Frei gave the visitors the lead, before Steven Gerrard smashed in a fine free-kick, which was not enough in the end.
Rodgers told ITV Sport: “We have to fight going forward to be in this competition next season. We are not too far off the top four and, if we get there, we will be better in it next season.
“The possibility of January signings is something to discuss with the owners, but there was a big investment in the summer.
“We have been unlucky with injuries to key players who are important to the way that we play, but that is to take nothing away from the players tonight who have put everything into the game.”
The Reds finished the game with 10 men, after Lazar Markovic was shown a straight red card. Rodgers admits he was not impressed with the referee’s decision.
“I thought it was an awful decision, I thought the referee was poor tonight to say the least. Young Markovic has come on and been bright and, although he has put his hand out, it was not a sending off.
“The referee has seen the lad lying on the floor holding his face, but I don’t know if he has seen the actual incident.”
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