In reviewing a paper I notice that the result presented there can be improved. However, the improvement may not be significant enough to appear in a good journal. Therefore, to me the best situation would be to have the authors' permission and to join the paper under review.
I would like to be advised of what I may do. I think I may write to the editor, and I would especially like to know how to ask politely. Would anyone have similar experience?
Edit: First of all thanks a lot for all suggestions! I just forgot to mention that my report was already sent one month ago. Would there be something extra I should take care of?
Edit 2: Again, many thanks to you guys! Actually contacting the authors is never an option in my mind. My options are only: (i) tell the editor (not recommending myself as a coauthor); (ii) write my own note. Here note means a short paper, since my new short proof may be of ten lines (the original proof is three pages), that's the reason I would like to consult you guys, and to see if there is a third option. At the moment, I prefer to do the first option, and I am thinking of whether in the email to editor I should include my new proof, or just say "I may further provide the details". How do you guys think?