  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.
