Inequalities for Weighted Means and Applications to Positive Definite Matrices

It is shown that if ai , ai)0 is1, . . . , n. with nis1 ais1 and if p)0, then n n y pq1.rr y1rp n 1rr ` r r ai aiF pH ai xqai . dx F ai ai 5 / is1 0 is1 is1 for rG1; and n 1rr n y pq1.rr y1rp n ` r a ar F pH a xqa . dx F a a i i i i i i / 5 is1 0 is1 is1 for rF1. Applications of these inequalities to positive definite matrices are also given.