Your answer would be the following one: their, there, buy, and by.
Explanation:
The first option is the possessive determiner because you can see that the word is followed by a noun phrase. Possessive determiners are words that are used in front of nouns to express possession. They occur with a noun and they do not replace noun phrases as pronouns do.
The second option is the adverb "there". In this case, the adverb of place has been used to refer to a place that has already been mentioned (the convention). Then, the third option is the verb "buy". The word is preceded by the indefinite particle and followed by an NP which functions as the direct object. Finally, the last option is the preposition "by". That PP takes an NP complement "their favorite author".
