The main results of the paper sharpen the classical well-known inequalities between power means. As a consequence, the inequality
is proved for all , , which was conjectured by W. Janous, M. K. Kuczma and M. S. Klamkin [Problem 1598, Crux Math. 16, 299–300 (1990), per bibl.]. The methods of the paper are analytic and use majorization and Schur-convexity. Some geometric applications are also obtained.