Sharp Estimates of the Generalized Euler-Mascheroni Constant

2017/12/23 by Huang, Ti-Ren, Han, Bo-Wen, Liu, You-Ling +1
#11Y60 #33B15 #40A05 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1712.08799

Abstract

Let a∈ (0, ∞), γ(a) be the Generalized Euler-Mascheroni Constant, and let amp;x_n=\frac1a+(1)/(a+1)+⋯+(1)/(a+n-1)-ln(a+n)/(a),
amp;y_n=\frac1a+(1)/(a+1)+⋯+(1)/(a+n-1)-ln(a+n-1)/(a). In this paper, we determine the best possible constants α_i, β_i (i=1,2,3,4) such that the following inequalities \beginalign* (1)/(2(n+a)-α₁)≤ &γ(a)-x_n< (1)/(2(n+a)-β₁), (1)/(2(n+a)-α₂)≤ &y_n-γ(a)< (1)/(2(n+a)-β₂), (1)/(2(n+a))+(α₃)/((n+a)²)≤ &γ(a)-x_n

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