Updating and Inserting data in one web service call

I am new to Web Service. Recently I have faced below question in an interview. For example if I have to expose a library application service through REST API. How would I implement a service to INSERT a new book details and update the user details with the book rent details in the single service call. Here the dilemma is whether to use PUT or POST verb for the service. Because we are doing both UPDATE and INSERT operations in the single service call.