Abstract
We present the necessary and sufficient conditions such that the functions involving \(R ( x ) =\psi ( x+1/2 ) -\ln x\) with a parameter are completely monotonic on \(( 0,\infty )\), find three new sequences which are fast convergence toward the Euler-Mascheroni constant, and give a positive answer to the conjecture proposed by Chen (J. Math. Inequal. 3(1):79-91, 2009), where ψ is the digamma function.
Highlights
The gamma and polygamma functions have attracted the attention of many researchers since they play
2 Lemmas In order to prove our results we need several lemmas, which we present
Motivated by inequalities ( . ) and ( . ) we propose two conjectures
Summary
A real-valued function f is said to be completely monotonic on the interval I if f has derivatives of all orders on I and satisfies (– )nf (n)(x) ≥. . f is said to be strictly completely monotonic on I if inequality Many results involving the quicker convergence toward the Euler-Mascheroni constant can be found in the literature [ – ]. Chen ([ ], Theorem ) proved that both H(n) and ((n+ / )/n) H(n) are strictly increasing and concave sequences, while ((n+ )/n) H(n) is a strictly decreasing and convex sequence, and conjectured that:. (ii) The function ((x + )/x) H(x) is strictly completely monotonic on ( , ∞). Where the last equality holds due to limt→∞(e–xtQ(t)) = limt→ (e–xtQ(t)) =.
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