Sharp Estimates of the Generalized Euler-Mascheroni Constant

الأصل · EN

Let a∈ (0, ∞), γ(a) be the Generalized Euler-Mascheroni Constant, and let align* &xₙ=1a+1/a+1++1/a+n-1-+n/a, &yₙ=1a+1/a+1++1/a+n-1-+n-1/a. align* In this paper, we determine the best possible constants αᵢ, βᵢ (i=1,2,3,4) such that the following inequalities align* 1/2(n+a)-α₁≤ &γ(a)-xₙ< 1/2(n+a)-β₁, 1/2(n+a)-α₂≤ &yₙ-γ(a)< 1/2(n+a)-β₂, 1/2(n+a)+α₃/(n+a)²≤ &γ(a)-xₙ<1/2(n+a)+β₃/(n+a)², 1/2(n+a-1)+α₄/(n+a-1)²< &yₙ-γ(a)≤1/2(n+a-1)+β₄/(n+a-1)². align* are valid for all integers n≥ 1.