Lawyers acting for Scott Snowden, 39, and Robert Jennings, 51, want judges at the Court of Criminal Appeal in Edinburgh to quash their clients' jail terms.
The two men were jailed in July 2013 for the murders of Thomas Sharkey Senior, 55, his son Thomas Jnr, 21, and his daughter Bridget, eight, in Helensburgh, Dunbartonshire, in July 2011.
Trial judge Lord Matthews told Snowden that he would have to serve at least 33 years in prison before he would become eligible for parole. Jennings was ordered to serve 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 that 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 which they say was inaccurate and prejudiced the case against their client.
On Tuesday, Snowden's advocate Donald Findlay QC told the Appeal Court judges that Lord Matthews committed mistakes when he summed up evidence at the end of the case.
He said: "He was obliged to ensure that 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 set a fire which killed Mr Sharkey, Thomas Jnr and Bridget at their home in Helensburgh. They were also convicted of trying to murder Mr Sharkey's wife Angela, 48, who survived the blaze.
On that occasion, the court heard that Mr Sharkey, who himself had previously served a prison term for drug dealing, was one of the people Snowden held a grudge against.
Snowden became enraged when Mr Sharkey intervened in a drugs debt that he was owed by the fire death victim's niece.
To exact his revenge, Snowden had the Mariners pub in Helensburgh, which was owned by Mr Sharkey, burned down before it was opened. The blaze left Mr Sharkey in financial difficulty.
Snowden did not stop there, however, and ordered Jennings to set fire to Mr Sharkey's home.
Jennings carried out the act while Snowden was on a family holiday in Mexico, seemingly giving him the perfect alibi.
Once at the Sharkey home, at about 5am on July 24, 2011, Jennings poured petrol through the letterbox of the only door to the house and set it alight before leaving the house.
The three people later died leaving Angela as the only survivor.
Passing sentence, trial judge Lord Matthews told them that it was the worst crime that he had ever been involved 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 that you had a cast iron alibi.
"It maybe thought that you were safe from prosecution, hoping that 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."
On Tuesday, Mr Findlay said that when Lord Matthews summed up the case to the jury, he was too biased in favour of the prosecution.
Mr Findlay argued that this meant his client's conviction was unsafe and that 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 before Lady Smith, Lord Carloway and Lord Brodie on Wednesday.