  This work extends the Mond-Pecaric method to functions with multiple
operators as arguments by providing arbitrarily close approximations of the
original functions. Instead of using linear functions to establish lower and
upper bounds for multivariate functions as in prior work, we apply sigmoid
functions to achieve these bounds with any specified error threshold based on
the multivariate function approximation method proposed by Cybenko. This
approach allows us to derive fundamental inequalities for multivariate
hypercomplex functions, leading to new inequalities based on ratio and
difference kinds. For applications about these new derived inequalities for
multivariate hypercomplex functions, we first introduce a new concept called
W-boundedness for hypercomplex functions by applying ratio kind multivariate
hypercomplex inequalities. W-boundedness generalizes R-boundedness for norm
mappings with input from Banach space. Additionally, we develop an
approximation theory for multivariate hypercomplex functions and establish
bounds algebra, including operator bounds and tail bounds algebra for
multivariate random tensors.
