Let x be an element of a finite group G. It is clear that 〈 x 〉 ≤ C G ⁢ ( x ) ≤ G {\langle x\rangle\leq C_{G}(x)\leq G} . For the cases where C G ⁢ ( x ) = G {C_{G}(x)=G} and C G ⁢ ( x ) = 〈 x 〉 {C_{G}(x)=\langle x\rangle} for any element x of G, the structure of G is easily decided. In this paper, we investigate the case where C G ⁢ ( x ) {C_{G}(x)} is maximal in G and obtain the structure of G.

In this paper, we employ iteration on operator version of the famous Young inequality and obtain more arithmetic-geometric mean inequalities and the reverse versions for positive operators. Concretely, we obtain refined Young inequalities with the Kantorovich constant, the reverse ratio type and difference type inequalities for arithmetic-geometric operator mean under different conditions.