  This paper gives particular emphasis to the nabla tempered fractional
calculus, which involves the multiplication of the rising function kernel with
tempered functions, and provides a more flexible alternative with considerable
promise for practical applications. Some remarkable inequalities for such nabla
fractional calculus are developed and analyzed, which greatly enrich the
mathematical theory of nabla tempered fractional calculus. Numerical results
confirm the validity of the developed properties once again, which also reveals
that the introduction of tempered function provides high value and huge
potential.
