Generalized Converses of Operator Jensens Inequalities with Applications to Hypercomplex Function Approximations and Bounds Algebra

Abstract

Mond and Pecaric proposed a powerful method, namd as MP method, to deal with operator inequalities. However, this method requires a real-valued function to be convex or concave, and the normalized positive linear map between Hilbert spaces. The objective of this study is to extend the MP method by allowing non-convex or non-concave real-valued functions and nonlinear mapping between Hilbert spaces. The Stone-Weierstrass theorem and Kantorovich function are fundamental components employed in generalizing the MP method inequality in this context. Several examples are presented to demonstrate the inequalities obtained from the conventional MP method by requiring convex function with a normalized positive linear map. Various new inequalities regarding hypercomplex functions, i.e., operator-valued functions with operators as arguments, are derived based on the proposed MP method. These inequalities are applied to approximate hypercomplex functions using ratio criteria and difference criteria. Another application of these new inequalities is to establish bounds for hypercomplex functions algebra, i.e., an abelian monoid for the addition or multiplication of hypercomplex functions, and to derive tail bounds for random tensors ensembles addition or multiplication systematically.