REFINED YOUNG INEQUALITY WITH KANTOROVICH CONSTANT

The Specht ratio S(h) is the optimal constant in the reverse of the arithmetic-geometric mean inequality, i.e., if 0 < m <= a, b <= M and h = M/m, then (1 - mu)a + mu b <= S(h)a(1-mu) b(mu) for all mu is an element of [0, 1]. Recently S. Furuichi proved that (1 - mu)a + mu b >= S(h(r))a(1-mu) b(mu) for a, b > 0, mu is an element of [0, 1], where h = b/a and r = min{mu, 1 - mu}. In this paper, we improve it by virtue of the Kantorovich constant, utilizing the refined scalar Young inequality we establish a weighted arithmetic-geometric-harmonic mean inequality for two positive operators. In the remainder of this work we focus on extending the refined weighted arithmetic-harmonic mean inequality to an operator version for another type of improvement.