Sharp inequalities for the harmonic numbers

Let Hn be the nth harmonic number, and let γ be the Euler-Mascheroni constant. We prove that for all integers n ≥ 1, the double-inequality α − ln(e 2 √ 2/(4n+2+√ 2) − 1) < Hn ≤ β − ln(e 2 √ 2/(4n+2+√ 2) − 1) holds with the best possible constants α = γ − 1 2 ln 2 = 0.2306 . . . and β = 1 + ln(e 2 √ 2/(6+√ 2) − 1) = 0.2331 . . . . We also establish inequality for the Euler-Mascheroni constant √ 2 96n3 − 13 + 18√ 2 1152n4 < µn − γ < √ 2 96n3 , where µn = Hn + ln(e 2 √ 2/(4n+2+√ 2) − 1) + 1 2 ln 2 .