A harmonic mean inequality for the polygamma function
نویسندگان
چکیده
منابع مشابه
A harmonic mean inequality for the digamma function and related results
We present some inequalities and a concavity property of the digamma function ψ = Γ′/Γ, where Γ denotes Euler’s gamma function. In particular, we offer a new characterization of Euler’s constant γ = 0.57721.... We prove that −γ is the minimum of the harmonic mean of ψ(x) and ψ(1/x) for x > 0. Mathematics Subject Classification (2010). 33B15, 39B62, 41A44.
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For m,n ∈ N, let fm,n(x) = ψ (x)+ ˆ ψ(x) ̃2 for x > 0. In the present paper, we prove that f1,2(x) is the only nontrivial completely monotonic function on (0,∞). Accurately, the functions f1,2(x) and fm,2n−1(x) are completely monotonic on (0,∞), but the functions fm,2n(x) for (m,n) 6= (1, 1) are not monotonic and does not keep the same sign on (0,∞).
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ژورنال
عنوان ژورنال: Mathematical Inequalities & Applications
سال: 2020
ISSN: 1331-4343
DOI: 10.7153/mia-2020-23-06