はじめまして what is the exact meaning?
I know that the English translation of はじめまして is "How do you do"
and どうぞよろしく is translated "Nice to meet you"
But searching for the roots of the meaning i try to translate it literally so
はじめましてどうぞよろしく is "we meet for the first time, by all means, please remember me" but I am not sure about まして.
Am I correct?

Question

はじめまして what is the exact meaning?
I know that the English translation of はじめまして is "How do you do"
and どうぞよろしく is translated "Nice to meet you"
But searching for the roots of the meaning i try to translate it literally so
はじめまして どうぞよろしく  is "we meet for the first time, by all means, please remember me" but I am not sure about まして.
Am I correct?

Answer

I think it  is an omission of 
"（私は）はじめ(てあなたにお会いし)まして、（今後もお世話になると思いますが）どうぞよろしく（お願いします。）"
"（I am meeting you) for the first time and,(from now, I need your help)please good to me (please). 
まして can  separate まし て.
ましis  cojnugated form of ます. 
て is omission of そして means " and" . 
But often we omission sentense after て because we can understand after words.It is idiomatic phrase.
ex)どういたしまして（お気になさらないでください。） お気をつけて（お帰り下さい。）

Answer

No you are not correct by what I am reading.
はじめまして means Nice to meet you.
どうぞよろしく means please to meet you.