Autor you might need an add on that sends notifications in the browser. However I do not know what mail service you have. Check out addons.mozilla.org.

auctor said

When new email messages come in, the number of new messages is not indicated in the email tab. Is there some way to set that up. Please advise. Thanks!

Thanks for your reply. Since I posted, I've been seeing the new email messages indicated in the email tab. So, I guess It's working fine.