Abstract
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.
Highlights
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
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
1 2 and the upper bound cannot be replaced by any larger scalar and any smaller scalar respectively
Summary
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. The double inequalities (3.1) and (3.2) come from the positivity of the functions ±Fp,m,n,q;cp,m,n,q (x) and their sharpness can be concluded from the limits ψ(m+n)(x) xm+n+1ψ(m+n)(x) lim. (2) if and only if ηk ≤ 0, the function −Fk,ηk (x) is completely monotonic on (0, ∞);. The sharp double inequality (2.4) implies that t ηk uk−1(t − u)k−1g(u)g(t − u)du − t2k−1g(t) ≤ 0 on (0, ∞) only if ηk ≤ 0, while t ηk uk−1(t − u)k−1g(u)g(t − u)du − t2k−1g(t) ≥ 0 on (0, ∞) only if ηk ≥. Taking p = 2k + 1, q = k, m = 2k, and n = k + 1 in Theorem 3.1 yields that the function (−1)k+1 ψ(2k)(x)ψ(k+1)(x) − c2k+1,2k,k+1,kψ(2k+1)(x)ψ(k)(x) and its negativity are completely monotonic on (0, ∞) if and only if (2k − 1)!k!
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