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.
