In this paper we give several remarks on [3] and a different proof of the inequalities in [8]. Also we introduce the concept of a lower perturbation function, and prove that some usual measures of noncompactness of operators and also some measures of non-strict-cosingularity of operators are lower perturbation functions. For each lower perturbation function $\delta$, we define functions $\nabla_\delta$ and $K_\delta$, and give the connection with lower semi-Fredholm operators.