The main results of the paper sharpen the classical well-known inequalities between power means. As a consequence, the inequality \[ \left(\sum_{i=1}^n x_i\right)^n \leq (n-1)^{n-1} \sum_{i=1}^n x_i^n + n\big(n^{n-1}-(n-1)^{n-1}\big)\prod_{i=1}^n x_i \] is proved for all \(x_1,\dots,x_n>0\), \(n\geq2\), which was conjectured by \textit{W. Janous, M. K. Kuczma} and \textit{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.