Inequalities for the harmonic numbers

We present various inequalities for the harmonic numbers defined by ${H_n=1+1/2 +ldots +1/n,(nin{bf N})}$ . One of our results states that we have for all integers n ???2: $$alpha , frac{log(log{n}+gamma)}{n^2} leq H_n^{1/n} -H_{n+1}^{1/(n+1)} < beta , frac{log(log{n}+gamma)}{n^2}$$ with the best possible constant factors $$alpha= frac{6 sqrt{6}-2 sqrt[3]{396}}{3 log(log{2}+gamma)}=0.0140ldots quadmbox{and} quadbeta=1.$$ Here, ?? denotes Euler??s constant.