Apple Mail on my iPhone is showing unread emails?

Welcome Rjtiedemann,

Thank you for using Apple Support Communities. Based on your post, your iPhone shows unread emails while your iPad and iMac do not. We'll do our best to assist.

It sounds like the email account is set up as a POP account on your iPhone. That means that when you check emails on one device, it does not sync over those changes to the server on your other devices. In order for your email account to show the same across all of your devices and the server, you'd want to use IMAP: What is the difference between POP and IMAP?

We'd recommend contacting your email provider to see if they have any IMAP settings for iOS. They can also probably walk through setting up the account on your iPhone or you're welcome to use: Add an email account to your iPhone, iPad, or iPod touch

Hoping this helps out, take care.