Learn English – Which polite formula to say “Likewise”

In the social standing of Karen in All About Eve, "aren't" is considered proper (ie standard) usage. So when "she" switches to "ain't I" she is using what Karen and most her ilk would consider the improper form. So, by using the so-called improper form, she is affirming, merely by changing to the "less elegant" word, that she is "being rude now."

"Aren't" and "Ain't" do have the same meaning in that sentence. She is not asking which word to use; she is deliberately using a word that she knows that Karen, who has "unyielding good taste," would find inelegant, objectionable, indicative of less education, and rude. So with one word, she assaults Karen's sophisticated tastes and, by the way, her Radcliffe education.

There are many ways to express this idea. The one I use most often at work when I would like someone to feel free to ask me questions is:

Please don't hesitate to contact me if you have any questions or concerns.

Another way you might express it is:

Please let me know if you have any questions.

To make it more polite you would request that the reader contact you by using "please"; Omitting the "please" sometimes makes it seem like a demand instead of a request. In the right context something like "Let me know if you have any questions." is fine. The "please" just makes sure the tone comes across as polite.

Also, if you want to be very deferential to someone like a professor, you could imply that the reason they might have a question is due to some fault of yours, for example you didn't explain something clearly or you may have misunderstood their message. For example,

Please ask if something I've written isn't clear.