Stewart Kevin Doran, 22, and his then 18-year-old friend, Bradley Davies, stamped and kicked busker Ben Bebbington, 43, to death beside a canal in Ditherington, Shrewsbury, in September 2012.
In May last year, having admitted Mr Bebbington's murder, Doran, of Bainbridge Green, Shrewsbury, was jailed for life, to serve a minimum term of 16 years, at Birmingham Crown Court.
Davies, who denied the killing but was found guilty by a jury, received a life term with a minimum of 14 years.
Doran's bid to have his minimum term reduced was rejected at the Court of Appeal yesterday. His lawyers argued his minimum term was too long, because there was an "objectionable disparity" between his 16 years and Davies' 14 years.
But the appeal attempt was rejected by Mr Justice Griffith Williams, Lord Justice Jackson and Judge Brian Barker, who said "no right-thinking member of the public" would disagree with the sentence.
They said Doran had played the leading role in the fatal assault and was the more criminally experienced of the pair.
Mr Justice Griffith Williams said the crown court judge had been best placed to assess the culpability of the two. He said: "In addition to the leading role of Doran, there was the consideration that he had previous convictions for battery and, in 2010, for an offence of assault, occasioning actually bodily harm.
"Davies had only one relevant conviction, an offence of battery committed in 2006. In our judgment, no right-thinking member of the public would consider that something had gone wrong with the administration of justice in this case.
"We are satisfied that the sentence was an entirely appropriate one."
Neither man can be freed at the end of their minimum terms unless the Parole Board is convinced they are safe to live in the community.
Mr Bebbington's sister Karen Higgins said the appeal process had been difficult for the family.
"There will always be harsh reminders like today of Ben's death but then there is not a day that goes by where we can say we got through without thinking about what has happened and this is especially true for Ben's mum who is living every mother's worst nightmare," she said.
"However, as a family we choose to stay focused on the positives such as the compassion of friends and the community as well the incredible support Ben's memory has raised for the wonderful Shrewsbury Ark," she added.