Two Sharp Inequalities for Power Mean, Geometric Mean, and Harmonic Mean

For p∈R, the power mean of order p of two positive numbers a and b is defined by Mp(a,b)=((ap+bp)/2)1/p,p≠0,  and  Mp(a,b)=ab,   p=0. In this paper, we establish two sharp inequalities as follows: (2/3)G(a,b)+(1/3)H(a,b)⩾M−1/3(a,b) and (1/3)G(a,b)+(2/3)H(a,b)⩾M−2/3(a,b) for all a,b>0. Here G(a,b)=ab and H(a,b)=2ab/(a+b) denote the geometric mean and harmonic mean of a and b, respectively