Apple has solved the compatibility issue with Gmail for its Mail app with OS X Mavericks 10.9.1 update. But one bug remains which prevent Mail app from loading new incoming emails. For many users, and depending which email providers you’re using, new email only appear when the Mail app is first opened. And you have to quit and open Mail app in order to see new emails in your inbox.
It is quite certain that Apple will solve this annoying problem with the upcoming OS X Maverick 10.9.2 release. Until then, Apple has a workaround to manually trigger new incoming emails without quitting and reopen Mail app: select Mailbox > Take All Accounts Offline. And then select Mailbox > Get All New Mail.
You can create shortcuts by adding “Take All Accounts Offline” and “Get All New Mail” commands as buttons on Mail toolbar. Go to View > Customize Toolbar. Drag “Take All Accounts Offline” and “Get All New Mail” to the toolbar and click Done.