(Refiles to recast third bullet headline and second paragraph)
* Asia's richest woman fighting three eldest children in court
* Children want Rinehart removed as trustee of multi-billion-dollar trust
* Judge reserves decision, rejects request for mediation
* Rinehart seeking funding for $10 bln Roy Hill iron ore project
By Narayanan Somasundaram
SYDNEY, Sept 12 (Reuters) - Lawyers for Asia's richest woman and three of her children held preliminary arguments on Wednesday over who should control a $4 billion trust, a bitter family feud that has captivated Australia.
Justice Paul Brereton of the New South Wales Supreme Court reserved a decision on whether the case against mining magnate Gina Rinehart should be thrown out of court or argued fully. A decision is not expected for some days.
At stake is Rinehart's position as the sole trustee of the trust that holds a near-one-quarter share in Hancock Prospecting Pty Ltd, one of the world's largest privately-owned mining companies.
Lawyers for Rinehart and her daughter Ginia Rinehart, the only child to side with her mother in the feud, sought to have the suit brought by the three elder children to remove their mother as a trustee thrown out of court.
Hancock Prospecting is developing what would be Australia's fourth-largest iron ore mine and generates hundreds of millions of dollars a year in royalties from tenements secured by Gina Rinehart's father, Lang Hancock, a legendary figure in Australian mining history.
The dispute has already caused a delay at Hancock's flagship Roy Hill iron ore mine, rail and port project in Western Australia, which is now running into some stiff headwinds from slowing China demand and soaring costs.
A halving in iron ore prices <.IO62-CNI=SI> over the past year has dented both investor appetite for such projects and the 58-year-old widow's fortune, estimated by Forbes in February at around $18 billion.
Rinehart has been putting some of her vast wealth to work with purchases of media companies Fairfax Media and Ten Network Holdings in the past year, causing some consternation among the chattering classes.
Her opposition to taxes and calls for miners to be allowed exemptions from laws restricting the use of foreign labour have also put her on a collision course with government and unions.
Recently, Rinehart warned Australia was becoming too expensive for mining firms which she said could hire workers for under $2 a day in Africa.
Rinehart, known as the Pilbara Princess, has a long history of controversy and has played out much of her life in the media spotlight.
She is the only child of Hancock, a larger-than-life character credited with discovering the world's largest deposit of iron ore in Pilbara, Western Australia.
Rinehart learned the business at her father's knee and, after a prolonged battle with his third wife following Hancock's death in 1992, cemented control of Hancock Prospecting.
Rinehart owns three-quarters of Hancock Prospecting and is the sole trustee of the family trust which holds a further 23 percent of the company for the benefit of her four children.
Last year, just days before the trust was due to vest, Rinehart changed the vesting date to 2068 and sought changes in the trust documents, prompting her three eldest children -- Hope Rinehart Welker, Bianca Rinehart and John Hancock -- to fight to have her removed as trustee.
E-mails earlier made public after Rinehart's efforts to keep the case behind closed doors showed Rinehart told the children the vesting of the trust would likely trigger crippling capital gains tax liabilities for them.
She also described the elder trio of children as being lazy and spoilt, and warned that their security would be at risk if they persisted with the action.
Lawyers for Rinehart, who was not in court, said she had since brought forward the vesting date for the trust and was prepared to hand over the assets, something the children did not want.
"The only remaining duty for the trustee is to divide the assets and hand it out," David Russell, counsel for the mining magnate said in court.
"The beneficiaries will not want to receive the shares for tax reasons. The offer is made in open court."
Earlier, an attempt by the youngest daughter, Ginia, to bring about a mediation was rejected by the justice and lawyer for the warring children.
"Ginia had a most genuine interest in moving the case to mediation without further public ventilation of this very unhappy family dispute," Ginia's lawyer, Francois Kunc, said.
The early skirmishes in the family feud delayed the sale of equity stakes in Roy Hill, Rinehart has previously said.
South Korean steel giant POSCO, Japanese trading company Marubeni, South Korea's STX Corp, and Taiwan's China Steel Corp hold a collective 30 percent stake in the project, with Hancock Prospecting holding the remainder.
Hancock is currently trying to raise about $7 billion in debt funding to get the massive project into production.
It is not clear what impact, if any, the removal of Rinehart as trustee of the family trust would have on Hancock Prospecting and its iron ore, coal and media empire. Shares in the company can only be held by Rinehart and her direct descendants and cannot be pledged as collateral. (Additional reporting and writing by Lincoln Feast; Editing by Raju Gopalakrishnan)