Hi randilp1,
I was wondering if you found a solution yet. I have the same problem with my new Mac book pro. Emails send to myself (trough gmail) arrive just fine when send from my iPhone or iPad. However, when send from my Mac book they very briefly arrive on my iPhone and iPad and then disappear from those inboxes again (They are not moved / filtered and the email does not show up in the send items folder on these devices. However, on the Mac book, the emails show up and remain in the inbox just fine, when sending the email to multiple recipients the same happens, but the email is received by other recipients just fine.
Thank you in advance for any insights you may have!