Kantorovich type inequalities for ordered linear spaces

Abstract

In this paper Kantorovich type inequalities are derived for linear spaces endowed with bilinear operations ◦1 and ◦2. Sufficient conditions are found for vector-valued maps Φ and Ψ and vectors x and y under which the inequality Φ(x)◦2 Φ(y) ≤ C + c / 2√Cc Ψ(x◦1 y) is satisfied. Complementary inequalities are also given. Some results of Dragomir [J. Inequal. Pure Appl. Math., 5 (3), Art. 76, 2004] and Bourin [Linear Algebra Appl., 416:890–907, 2006] are generalized. The inequalities are applied to C∗-algebras and unital positive maps.