Sorry, your question is a but confusing.
The iPhone does not have an iTunes App.
It has an iTunes Store app and a Music app, neither of which offer a login option directly. Which one are you referring to?
Both Apps use the account logged in at Settings ➜ Account ➜ Media & Purchases.
If there is no Apple Account signed in there it should offer a Sign In option, otherwise it will have the Sign Out option in red at the top, a View Account button below that and a Cancel button at the bottom of the short menu.
If the account is correctly logged in, the iTunes Store app will show the Music tab page of the store and the Music app will just show the Home Tab by default.
If you explain exactly try what you are trying to do, someone may be able to advise a course of action. Also why is it wrong for it to go back to the Home Screen after signing in?
What can't you do? Or what is the issue exactly?