Is citing in an abstract required to avoid plagiarism?
In the abstract, you want to inform the reader about the paper that you have written, you want to convince the reader that is it worthy of their time to actually read some of it. You should realize that for the bulk of the readers, the abstract is all that they will ever read of your paper and as such, the abstract should stand alone, it should be a self-contained story. Including references in an abstract will ask for an additional investment of time from the potential reader, which might chase them off.

Plagiarism in an abstract arises when you copy text from other sources and include it in the abstract itself. In such cases, you present work written by others as something that you have done yourself, which is clearly wrong.

Plagiarism of another nature could arise when you imply that previous results from others were actually performed by you. As the authors of the original work will have an interest in your paper, rest assured that they will be the first to send you a message (if it was not picked up in the peer review process) asking you why you presented yourself as the author of work that you did not do. 

Some final thoughts:
- An abstract is supposed to be short, so there is no room for things that do not belong there (a thorough literature review is one of these things)
- Clarity and honesty are essential. If you do not copy from other people and if you are Always clear (beyond a reasonable doubt) in your descriptions of the sources of the material that you present, you should be fine