Lawyers acting for Scott Snowden, 39 and Robert Jennings, 51, want judges at the Court of Criminal Appeal in Edinburgh to quash their clients' prison terms.
The two men were jailed in July 2013 for the murders of Thomas Sharkey Sr, 55, his son Thomas Jnr, 21, and his daughter Bridget, eight, in Helensburgh in 2011.
Trial judge Lord Matthews told Snowden that he would have to serve at least 33 years before he became 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 because they say Lord Matthews favoured the prosecution over the defence in his speech to the jury.
Jenning's legal team believe the judge was wrong to allow evidence to go to the jury which they say was inaccurate and prejudiced the case against their client.
The pair set a fire which killed Mr Sharkey, Thomas Jnr and Bridget at their home. They were also convicted of trying to murder Mr Sharkey's wife Angela, 48, who survived.
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.