Two senior Saudi princes have failed to extract themselves from English justice after the court of appeal upheld a ruling that they are not entitled to sovereign immunity in a case involving their London-based business interests.

Lawyers for the princes were informed that they had lost their claim at the start of a two-day hearing on Wednesday into a related attempt to impose reporting restrictions on the sensitive commercial dispute.

The Guardian and Financial Times have requested documents detailing disputed transactions involving Saudi interests in Beirut and Nairobi. The case involves a London registered company, FI Call Ltd.

The Saudi royals are Prince Mishal bin Abdulaziz Al Saud, a former defence minister who is the brother of King Abdullah and chairman of the country's influential allegiance council, and his son Prince Abdulaziz bin Mishal bin Al Saud. They and Global Torch Ltd, the British Virgin Island company they are said to control, are in dispute with a Jordanian businessman, Faisal Almhairat, and his Seychelles-based firm Apex Global Management.

The three appeal court judges, Lord Justice Kay, Lord Justice Richards and Lord Justice Briggs, are due to give the reasons for their sovereign immunity decision in due course.

Lawyers for the Saudis are relying on the Human Rights Act to obtain an interim ban on the reporting of court documents related to the case. Mark Warby QC, for the princes and Global Torch, said the principle of open justice "was not a trump card" that automatically removed all the other "fundamental rights" of claimants in a court case.

He said the allegations against the princes were very grave and were an attack on their reputations and their rights to a private and family life under article 8 of the European convention on human rights.

Warby said he was seeking an interim ban on the release of sealed court documents until the trial was heard in full. Some of the claims in the petitions, he alleged, amounted to an "arguable case" of extortion.

"It's obvious that allegations of this gravity are going to have an effect – in the old language of libel, they [the Saudis] are going to be shunned and avoided."

But Guy Vassall-Adams, counsel for the Guardian and Financial Times, told the appeal court: "Global Torch has chosen to bring proceedings in this jurisdiction. This is an open justice jurisdiction.

"They [the Saudis] have to accept that these damaging allegations will be heard in open court in the usual way. The protection they are entitled to is a judgment delivered in public which will refute unfounded allegations.

"That's how a legal system works in a democracy under the rule of law."

He said parliament had sanctioned fair and accurate reporting of defamatory allegations that arise in court proceedings. "People understand that allegations are not proven matters."

Vassall-Adams said allegations of blackmail or extortion were being used as "a crutch … to support the [Saudis'] application for a private hearing".

"Private hearings are not necessary to do justice in this case," he said. "It's very clear the courts are extremely reluctant to allow article 8 rights to prevail over the principle of open justice."

Robert Howe QC, for Almhairat and Apex, said the Saudis' application was "an illegitimate and ill-judged" attempt to amend the principle of open justice through arguments used in privacy and defamation cases.

"This is against a backdrop of a flood of attempts by wealthy litigants to try and close the court doors" by applying for privacy for their claims. Merely stating that there was an "arguable case" that allegations were false was not sufficient to order what was in effect a privacy hearing, Howe said.

Public faith in the administration of justice has already been damaged because normal reporting of the case has already been prevented, he said. No request has been served on the court to strike out the evidence.

"The matter of the allegations do raise important issues of widespread public interest," Howe said. "My client has had the misfortune to become embroiled in a very difficult dispute with some very rich individuals … It's a basic constitutional position that [such a hearing] should be held in public."

The case continues.