Lawyers acting for Scott Snowden and Robert Jennings want judges at the Court of Criminal Appeal in Edinburgh to quash their clients' jail terms.
The two men were locked up in July 2013 for the murders of Thomas Sharkey, 55, his son Thomas, 21, and eight-year-old daughter Bridget in Helensburgh, Dunbartonshire, in July 2011.
Trial judge Lord Matthews told Snowden, 39, he would have to serve at least 33 years in prison before he would become eligible for parole. Jennings, 51, was sentenced to at least 29 years.
Now the pair want Lord Carloway, Lady Smith and Lord Brodie to quash their murder convictions. Snowden's legal team believe he should walk free from prison because they say Lord Matthews favoured the prosecution over the defence in his closing speech to the jury.
Jenning's legal team believe the judge was also wrong to allow a piece of evidence to go to the jury that they say was inaccurate and prejudiced the case against their client.
Yesterday, Snowden's advocate, Donald Findlay, QC, told the Appeal Court judges Lord Matthews committed mistakes when he summed up evidence at the end of the case.
He said: "He was obliged to ensure there was a fair and balanced summing up of evidence, representing both the position of the Crown and the defence.
"While the learned trial judge rehearsed the Crown case, he failed to remind the jury of a number of important aspects of the defence case and the criticisms offered by the defence of the prosecution case."
The pair were convicted of setting a fire that killed the Sharkeys at their home in Helensburgh. Snowden and Jennings were also convicted of trying to murder Mr Sharkey's wife Angela, 48, who survived the blaze.
Passing sentence, Lord Matthews told them it was the worst crime he had ever been involved in with during his legal career.
He added: "These crimes were cowardly enough in themselves, but it is also a common feature of them that you cynically recruited others to do your dirty work for you making sure you had a cast-iron alibi.
"It may be thought that you were safe from prosecution, hoping those who knew about your conduct would not alert the authorities. If that is what you thought, the fatal fire put an end to such aspirations."
Mr Findlay told the appeal judges that when Lord Matthews summed up the case to the jury, he was too biased in favour of the prosecution.
The QC argued this meant his client's conviction was unsafe and it should be quashed. Mr Findlay added: "He ran the risk of making the pieces of evidence seem more important to the jury than they actually were."
The appeal court hearing continues.