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COMPLETE MONOTONICITY for a NEW RATIO of FINITE MANY GAMMA FUNCTIONS Feng Qi

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COMPLETE MONOTONICITY for a NEW RATIO of FINITE MANY GAMMA FUNCTIONS Feng Qi COMPLETE MONOTONICITY FOR A NEW RATIO OF FINITE MANY GAMMA FUNCTIONS Feng Qi To cite this version: Feng Qi. COMPLETE MONOTONICITY FOR A NEW RATIO OF FINITE MANY GAMMA FUNCTIONS. 2020. hal-02511909 HAL Id: hal-02511909 https://hal.archives-ouvertes.fr/hal-02511909 Preprint submitted on 19 Mar 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. COMPLETE MONOTONICITY FOR A NEW RATIO OF FINITE MANY GAMMA FUNCTIONS FENG QI Dedicated to people facing and fighting COVID-19 Abstract. In the paper, by deriving an inequality involving the generating function of the Bernoulli numbers, the author introduces a new ratio of finite many gamma functions, finds complete monotonicity of the second logarithmic derivative of the ratio, and simply reviews complete monotonicity of several linear combinations of finite many digamma or trigamma functions. Contents 1. Preliminaries and motivations 1 2. A lemma 3 3. Complete monotonicity 4 4. A simple review 5 References 7 1. Preliminaries and motivations Let f(x) be an infinite differentiable function on (0; 1). If (−1)kf (k)(x) ≥ 0 for all k ≥ 0 and x 2 (0; 1), then we call f(x) a completely monotonic function on (0; 1). See the review papers [23, 32, 42] and [40, Chapter IV]. The classical gamma function Γ(z) can be defined by Z 1 Γ(z) = tz−1e−tdt; <(z) > 0 0 or by n!nz Γ(z) = lim n ; z 2 n f0; −1; −2;::: g: n!1 Q C k=0(z + k) See [1, Chapter 6], [14, Chapter 5], the papers [22, 41], and [39, Chapter 3]. In the literature, the logarithmic derivative Γ0(z) (z) = [ln Γ(x)]0 = Γ(z) 2010 Mathematics Subject Classification. Primary 33B15; Secondary 26A48, 26A51, 26D15, 44A10. Key words and phrases. complete monotonicity; ratio; gamma function; digamma function; trigamma function; inequality; Bernoulli number; generating function; logarithmic derivative; linear combination. This paper was typeset using AMS-LATEX. 1 2 F. QI and its first derivative 0(z) are respectively called the digamma and trigamma functions. See the papers [25, 27] and closely related references therein. This paper is motivated by a sequence of papers [2, 10, 11, 16, 18, 26, 30, 31, 36]. For detailed review and survey, please read the papers [18, 26, 30, 31, 36] and closely related references therein. In the paper [2], motivated by [10, 11], the function Γ(nx + 1) pkx(1 − p)(n−k)x (1.1) Γ(kx + 1)Γ((n − k)x + 1) was considered, where p 2 (0; 1) and k; n are nonnegative integers with 0 ≤ k ≤ n. In [16, Theorem 2.1] and [36], the function Γ1 + x Pn λ n k=1 k Y λkx Qn pk (1.2) Γ(1 + λkx) k=1 k=1 was studied independently by those authors, where n ≥ 2, λk > 0 for 1 ≤ k ≤ n, Pn pk 2 (0; 1) for 1 ≤ k ≤ n, and k=1 pk = 1. In [18], the q-analogue Γ 1 + x Pn λ n q k=1 k Y λkx Qn pk (1.3) Γq(1 + λkx) k=1 k=1 of the function in (1.2) was investigated, where q 2 (0; 1), n ≥ 2, λk > 0 for Pn 1 ≤ k ≤ n, pk 2 (0; 1) for 1 ≤ k ≤ n with k=1 pk = 1, and Γq(z) is the q- analogue of the gamma function Γ(x). For information on q-analogues of the gamma function Γ(x), digamma function (z), and trigamma function 0(x), please refer to [18, 21, 38, 44] and closely related references therein. In [15, Theorem 2.1] and [30, Theorem 4.1], the functions Qm Qn i=1 Γ(1 + νix) j=1 Γ 1 + τjx Qm Qn (1.4) i=1 j=1 Γ 1 + λijx and Qm Qn i=1 Γ(1 + νix) j=1 Γ 1 + τjx Qm Qn ρ (1.5) i=1 j=1 Γ 1 + λijx Pn were respectively considered, where ρ 2 R and λij > 0, νi = j=1 λij, τj = Pm i=1 λij for 1 ≤ i ≤ m and 1 ≤ j ≤ n. In [26], the function θ θ Qm ν Qn τj [Γ(1 + νix)] i Γ 1 + τjx i=1 j=1 (1.6) ρλθ Qm Qn ij i=1 j=1 Γ 1 + λijx Pn Pm was discussed, where ρ, θ 2 R and λij > 0, νi = j=1 λij, τj = i=1 λij for 1 ≤ i ≤ m and 1 ≤ j ≤ n. In the paper [31], the function Pn θ ( λk) n !%x Γ1 + x Pn λ k=1 k=1 k Y λk n θ pk (1.7) Q ρλk k=1[Γ(1 + λkx)] k=1 was investigated, where n ≥ 2, ρ, %; θ 2 R, λk > 0 for 1 ≤ k ≤ n, and pk 2 (0; 1) for Pn 1 ≤ k ≤ n with k=1 pk = 1. A NEW RATIO OF FINITE MANY GAMMA FUNCTIONS 3 There are a lot of literature on various ratios of gamma functions. For a much complete list of related references before 2010, please refer to the review and survey articles [7, 19, 20, 33, 34] and closely related references. One can also find new results on several ratios of gamma functions in [5, 43], for example. In this paper, motivated by the above seven functions (1.1), (1.2), (1.3), (1.4), (1.5), (1.6), and (1.7), we will consider the function n ρw λθ Q [Γ(1 + λ x)] k k F (x) = k=1 k ; x 2 (0; 1); (1.8) (Pn w λ )θ Pn k=1 k k Γ 1 + x k=1 wkλk Pn where n ≥ 2, ρ ≥ 1, θ ≥ 0, wk; λk > 0 for 1 ≤ k ≤ n, and k=1 wk = 1. 2. A lemma For stating and proving our main results, we need the following lemma. Lemma 2.1. Let x H(x) = ; x 2 : ex − 1 R Pn Let α ≥ 0, n ≥ 2, λk 2 (0; 1) and wk 2 (0; 1) for 1 ≤ k ≤ n, and k=1 wk = 1. Then n !α+1 ! n X x X α+1 x wkλk H Pn ≤ wkλk H : (2.1) wkλk λk k=1 k=1 k=1 Proof. It is well known that the function H(x) is the generating function of the Bernoulli numbers. See [17, 24] and [39, Chapter 1]. In [26, Theorem 3.1], it was α 1 proved that the function x H x is convex on (0; 1) if and only if α ≥ 1. If f(x) is a convex function on an interval I ⊆ R and if n ≥ 2 and xk 2 I for 1 ≤ k ≤ n, then n ! n X X f wkxk ≤ wkf(xk); (2.2) k=1 k=1 Pn where wk 2 (0; 1) for 1 ≤ k ≤ n and k=1 wk = 1. If f(x) is a concave function, the inequality (2.2) is reversed. In the literature, the inequality (2.2) is called Jensen's discrete inequality for convex functions. See [12, Section 1.4] and [13, Chapter I]. α 1 Consequently, when α ≥ 1 and n ≥ 2, replacing f(x) in (2.2) by x H x yields n !α ! n X 1 X α 1 wkxk H Pn ≤ wkxk H wkxk xk k=1 k=1 k=1 Pn for xk 2 (0; 1) and wk 2 (0; 1) with k=1 wk = 1. Further replacing xk in the λk above inequality by x for x 2 (0; 1) yields the inequality (2.1). The proof of Lemma 2.1 is complete. Remark 2.1. In the papers [2, 16, 18, 26, 30, 31, 36], inequalities n !α+1 n X x X α+1 x λk H Pn ≥ λk H ; (2.3) λk λk k=1 k=1 k=1 m n m n X x X x X X x να+1H + τ α+1H ≥ 2 λα+1H ; (2.4) i ν j τ ij λ i=1 i j=1 j i=1 j=1 ij 4 F. QI or their special cases were used, where α ≥ 0, x > 0, λk > 0, λij > 0 for 1 ≤ i ≤ m Pn Pm and 1 ≤ j ≤ n, νi = j=1 λij, and τj = i=1 λij. The inequalities (2.3) and (2.4) α 1 can be deduced from convexity of the function x H x on (0; 1). This has been reviewed in the paper [26]. 3. Complete monotonicity Now we are in a position to state and prove our main results. Pn Theorem 3.1. If n ≥ 2, ρ ≥ 1, θ ≥ 0, wk; λk > 0 for 1 ≤ k ≤ n, and k=1 wk = 1, then the function F (x) defined by (1.8) has the following properties: (1) the function F (x) has a unique minimum on (0; 1); (2) the logarithmic derivative [ln F (x)]0 is an increasing function from (0; 1) onto 0" n n !θ+1# 1 X θ+1 X @ ρ wkλk − wkλk (1); 1A ; k=1 k=1 (3) the second derivative [ln F (x)]00 is a completely monotonic function on (0; 1).
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