  Mond and Pecaric introduced a method to simplify the determination of
complementary inequalities for Jensen's inequality by converting it into a
single-variable maximization or minimization problem of continuous functions.
This principle has significantly enriched the field of operator inequalities.
Our contribution lies in extending the Mond-Pecaric method from single-input
operators to multiple-input operators. We commence by defining normalized
positive linear maps, accompanied by illustrative examples. Subsequently, we
employ the Mond-Pecaric method to derive fundamental inequalities for
multivariate hypercomplex functions bounded by linear functions. These
foundational inequalities serve as the basis for establishing several
multivariate hypercomplex function inequalities, focusing on ratio
relationships. Additionally, we present similar results based on difference
relationships. Finally, we apply the derived multivariate hypercomplex function
inequalities to establish Sobolev embedding via Sobolev inequality for
hypercomplex functions with operator inputs.
