Abstract
In the paper, the authors extend a function arising from the Bernoulli trials in probability and involving the gamma function to its largest ranges, find logarithmically complete monotonicity of these extended functions, and, in light of logarithmically complete monotonicity of these extended functions, derive some inequalities for multinomial coefficients and multivariate beta functions. These results recover, extend, and generalize some known conclusions.
Highlights
Let us denote by Pn,k(p) the probability of achieving exactly k successes in n Bernoulli trials with success probability p
In Section, in light of logarithmically complete monotonicity of Q(x) and Q(x), we will offer some inequalities for multinomial coefficients
In light of logarithmically complete monotonicity of Q(x) and Q(x), we offer some inequalities for multinomial coefficients
Summary
Let us denote by Pn,k(p) the probability of achieving exactly k successes in n Bernoulli trials with success probability p. Recall from [18, Chapter XIII], [37, Chapter 1], and [38, Chapter IV] that an infinitely differentiable and nonnegative function f (x) is said to be completely monotonic on an interval I if and only if 0 ≤ (−1)m−1f (m−1)(x) < ∞, m ≥ 2, x ∈ I. Recall from [5, 11, 27, 37] that a logarithmically completely monotonic function must be completely monotonic on the same defined interval, but not . In Section , we will verify that the function Q(x) is logarithmically completely monotonic on (0, ∞). In Section , we will show that the function Q(x) is logarithmically completely monotonic on (0, ∞). In Section , in light of logarithmically complete monotonicity of Q(x) and Q(x), we will offer some inequalities for multinomial coefficients.
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