Estimations of the weighted power mean by the Heron mean (Research on structure of operators using operator means and related topics)

Abstract

As the means generalizing the arithmetic and the geometric ones, the power mean and the Heron mean are known. For positive real numbers a and b, the weighted power mean P_{t, q}(a, b) and the weighted Heron mean K_{t, q}(a, b) for tin [0, 1] and qin mathbb{R} are defined by P_{t, q}(a, b)={(1-t)a^{q}+tb^{q}}^{frac{1}{q}} and K_{t, q}(a, b)= (1-q)a^{1-t}b^{t}+q{(1-t)a+tb}, respectively. In this report, as a generalization of Wu and Debnath's result on non-weighted means (the case t= frac{1}{2}), we get estimations of the weighted power mean by the weighted Heron mean. We also obtain the results for bounded linear operators on a Hilbert space, and some determinant and trace inequalities of matrices by using our main results.