Is "put to the shame" not appropriate to be mentioned within a book foreword? is it considered rude? or it is just a normal expression and anyone can use it in any situation including formal situations?
Do you mean put to shame? There's no the in that idiom. If we're speaking about the same thing, meaning to embarrass or make someone feel ashamed, or also to one-up someone, then no, there is no rudeness inherent in this idiom.
What Wrey said Put to shame just means better than as in "your shiny range rover puts my rusting pick up truck to shame" You can also have "Put the shame on" which is an Irish idiom - "that lad or yours puts the shame on you"... ie you should be ashamed of how he behaves... but this isn't in common use outside of Irish communities
