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.