LOWER BOUND OF SECTIONAL CURVATURE OF MANIFOLD OF BETA DISTRIBUTIONS AND COMPLETE MONOTONICITY OF FUNCTIONS INVOLVING POLYGAMMA FUNCTIONS

In the paper, by convolution theorem for the Laplace transforms and analytic techniques, the author finds necessary and sufficient conditions for complete monotonicity, monotonicity, and inequalities of several functions involving polygamma functions. By these results, the author derives a lower bound of a function related to the sectional curvature of the manifold of the beta distributions. Finally, the author poses several guesses and open problems related to monotonicity, complete monotonicity, and inequalities of several functions involving polygamma functions.

Recall from Chapter XIII in [9], Chapter 1 in [21], and Chapter IV in [22] 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 ∈ {0} ∪ N, where N denotes the set of all positive integers, then we call f (x) a completely monotonic function on I. Theorem 12b in [22, p. 161] characterized that a function f (x) is completely monotonic on (0, ∞) if and only if where σ(s) is non-decreasing and the integral in (1.1) converges for x ∈ (0, ∞). The integral representation (1.1) means that a function f (x) is completely monotonic on (0, ∞) if and only if it is a Laplace transform of a non-decreasing measure σ(s) on (0, ∞). In [7,Proposition 3] and [8,Proposition 13], the sectional curvature K(x, y) of the manifold of the beta distributions was given by In [7,Proposition 4] and [8,Proposition 14], the following limits were computed: In [14,Theorem 4.1] and [16,Theorem 4], the author presented that (1) if and only if α ≥ 2, the function H α (x) = Φ (x) + αΦ 2 (x) is completely monotonic on (0, ∞) (2) if and only if α ≤ 1, the function −H α (x) is completely monotonic on (0, ∞); (3) the double inequality −2 < Φ (x) Φ 2 (x) < −1 is valid on (0, ∞) and sharp in the sense that the constants −2 and −1 cannot be replaced by any bigger and smaller numbers respectively.
In [14,Theorem 1.1], the author found the following necessary and sufficient conditions and limits: (2) if β ≤ 1, the function H β (x) is increasing on (0, ∞), with the limits In [10], the author considered the functions and J 0,µ0 (x) = H µ0 (x) for λ 0 = α and µ 0 = β. In [10, Theorems 3.1 and 4.1], the author discovered the following necessary and sufficient conditions, limits, and double inequality: (4) if µ k ≤ 1, the function J k,µ k (x) is increasing on (0, ∞), with the limits is valid on (0, ∞) and sharp in the sense that the lower and upper bounds cannot be replaced by any larger and smaller numbers respectively. In the paper [15], the author considered the functions In [15, Theorems 3.1 and 4.1], the author presented the following necessary and sufficient conditions, limits, and double inequalities: (1) if and only if θ k ≥ 3(2k+2)! k!(k+1)! , the function G k,θ k (x) is completely monotonic on (0, ∞); (2) if and only if θ k ≤ 0, the function −G k,θ k (x) is completely monotonic on (0, ∞); (3) if and only if τ k ≥ 2, the function G k,τ k (x) is decreasing on (0, ∞); the function G k,τ k (x) is increasing on (0, ∞); (6) the following two limits are valid: 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.
In [7,Proposition 5], the sectional curvature K(x, y) was proved to be negative and bounded from below. On 19 February 2020, Alice Le Brigant, the first author of the papers [7,8] told the author of this paper via e-mails and the ResearchGate that the lower bound of K(x, y) should be − 1 2 .
In this paper, we consider the function on (0, ∞) and prove the sharp double inequality which verifies the first conjecture in Conjecture 1.1 along the half-line x = y > 0 in the first quadrant on R 2 .

Lemmas
The following lemmas are necessary in this paper.
Proof. These two limits can be found in [5, p. 9896 Then the following conclusions are valid: (1) the function g(t) (a) satisfies the identity , and logarithmically concave on (−∞, ∞); (2) the function g(2t) g 2 (t) is increasing from (−∞, 0) onto (0, 1) and decreasing from (0, ∞) onto (0, 1); (3) the double inequality is valid on (0, ∞) and sharp in the sense that the lower bound 0 and the upper bound 1 cannot be replaced by any larger scalar and any smaller scalar respectively; (4) for any fixed t > 0, the function g(st)g((1 − s)t) is increasing in s ∈ 0, 1 2 . Proof. The verification of the identity (2.3) is straightforward.
The proof of Lemma 2.7 is complete.
For k, ∈ N and a ≥ 0, we have and where we used the limit (2.2).

Necessary and sufficient conditions of complete monotonicity
For verifying the lower bound in the double inequality (1.3), we find a lower bound for the second factor in (1.2) and more.
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.
Proof. The sufficient conditions were proved in [3, Theorem 4.1]. The first derivative of the function F p,m,n,q;cp,m,n,q (x) is and, for q ≥ 1, If ±F p,m,n,q;cp,m,n,q (x) is completely monotonic, then ±F p,m,n,q (x; c p,m,n,q ) ≤ 0 which are equivalent to c p,m,n,0 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 November 2020 doi:10.20944/preprints202011.0315.v1 for m+n = p+q, where we used the limits (2.1) and (2.2) in Lemma 2.1. Moreover, for p > m ≥ n > q > 0 such that m + n = p + q, we have Hence, necessary conditions are proved.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 November 2020 doi:10.20944/preprints202011.0315.v1 The double inequalities (3.1) and (3.2) come from the positivity of the functions ±F p,m,n,q;cp,m,n,q (x) and their sharpness can be concluded from the limits where we used the limits (2.1) and (2.2) in Lemma 2.1 once again. The proof of Theorem 3.1 is complete.
Then the following conclusions are true: k+1 , the function F k,ϑ k (x) is increasing on (0, ∞); (5) the following limits are valid: 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.
Proof. Taking q = 0, m = n = k, and p = 2k in Theorem 3.1 leads to that the function is completely monotonic on (0, ∞) if and only if c 2k,k,k,0 ≤ [(k−1)!] 2 (2k−1)! . This result is equivalent to that the function F k,η k (x) is completely monotonic on (0, ∞) if and For completeness, in what follows, we will prove this result in details once again.
If F k,η k (x) is completely monotonic on (0, ∞), then its first derivative which can be rewritten as If −F k,η k (x) is completely monotonic on (0, ∞), then its first derivative → 0 as x → 0 + , where we used the limit (2.1). From the integral representation for (z) > 0 and n ∈ N, see [1, p. 260, 6.4.1], it follows that (−1) k+1 ψ (k) (x) for all k ∈ N are completely monotonic on (0, ∞). Further considering the fact that the sum of finite completely monotonic functions is also completely monotonic, we see that the necessary condition η k ≤ 0 is also sufficient for −F k,η k (x) to be completely monotonic on (0, ∞).
It is not difficult to verify that for v ∈ (0, ∞) and k ∈ N. This implies that By Lemma 2.1, we obtain The proof of Theorem 3.2 is complete.

A completely monotonic function involving tetragamma function
For verifying the lower bound in the double inequality (1.3), we establish an upper bound for the third factor in (1.2) and more.
is completely monotonic on (0, ∞). Consequently, the double inequality is valid on (0, ∞) and sharp in the sense that the lower bound 0 and the upper bound 2 cannot be replaced by any greater number and any less number.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 November 2020 doi:10.20944/preprints202011.0315.v1 Since K(x, y) < 0 was proved in [7,Proposition 5], by the express (1.2), the lower bound in (4.1) is immediate. The upper bound of (4.1) comes from the complete monotonicity of the function I ν (x). The sharpness of the double inequality (4.1) can be deduced from the limits where we used the formulas (4.2) and (4.3), the limit (2.1) in Lemma 2.1, and the limits (2.6) and (2.8) in Lemma 2.8. The proof of Theorem 4.1 is complete.

The lower bound of sectional curvature
In this section, we prove the double inequality (1.3) and its sharpness.
Theorem 5.1. For x > 0, the double inequality 0 > K(x) > − 1 2 is valid on (0, ∞) and sharp in the sense that the lower bound − 1 2 and the upper bound 0 cannot be replaced by any larger scalar and any smaller scalar respectively.
Proof. By the double inequality (3.3) for k = 1 in Theorem 3.2, we obtain Combining this double inequality with the double inequality (4.1) gives where we used the expression (1.2) for K(x). By the limit (2.1), we obtain By the limit (2.2), we obtain where, by the second limit in (2.5), and, by the formulas (4.2) and (4.3) and by the limits (2.2) and (2.7), The proof of Theorem 5.1 is complete.

Several remarks, guesses, and open problems
Finally, we list several remarks, guesses, and open problems related to monotonicity, complete monotonicity, and inequalities of several functions involving polygamma functions. Remark 6.1. Theorem 3.1 has been generalized to divided cases in the papers [2,23].