In reply to Turdus torquatus:
I connect to my work email on an Xperia Z3 Compact using IMAP and I get notifications. You can't get true push unless you use Exchange (as noted) as POP/IMAP are not push technologies (though IMAP is better than POP as it supports folder structures on the server etc), but you can set the check frequency up a bit.
In the email app, go into settings->accounts->your account->check frequency and set a reasonably frequent one (ignore the "push" option). Every 5 (the most frequent) will do the job. If you prefer to use less data, every 15 is probably adequate.
Edit: read it again and it seems you already do that. That's all you will be able to do unless you use a protocol supporting push, i.e. Exchange or Gmail.
Edit again: It seems from more research the iPhone supports this:
http://en.wikipedia.org/wiki/Push-IMAP and Android doesn't (despite having a menu option for it on the Xperia).
Neil
Post edited at 12:30