We have got a deprecated domain www.deprecateddomain.com. Specific fact is that we have got reverse proxy working and redirecting all requests from this domain to the new one www.newdomain.com. The problem is when you type "deprecateddomain.com" in google search, there is a link to www.deprecateddomain.com in search results besides results with "newdomain.com". It means that there is such entries in google index. Our customer don't want to see links to old site. We were suggested to create fake robots.txt with Disallow: / directive for www.deprecateddomain.com and reverse proxy rules to get this file from some directory. But after investigation the subject I started hesitating that it will help. Will it remove entries with old domain from index? Why not to just create the request in search console to remove www.deprecateddomain.com from index? In my opinion it might help. Anyway, I'm novice in this question. Could you give me advice what to do?
Google takes time to remove old/obsolete entries from its ranking, especially on low visited or low value pages. You have no control on it. Google needs to revisit each page to see the redirection you have implemented.
So DO NOT implement a disallow on the old website, because it will make the problem worse. Bots won't be able to crawls those pages and see the redirection you have implemented. So they will stay longer in the rankings.
You must also make sure you implement a proper 301 redirection (i.e. a permanent one, not a temporary) for all pages of the old website. Else, some pages may stay in the ranking for quite some time.
If some pages are obsolete and should be deleted rather than redirected, return a 404 for them. Google will remove them quickly from its index.