My Mail Inbox needs rebuilding after every startup.

My Mail Inbox needs rebuilding after every startup: deleting messages does not remove them from the inbox list until I have rebuilt the inbox. Also new mail is not flagged on the Dock icon. The inbox is not over-full (usually no 6 messages left in it at end of day). I have checked all my settings. Running Mavericks on a 2012 iMac. Any suggestions please?