Excellent suggestions. Unfortunately (for me) I tried both suggestions last week. Since others read these posts, I will give as much information as I can in case someone may benefit. I have often found interesting information from other people's "woes".
1) The only differences I could detect were that the 'good' server runs PHP 5.3.23 and the 'bad' server runs PHP 5.3.14
Since phpinfo.php sends so much information, I can never be sure exactly what could effect my situation. I did an electronic search for any parameter containing 'mail' and I found six (6) directives containing 'mail' such as "mail.force_extra_parameters" which I understand and "Path to sendmail " which I really don't think I understand; i.e. relation to php command mail(). In any case, both servers were had identical directives.
2) I have asked Support at Lunarpages. For years, they have been very good and I can't really complain. On this one, all I get are messages such as "We have passed this question one to xxx."
I am going to try a workaround; namely redirect the command to a different server. Since that could be interpreted as hacking, I may get into trouble.