Mail-app not using Push
This may solve your issue.


Open Settings app.
Open Mail, Contacts, Calendars
Under the Accounts heading, open "Fetch New Data".
Be sure that Push is set to on.
Open Advanced at the bottom.
Under each individual account, make sure that it is set to Push.  If push is not an option, then your email provider does not support push.  The only ones that I am sure of are iCloud, Microsoft Exchange, and Gmail (if setup as an Exchange account).