iRule variable match is case sensitive

The "tolower" function converts a string to lowercase, it does not make string match case insensitive.

Correct syntax is:

if (tolower(http_uri) starts_with "/authentication") {
  use pool IIS_Pool_80
} else {
  use pool beta.pmi.org
 }

BTW, the best way how to match one of several strings is to use a class and the "one of" operator. It is more efficient than several string match experssions ORed and also more efficient than a chain of if-else statements. One of your previous rule examples could be rewritten like this:

class allowed_uris {
  "/certapp"
  "/authentication"
  "/lms"
}
rule r1 {
  if(tolower(http_uri) starts_with one of allowed_uris) {
    use pool IIS_Pool_80
  } else {
    use pool beta.pmi.org
  }
}