key: cord-0010697-e0c2f8f4
authors: Qi, Feng; Lim, Dongkyu
title: Monotonicity properties for a ratio of finite many gamma functions
date: 2020-05-01
journal: Adv Differ Equ
DOI: 10.1186/s13662-020-02655-4
sha: 8102484ac9d8f1c0add0fa405fd19d56d4d7ef60
doc_id: 10697
cord_uid: e0c2f8f4

In the paper, the authors consider a ratio of finite many gamma functions and find its monotonicity properties such as complete monotonicity, the Bernstein function property, and logarithmically complete monotonicity.

Let f (x) be an infinite differentiable function on an infinite interval (0, ∞).

(1) If (-1) k f (k) (x) ≥ 0 for all k ≥ 0 and x ∈ (0, ∞), then we call f (x) a completely monotonic function on (0, ∞). See the review papers [22, 31, 36] and [35, Chapter IV]. [3, 4, 7, 24] and [33, Chap. 5] .

(3) If f (x) is a completely monotonic function on (0, ∞), then we call f (x) a Bernstein function on (0, ∞). See the paper [28] and the monograph [33] . The classical gamma function Γ (z) can be defined by , z ∈ C \ {0, -1, -2, . . .}.

See [1, Chap. 6] , [15, Chap. 5] , the paper [18] , and [34, Chap. 3] . In the literature, the logarithmic derivative

and its first derivative ψ (z) are respectively called the digamma and trigamma functions. See the papers [5, 6, 10, 25, 26] and closely related references therein.

This paper is motivated by a series of papers [2, 11, 12, 16, 19, 21, 27, 29, 32] . For detailed review and survey, please read the papers [19, 27, 29, 32] and closely related references therein. In the paper [2] , motivated by [11, 12] , the function

was considered, where p ∈ (0, 1) and k, m are nonnegative integers with 0 ≤ k ≤ m. In [16, Theorem 2.1] and [32] , the function

of the function in (2.2) was investigated in [19] , where q ∈ (0, 1), m ≥ 2, λ i > 0 for 1 ≤ i ≤ m, p i ∈ (0, 1) for 1 ≤ i ≤ m with m i=1 p i = 1, and Γ q is the q-analogue of the gamma function Γ .

The functions

were respectively considered in [17, Theorem 2.1] and [29, Theorem 4.1] , where ρ ∈ R and λ ij > 0, ν i = n j=1 λ ij , τ j = m i=1 λ ij for 1 ≤ i ≤ m and 1 ≤ j ≤ n. In [27] , the function (2.6) was discussed, where ρ, θ ∈ R and λ ij > 0,

In this paper, stimulated by the above six functions (2.1), (2.2), (2.3), (2.4), (2.5), and (2.6), we consider a new function

In this section, we now start out to find and prove some monotonicity properties for the function Q(x) = Q m,a,p,ρ, ,θ (x) defined in (2.7). Our main results in this section can be stated in the following theorem.

is completely monotonic on (0, ∞); (2) when ρ = 1, = 0, and θ = 0, the function

is increasing on (0, ∞) and its logarithmic derivative

is a Bernstein function on (0, ∞); (3) when ρ = 1, ≥ 1, and θ = 0, the function Q m,a,p,1, ,0 (x) is logarithmically completely monotonic on (0, ∞); (4) when (ρ, , θ ) ∈ S and

the function Q m,a,p,ρ, ,θ (x) has a unique minimum on (0, ∞).

Proof Direct calculation gives

As in [27, 29, 32] , from 

where τ > 0 and h(t) = t e t -1 is the generating function of the classical Bernoulli numbers, see [20, 23] and [34, Chap. 1] . Accordingly, we have

where α ≥ 0, x > 0, λ ij > 0 for 1 ≤ i ≤ m and 1 ≤ j ≤ n, ν i = n j=1 λ ij , and τ j = m i=1 λ ij . As remarked in [27, Remark 4.1], setting n = m and λ 1k = λ k1 = λ k > 0 for 1 ≤ k ≤ m and letting λ ij → 0 + for 2 ≤ i, j ≤ m in inequality (3.2) result in 

where ψ(1) = -0.577 . . . , and

Let ξ = (ξ 1 , ξ 2 , . . . , ξ m ) such that m i=1 ξ i = 1 and ξ i ∈ (0, 1) for 1 ≤ i ≤ m and m ≥ 2. Then the first derivative of the function This can be further rewritten as

(3.5)

Considering inequality (3.5) reveals that (1) when θ = 0, we have

(2) when θ > 0 and ρ ≤ 1, we have

Hence, when θ = 0 and ρ < 1 or when θ > 0 and ρ ≤ 1, we obtain lim x→∞ ln Q m,a,p,ρ, ,θ (x) = ∞;

when θ = 0 and ρ = 1, we have

Let f be a convex function on an interval I ⊆ R, let m ≥ 2 and x i ∈ I for 1 ≤ i ≤ m, and let q i > 0 for 1 ≤ i ≤ m. Then 

Accordingly,

Consequently, when θ = 0, ρ = 1, and ≥ 1, the function Q m,a,p,ρ, ,θ (x) is logarithmically completely monotonic on (0, ∞). 

Let m, n ≥ 2, ρ, , θ ∈ R, let λ = (λ ij ) m×n with 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 p = (p ij ) m×n with m i=1 n j=1 p ij = 1 and p ij ∈ (0, 1) for 1 ≤ i ≤ m and 1 ≤ j ≤ n. Define on the infinite interval (0, ∞). Can one find monotonicity properties for the function Q m,n;λ;p;ρ; ;θ (x) defined in equation (4.1)?

Remark 4.1 This paper is a slightly revised version of the electronic preprint [30] .

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables

Complete monotonicity of a function related to the binomial probability

Some properties of a class of logarithmically completely monotonic functions

Integral representation of some functions related to the gamma function

On the complete monotonicity of quotient of gamma functions

The q-deformed gamma function and q-deformed polygamma function

A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function

Two new proofs of the complete monotonicity of a function involving the psi function

Sharp inequalities for polygamma functions

Some monotonicity properties and inequalities for Γ and ζ -functions

On a uniformly integrable family of polynomials defined on the unit interval

A family of inequalities related to binomial probabilities

Analytic Inequalities. Die Grundlehren der mathematischen Wissenschaften

Classical and New Inequalities in Analysis. Kluwer Academic

NIST Handbook of Mathematical Functions

Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex

Complete monotonicity of a ratio of gamma functions and some combinatorial inequalities for multinomial coefficients

Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities

A logarithmically completely monotonic function involving the q-gamma function

A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers

Complete monotonicity for a new ratio of finite many gamma functions (2020). HAL preprint

On complete monotonicity for several classes of functions related to ratios of gamma functions

Two closed forms for the Bernoulli polynomials

A complete monotonicity property of the gamma function

Complete monotonicity of divided differences of the di-and tri-gamma functions with applications

Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function

From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions

Integral representations and properties of some functions involving the logarithmic function. Filomat

A ratio of many gamma functions and its properties with applications

Monotonicity properties for a ratio of finite many gamma functions (2020). HAL preprint

Completely monotonic degrees for a difference between the logarithmic and psi functions

Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions

Bernstein Functions-Theory and Applications

Special Functions: An Introduction to Classical Functions of Mathematical Physics

The Laplace Transform

A class of completely mixed monotonic functions involving the gamma function with applications

The authors are thankful to anonymous referees for their careful corrections, helpful suggestions, and valuable comments on the original version of this paper.

The second author was partially supported by the National Research Foundation of Korea under Grant NRF-2018R1D1A1B07041846, South Korea.

Not applicable.

None of the authors has any competing interests.