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For p&#x2208;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&#x2260;0,&#x02009;&#x02009;and&#x02009;&#x02009;Mp(a,b)=ab,&#x02009;&#x02009; p=0. In this paper, we establish two sharp inequalities as follows: (2/3)G(a,b)+(1/3)H(a,b)&#x02A7E;M&#x2212;1/3(a,b) and (1/3)G(a,b)+(2/3)H(a,b)&#x02A7E;M&#x2212;2/3(a,b) for all a,b&#x003E;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

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