Follow along with the video below to see how to install our site as a web app on your home screen.
Note: This feature may not be available in some browsers.
What e-mail app are you using ? The stock one or Gmail ? If you use the Gmail app, make sure under >>>Settings>>>Accounts and Sync>>>Auto-sync is turned on and your e-mail address shows "Sync is on" !!
In the Gmail app, under >>>settings>>>Account Settings>>>your e-mail address towards the bottom, you have set your "Sync inboxes and lables" correctly and also checked "E-Mail notifications".....As well as "Lables to notify"

That's strange, I wonder why ?? Maybe it has something to do with how the settings are for Gmail on someones computer..... I mean, there are lots of Gmail settings and maybe one of these settings is causing the trouble.
Maybe, afterall, it's not a phone or app issue, maybe a Gmail settings issue, a setting one would not think off under normal circumstances.....