  The T product operation between two three order tensors was invented around
2011 and it arises from many applications, such as signal processing, image
feature extraction, machine learning, computer vision, and the multiview
clustering problem. Although there are many pioneer works about T product
tensors, there are no works dedicated to inequalities associated with T product
tensors. In this work, we first attempt to build inequalities at the following
aspects: (1) trace function nondecreasing and convexity; (2) Golden Thompson
inequality for T product tensors; (3) Jensen T product inequality; (4) Klein T
product inequality. All these inequalities are related to generalize celebrated
Lieb concavity theorem from matrices to T product tensors. This new version of
Lieb concavity theorem under T product tensor will be used to determine the
tail bound for the maximum eigenvalue induced by independent sums of random
Hermitian T product, which is the key tool to derive various new tail bounds
for random T product tensors. Besides, Qi et. al introduces a new concept,
named eigentuple, about T product tensors and they apply this concept to study
nonnegative (positive) definite properties of T product tensors. The final main
contribution of this work is to develop the Courant Fischer Theorem with
respect to eigentuples, and this theorem helps us to understand the
relationship between the minimum eigentuple and the maximum eigentuple. The
main content of this paper is Part I of a serious task about T product tensors.
The Part II of this work will utilize these new inequalities and Courant
Fischer Theorem under T product tensors to derive tail bounds of the extreme
eigenvalue and the maximum eigentuple for sums of random T product tensors,
e.g., T product tensor Chernoff and T product tensor Bernstein bounds.
