A note on variants of Buchstab's identity

The author proves variants of Buchstab’s identity on sieve functions, refining the previous work on new iteration rules of Brady. The main tool used in the proof is a special form of combinatorial identities related to the binomial coefficients. As a by–product, the author obtains better inequalities of $F_{\kappa }(s)$ and $f_{\kappa }(s)$ for dimensions $\kappa >1$.