To configure your push vs pull (fetch) mail settings on iPhone:
First, tap your Settings icon, then scroll down and tap Mail Contacts Calendars:
To check your settings, tap on Fetch New Data:
Within the Fetch New Data settings you can choose which accounts automatically updates vs. those that require you to manually pull the screen to refresh:
I recommend that Fetch be set to manual to help prevent your battery from being drained unless you have an account that you require notify you as new emails are received.