Lower Bound of Sectional Curvature of Fisher–Rao Manifold of Beta Distributions and Complete Monotonicity of Functions Involving Polygamma Functions

In the paper, by virtue of 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 Fisher–Rao manifold of beta distributions.


Motivations
In the literature [1,Section 6.4], the function Γ(z) = The Fisher-Rao manifold of beta distributions refers to the space of parameters of the family of beta distributions, equipped with the Fisher-Rao metric. It is of interest in information geometry, an expanding field that studies statistical objects from a geometrical point of view.
In [2,Proposition 4], [3,Proposition 14], and [4,Proposition 15], the limits were computed. In [2,Proposition 5] and [4,Theorem 6], the sectional curvature K(x, y) was proved to be negative and bounded from below. 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 [6, p. 9896 Proof. It is easy to see that Direct differentiation yields for t > 0 and u ∈ (0, t). Accordingly, all the functions W (t, u) for 2 ≤ ≤ 4 are positive, increasing, and convex in u ∈ (0, t). Therefore, the function W 1 (t, u) is positive and increasing in u ∈ (0, t). Hence, the function W t (u) is increasing in u ∈ (0, t). As a result, the inequality W t (u) < 2 is sharp for t > 0 and u ∈ (0, t). The proof of Lemma 2.3 is complete.
where we used the limit (2.2). It is also straightforward that where we used the limits (2.2) and (2.4). The proof of Lemma 2.4 is complete.

Necessary and Sufficient Conditions of Complete Monotonicity
For verifying the lower bound in the double inequality (1.4), we find a lower bound for the second factor in (1.3) and more.
Theorem 3.1. Let p > m ≥ n > q ≥ 0 be integers such that m + n = p + q and let 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. Let Theorem 4.1 in [5] reads that (1) the function F p,m,n,q;cp,m,n,q (x) for q ≥ 1 or q = 0, and (2) the function −F p,m,n,q;dp,m,n,q (x) for q ≥ 1 are both completely monotonic on (0, ∞). It is clear that where we used p + q = m + n and 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. The double inequalities (3.3) and (3.4) come from the positivity of the functions ±F p,m,n,q;c (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: (4) if and only if ϑ k ≤ 2k+1 k+1 , the function F k,ϑ k (x) is increasing on (0, ∞); (5) the following limits are valid: (3.6) 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 the fact that the function where we used the limit (2.1). The integral representation for (z) > 0 and k ∈ N, see [1, p. 260, 6.4.1], means that the functions ψ (k) (x) for all k ∈ N are completely monotonic on (0, ∞). Further considering the fact that the sum of finitely many 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, ∞). Direct computation gives Taking p = 2k + 1, q = k, m = 2k, and n = k + 1 in Theorem 3.1 yields that the function and its opposite is completely monotonic on (0, ∞) if and only if and c ≥ (2k)!(k + 1)! (2k + 1)!k! = k + 1 2k + 1 respectively. Therefore, we conclude that, (1) if and only if ϑ k ≥ 2, the derivative F k,ϑ k (x) ≤ 0, and then the function F k,ϑ k (x) is decreasing, on (0, ∞); (2) if and only if ϑ k ≤ 2k+1 k+1 , the derivative F k,ϑ k (x) ≥ 0, and then the function F k,ϑ k (x) is increasing, on (0, ∞).
The double inequality (3.7) follows from the decreasing property of the function F k,2 (x) on (0, ∞) and the limits in (3.5) and (3.6) for ϑ k = 2. The proof of Theorem 3.2 is complete.

A Completely Monotonic Function Involving Tetragamma Function
For verifying the lower bound in the double inequality (1.4), we establish an upper bound for the third factor in (1.3) 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.
Proof. Utilizing the duplication formula in [1, p. 259, 6.3.8] gives and Let Then, by the integral representation (3.8) and Lemma 2.2, we obtain Employing Lemma 2.3 and the positivity of g(t) yields that, when ν ≥ 2, the function 4I ν (x) is completely monotonic on (0, ∞). By the expression in (4.1), if I ν (x) is completely monotonic on (0, ∞), then I ν (x) ≥ 0, which is equivalent to The sharpness of the double inequality (4.2) can be deduced from the limit in (4.5) and the limit

The Lower Bound of Sectional Curvature
In this section, we prove the double inequality (1.4) and its sharpness.  The sharpness follows. The proof of Theorem 5.1 is complete.