Vector state inequalities related to the Hölder–McCarthy inequality and applications

We present vector state inequalities related to the H?lder?McCarthy inequality for means of Hilbert space operators. Some particular cases of our result are improvements of known inequalities. As applications, we obtain order preserving results in the cone of positive operators. In particular, we show that if A is a positive operator whose spectrum contained in the interval [m,M] and if A ? B, then Ap ? Bp + (M ? m)pI for every p ? [1, 2]. We give some examples to compare our estimation to the existing results.