Pukki, who scored 29 goals as Norwich were promoted from the Championship last season, has five in just three games this term to match Raheem Sterling as the Premier League's top scorer.
Barnes is alongside Sterling's Manchester City team-mate Sergio Aguero on four, making this only the second Premier League season with four players at better than a goal a game for the first three games. In 2010-11, Chelsea pair Didier Drogba and Florent Malouda were on four apiece along with Theo Walcott, then at Arsenal, and Newcastle's Andy Carroll.
Only Edin Dzeko, for champions Manchester City in 2011-12, has ever scored six in the first three games, with Manchester United's Wayne Rooney that same season and Robbie Fowler for Liverpool in 1994-95 matching Sterling and Pukki with five. In all, 24 players prior to this season had notched four or more by this stage – the most recent being Riyad Mahrez in Leicester's title-winning 2015-16 campaign.
The 24 players went on to score an average of 15.83 goals over the full season, ranging from Rooney's 27 down to nine, Walcott's final total as well as that of Teddy Sheringham for Portsmouth in 2003-04.
The benchmark, then, would effectively be 16 goals – but there are reasons to believe most of this season's fast starters can exceed that expectation.
Aguero has scored at least 20 in each of the last five seasons while the previous players with five or more through three games have averaged 22 for the season. Fowler finished with 25, two fewer than Rooney, though Dzeko managed only 14. Sterling has notched 18 and 17 in the last two seasons, while Pukki has shown no sign of his second-tier form deserting him.
Burnley striker Barnes is a harder case to judge – he was not a prolific Premier League goalscorer until midway through last season but has been a constant threat in 2019. Nine of his 12 goals last season came from New Year onwards, meaning he already has 13 this calendar year compared to 23 previously in the top flight.
There were two 10-goal seasons and four of 11 goals – the most frequent final tally, along with 18. Six players, or one out of every four, reached 20 or more for their respective seasons – Andy Cole the same season as Fowler, Dwight Yorke in 1999-2000 and Robin Van Persie the season after Rooney, all with United, and then Diego Costa for Chelsea in 2014-15.
Some of the lower-scoring seasons in years past show how a hot start can fall away – Walcott, for example, hit a hat-trick against minnows Blackpool and netted at Blackburn before the end of August but then did not score again before Christmas.
Carroll's 13-goal season was perhaps the strangest of all. Starting with a hat-trick against Aston Villa, he had 11 by Boxing Day for Newcastle but after his big-money January move to Liverpool he scored only twice more – both in a 3-0 win over Manchester City on April 11.