Manager David Moyes says Manchester United are among the favourites for the Champions League but need to improve.
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The three-time European champions have suffered from indifferent form in the Premier League, but finished top of Champions League Group A after a 1-0 home victory over Shakhtar Donetsk.
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"I think we could be classed as one of the favourites," said Moyes.
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"We'll have to improve in order to progress. A club like Manchester United has to be trying to win the trophy."
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The Scot, who replaced two-time Champions League winner Sir Alex Ferguson in the summer, added: "You have to get better as the tournament goes on.
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"It's a tough competition, a hard one, but we will take it one game at a time."
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Phil Jones scored the only goal against Shakhtar as United avoided a third successive home defeat, something that had not happened since 1962.
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Domestic league losses to Everton and Newcastle led to criticism of Moyes and his players, while defender Jonny Evans said the side had been affected by a lack of confidence.
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The current champions have already lost three times at Old Trafford in the Premier League this season, as many home defeats as during the whole of their title-winning campaign in 2012-13.
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United are ninth in the table after 15 games, 13 points behind leaders Arsenal.