I think Arsenal will beat Nottingham Forest, Ange is a step down from their previous coach and Arsenal needs to recover from the loss they suffered to Liverpool.
Liverpool should beat Brentford and I will like to see how Alexander Isak fare in his first match, I also think that Chelsea will win their match but I am not sure what will happen between Manchester United and Manchester City in the derby.
Your mistake, it is Burnley vs Liverpool and the match result is favorable to Liverpool by all means. We know Burnley to be a relegation team and performance isn't well equipped to be able to defeat title defenders. Liverpool are going to achieve success in the fixture, it is not a hard match, before the end of first half i expect Liverpool to already make important strikes to give them goals ahead of their opponent. Liverpool will try to avoid losing, knowing that maintaining the lead is very important, Chelsea are also hitting up for the season, same with other teams despite a poor start.
Manchester City vs Manchester United, it is going to be a difficult game for them both, we don't expect anything else than a competitive match in the fixture, anybody can become the winner or they both share points and move unto capitalizing in the next fixture.