Estimations of power difference mean by Heron mean (The research of geometric structures in quantum information based on Operator Theory and related topics)

Abstract

In this report, we discuss estimations of power difference mean by Heron mean. We obtain the greatest value $alpha$=$alpha$(q) and the least value $beta$=$beta$(q) such that the double inequality K_{$alpha$}(a, b)<J_{q}(a, b)<K_{$beta$}(a, b) holds for any a, b>0 and q in mathbb{R}, where J_{q}(a, b)=overline{q}^{mathrm{L}{frac{a^{q+1}-b^{q+1}{a^{mathrm{q}-bmathrm{q}+1 is the power difference mean and K_{q}(a, b)=(1-q)displaystyle sqrt{ab}+qfrac{a+b}{2} is the Heron mean. We also get similar inequalities for bounded linear operators on a Hilbert space.