Formal:
* Please inform us of...
* Kindly advise us on...
* We would appreciate it if you could tell us...
* We would be grateful if you could provide us with...
Informal:
* Tell us...
* Let us know...
* Give us a heads up about...
* Keep us in the loop about...
* Fill us in on...
More specific:
* Let us know if you have any questions.
* Let us know what you think.
* Let us know what your decision is.
* Let us know if you are interested.
* Let us know when you are available.
Please provide more context for a more accurate and suitable alternative. For example, what are you asking someone to "let you know"?