Apple mail inbox not updating
Before going any further, you should double-check to make sure you have installed any software updates and mail updates that are waiting, those updates may very well fix the issue you are experiencing with new emails not showing up.
The easiest solution for many has been to simply relaunch Mail app every time users want to check their mail with the trouble providers.
This does work to reestablish a connection to the remote mail servers, but it’s a tad clunky to say the least.
Apple releases updates to their operating system any time they identify a fix to an issue.
That should resolve whatever Mail problems have occurred in OS X after installing OS X 10.10.4 or OS X 10.11 El Capitan, but if you continue to encounter trouble, specifically with SMTP mail accounts, these solutions should do the trick.
Not all users encounter this issue with Mail app after installing an OS X update, and though the reason is unclear, it’s possible the problematic setups had configured an email account some time ago, and could be using legacy settings for a specific mail server.
BT mail worked fine on my i Phone and PC but NOT on the i Pad.
Solved the issue by simply deleting the 'Mail' app and reinstating from the app store ..