Some of the above results have been applied in [2, 33, 37, 54] to multinomial probability, to the Bernstein estimators on the simplex, to constructing combinatorial inequalities for multinomial coefficients, to constructing inequalities for multivariate beta functions, and the like.

Complete monotonicity of a linear combination of finite trigamma functions. In the paper [16] , the authors discussed complete monotonicity of the linear combination m k=1 a k ψ(b k x + δ) for δ ≥ 0 and a k , b k > 0. Now we discuss complete monotonicity of a linear combination of finite trigamma functions.

Theorem 5.1. Let λ ij > 0 for 1 ≤ i ≤ m and 1 ≤ j ≤ n, let ν i = n j=1 λ ij and τ j = m i=1 λ ij for 1 ≤ i ≤ m and 1 ≤ j ≤ n, and let ρ, θ ∈ R. If ρ ≤ 2 and θ ≥ 0, then the linear combination

is completely monotonic on (0, ∞).

Proof. Employing the integral representation Consequently, if θ ≥ 0 and ρ ≤ 2, the function P (t) is completely monotonic on (0, ∞). The proof of Theorem 5.1 is complete.

A new ratio of many gamma functions and its properties. Let λ ij > 0 for 1 ≤ i ≤ m and 1 ≤ j ≤ n, let ν i = n j=1 λ ij and τ j = m i=1 λ ij for 1 ≤ i ≤ m and 1 ≤ j ≤ n, and let

for ρ, θ ∈ R. It is clear that, when θ = 0, the function F (t) becomes f (t) defined in (5.4).

Theorem 5.2. The function F (t) has the following properties:

(1) If ρ ≤ 2 and θ ≥ 0, the second derivative [ln F (t)] ′′ is completely monotonic and maps from (0, ∞) onto the interval

and, consequently, is a Bernstein function on (0, ∞).

where γ = 0.57721566 . . . is the Euler-Mascheroni constant. Proof. Taking the logarithm of F (t) in (5.6) and differentiating give

and [ln F (t)] ′′ = P (t), where P (t) is defined in (5.5) . From Theorem 5.1, it follows immediately that, if ρ ≤ 2 and θ ≥ 0, the second derivative [ln F (t)] ′′ is completely monotonic on (0, ∞) and, consequently, that the function F (t) is logarithmically convex on (0, ∞).

It is easy to obtain that lim t→0 +

Since P (t) = [ln F (t)] ′′ is completely monotonic on (0, ∞), the logarithmic deriv- 

where the last term is equal to

∞, θ > 0 or ρ < 2. 

λ θ ij e −λij ts ds

where, when ρ = 2 and θ = 0, we used the fact [54] that The proof of Theorem 5.2 is complete.

Finally, basing on inequalities (4.2) and (4.3) in Theorem 4.2, motivated by Theorems 5.1 and 5.2, we would like to suggest to consider two linear combinations L 1 (t), L 2 (t) and two ratios R 1 (t), R 2 (t) defined by λ θ ijk ψ ′ 1 + λ ijk t ,

where t > 0, θ ≥ 0, and λ ijk > 0 for 1 ≤ i ≤ ℓ, 1 ≤ j ≤ m, and 1 ≤ k ≤ n.

For the sake of saving the space and shortening the length of this paper, we would not like to write down our guesses on possible conclusions and their detailed proofs of the functions L 1 (t), L 2 (t), R 1 (t), and R 2 (t) on (0, ∞).

Remark 7.1. The function 1 e x −1 has been investigated from viewpoints of analytic combinatorics and analytic number theory in [13, 19, 68, 70] and closely related references therein. There have been so many papers dedicated to research of functions involving the exponential function, please refer to the papers [7, 10, 11, 15, 17, 18, 22, 25, 26, 28, 34, 35, 40, 42, 43, 44, 46, 49, 48, 57, 60, 61, 62, 63, 64, 65, 71, 72] and closely related references therein.

Remark 7.2. There are so many papers dedicated to study of ratios of many gamma functions, please refer to the papers [12, 14, 38, 39, 45, 51, 53, 55, 56, 58] and closely related references therein.

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