I beg to

is an elliptical expression for I beg leave to; as, I beg to inform you.

See also: Beg

References in classic literature ?
SIR -- I beg to inform you that I was yesterday called into a neighbor's in Vauxhall Walk to attend a young lady who had been suddenly taken ill.
It is very true," said the Poodle, with austere dignity, "that I am small; but, sir, I beg to observe that I am all dog.