iRule to modify the URI without a client redirect

You may want to look at this page:

https://devcentral.f5.com/wiki/iRules.HTTP__uri.ashx

Using a redirect will change the URL on the browser.

Changing the Host header only affects what the server gets.

You will need the IP address of la.mysite.com, say it is 1.2.3.4

You will need an irule like this:

when HTTP_REQUEST {
  if { [HTTP::uri] equals "/" } {
     the node command directs the request to the server
     whether or not it is behind the BigIP.  Make sure the BigIP
     has a route to that server.
    node 1.2.3.4
     In case you don't have SNAT on the virtual, you may need it here.
    snat automap
     In case the server complains, set the host header.
    HTTP::header replace Host "la.mysite.com"
     Finally change the URI while on its way to server.
    HTTP::uri /documents/china.pdf
  }
}

Here is the page for the SNAT command:

https://devcentral.f5.com/wiki/iRules.snat.ashx

Here is the page for the node command:

https://devcentral.f5.com/wiki/iRules.node.ashx