I am referring above all to mezuzahs placed on rented spaces in the Diaspora before 30 days have elapsed. In the case of a rented space outside Eretz Yisroel, mezuzah does not become an obligation until 30 days have elapsed from when one moves in. Therefore, one customarily would not say a bracha if affixing the mezuzot before 30 days. (Source: https://judaism.stackexchange.com/a/132/1516.)
From that source: "If one wishes to affix the Mezuzah but delay recitation of the blessing until the thirtieth day, it should be touched before reciting the blessing. However, it is preferable to remove and reaffix the Mezuzah before reciting the blessing." Note that this does not indicate whether it would be /permissible/ to make a blessing if affixing before the 30 days. Furthermore, there are some sources ( http://dinonline.org/2011/05/30/mezuzah-before-thirty-days/ ) which suggest that mezuze is an obligation before 30 days if one plans to stay there longer. This would imply that a blessing could be made--although again they are not explicit.
What is the answer? Can one make a blessing when putting up a mezuzah before 30 days on a rental outside Eretz Yisroel? (Assume one will be living there long-term.)