富永 雅

In our previous paper, we obtained a reverse Holder's type inequality which gives an upper bound of the difference: (Sigma a(k)(p))(1/p)(Sigma b(k)(q))(1/q) - lambdaSigma a(k)b(k) with a parameter lambda > 0, for n-tuples a = (a(1),..,a(n)) and b = (b(1),....b(n)) of positive numbers and for p > 1, q > 1 satisfying 1/p + 1/q = 1. In this paper for commutative positive operators A and B on a Hilbert space H and a unit vector x is an element of H, we give an upper bound of the difference <A(p)x, x>(1/p)<B(q)x, x>(1/q) - lambda<ABx, x>. As applications, considering special cases, we induce some difference and ratio operator inequalities. Finally, using the geometric mean in the Kubo-Ando theory we shall give a reverse Holder's type operator inequality for noncommutative operators.

We prove a Golden-Thompson type inequality via Specht's ratio: Let H and K be selfadjoint operators on a Hilbert space H satisfying M1 greater than or equal to H, K greater than or equal to mI for some scalar M > m. Then M-h(1) (1 - lambda)e(tH) + lambdae(tK))(1/l) greater than or equal toe((1-lambda)H+lambdaK) greater than or equal to M-h(1)M--1(h)(t)(-1/l) ((1 - lambda)e(tH) + lambdae(tK))(1/l) holds for all t > 0 and 0 less than or equal to lambda less than or equal to 1, where h = e(M-m) and (generalized) Specht's ratio M-h(t) is defined for h > 0 as [GRAPHICS] (h not equal 1) and M1(1) = 1.

Using a technique due to Ozeki, we give an upper bound of (Sigmaa(k)(p))(1/p) (Sigmab(k)(q))(1/q) - lambda Sigmaa(k)b(k) for lambda > 0, for p > 1, q > 1 satisfying 1/p + 1/q = 1, and for n - tuples a = (a(1),...,a(n),) and b = (b(1),..., b(n)) of positive numbers under certain conditions. This yields a complement of Holder's inequality. The estimation with a parameter lambda enables us to unify the discussions on difference and ratio inequalities derived from Holders inequality.

The aim of this work is to generalize the main inequalities in [9] as follows: Let A be a Hermitian matrix, let Phi be a normalized positive linear map, let f and g be real valued continuous functions and let F(u, v) be a real valued function matrix non-decreasing in its first variable. Real constants alpha and beta such that alpha1 less than or equal to F[Phi>(*) over bar * (f(A)),g(Phi>(*) over bar * (A))] less than or equal to beta1 are determined. If f is a concave (resp. convex) function then the determination of beta (resp. alpha) is reduced to solving a single variable maximization (resp. minimization) problem. Some applications of these results to the power function, the means and the Hadamard product are also given.