The only way to make ones application goes to background programatically is to open another app (Safari) calling its URL. But I don't think that is what you have in mind (since you are saying you have a radio app).
So if you want to imitate the same action which happens when user pressing home button, it is not possible with iOS SDK. From iOS human interface guildelines
Don’t Quit Programmatically
Never quit an iOS application programmatically because people tend to
  interpret this as a crash. However, if external circumstances prevent
  your application from functioning as intended, you need to tell your
  users about the situation and explain what they can do about it.
Apple also says in the documentation
People, not applications, should initiate and control actions.
  Although an application can suggest a course of action or warn about
  dangerous consequences, it’s usually a mistake for the app to take
  decision-making away from the user. The best apps find the correct
  balance between giving people the capabilities they need while helping
  them avoid dangerous outcomes.
So I don't think it is possible using public API.