On power of singular cardinals

Using elementary methods we find bounds for the function. Using only ZFC without additional assumptions, when e.g., is strong limit of uncountable confinality: (1) If there is no weakly inaccessible below (formula presented), then there is no such cardinal below (formula presented) (2) If (formula presented) is the first cardinal such that (formula presented) with (formula presented) then (formula presented) K when K is the first cardinal such that (formula presented) with cofinality (formula presented) We shall also reprove some of Galvin and Hajnafs results. We do not require any knowledge of earlier results on the subject.