Email for iOS

The first thing to do is to check with your email provider. Open their webmail in a web browser and see if the emails are in the their trash/spam folder. If yes, you need to adjust their settings to let you mail through (Give them a call, they'll tell you how to proceed).

If not, your iPhone may be filtering the mail. Do you have any email rules set up? If yes, disable them and see if you emails come through.

Lastly, if you have a computer (besides your iPhone) with the same email account, do you see the same behavior?