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For positive real numbers a and b , the weighted power mean Pt,q(a,b) and the weighted Heron mean Kt,q(a,b) are defined as follows: For t ∈ [0,1] and q ∈ R , Pt,q(a,b) = {(1− t)aq + tbq} q and Kt,q(a,b) = (1− q)a1−tbt + q{(1− t)a+ tb} . These means generalize the arithmetic and geometric ones. In this paper, as a generalization of Wu and Debnath’s result on non-weighted means (the case t = 2 ), we get estimations of the weighted power mean by the weighted Heron mean. In other words, we obtain the greatest value α1 = α1(t,r) and the least value α2 = α2(t,r) such that the double inequality Kt,α1 (a,b) < Pt,r(a,b) < Kt,α2 (a,b) holds for t ∈ (0,1) and r ∈ R . We can also obtain the results for bounded linear operators on a Hilbert space. Moreover, our main results lead some determinant and trace inequalities of matrices. Mathematics subject classification (2010): 26E60, 15A45, 47A63.

Estimations of the weighted power mean by the Heron mean and related inequalities for determinants and traces