Optimal sublinear inequalities involving geometric and power means

summary:There are many relations involving the geometric means $G_n(x)$ and power means $[A_n(x^{\gamma })]^{1/\gamma }$ for positive $n$-vectors $x$. Some of them assume the form of inequalities involving parameters. There then is the question of sharpness, which is quite difficult in general. In this paper we are concerned with inequalities of the form $(1-\lambda )G_n^{\gamma }(x)+\lambda A_n^{\gamma }(x)\geq A_n(x^{\gamma })$ and $(1-\lambda )G_n^{\gamma }(x)+\lambda A_n^{\gamma }(x)\leq A_n(x^{\gamma })$ with parameters $\lambda \in \Bbb R$ and $\gamma \in (0,1).$ We obtain a necessary and sufficient condition for the former inequality, and a sharp condition for the latter. Several applications of our results are also demonstrated