Some Inequalities Involving Weighted Power Mean

In this paper, we first show some inequalities on weighted power mean. When a, b > 0, p ≥ 1and 0 <v ≤ τ< 1, we have vτ≤a♯p,vb-a♯vba♯p,τb-a♯τb≤1-v1-τ \frac{v}{\tau } \le \frac{{a{\sharp _{p,v}}b - a{\sharp _v}b}}{{a{\sharp _{p,\tau }}b - a{\sharp _\tau }b}} \le \frac{{1 - v}}{{1 - \tau }} and vτ≤a♯p,vb-a!vba♯p,τb-a!τb≤1-v1-τ. \frac{v}{\tau } \le \frac{{a{\sharp _{p,v}}b - a{!_v}b}}{{a{\sharp _{p,\tau }}b - a{!_\tau }b}} \le \frac{{1 - v}}{{1 - \tau }}. Further, we obtain the range of corresponding inequalities involving the m power form of weighted power mean in the same form as above for m ∈ ℕ⁺ or p ≥ m> 0, m ≤ p< 0. As further applications, we provide some inequalities about matrices and determinants, respectively.