In this paper, we present a version of horizontal weighted Hardy-Rellich type
and Caffarelli-Kohn-Nirenberg type inequalities on stratified groups and study
some of their consequences. Our results reflect on many results previously
known in special cases. Moreover, a new simple proof of the Badiale-Tarantello
conjecture [2] on the best constant of a Hardy type inequality is provided. We
also show a family of Poincar\'e inequalities as well as inequalities involving
the weighted and unweighted $p$-sub-Laplacians.