In an attempt to give his arguments more credence, Icahn disclosed that he invested another $500 million in Apple Inc. in a series of purchases made Thursday.
With the latest shopping spree, Icahn has spent $1 billion on Apple stock during the past weeks to raise his total holdings in the Cupertino, California company to $3.6 billion since he began buying the shares five months ago when they were still trading below $450.
Apple's stock rose $4.67 to close at $556.18 on Thursday. Despite the recent gains, the shares remain about 20 percent below their peak reached 16 months ago.
Icahn, a longtime activist investor, has been lobbying Apple's board to boost the stock by funneling more of the company's $147 billion in cash toward share repurchases. Meeting Icahn's demands would require Apple to spend more on its stock than it planned, something the board so far has been unwilling to do.
The impasse has set the stage for a Feb. 28 showdown at Apple's annual meeting, where shareholders will vote on a non-binding proposal recommending that the board follow Icahn's advice. Apple is urging its shareholders to vote against the proposal, saying it is already reviewing ways to return more cash to shareholders.
Icahn tried to sway Apple shareholders to his side Thursday with the release of an open letter that represented his most detailed explanation so far about why he believes the company is worth substantially more than its current market value of about $500 billion.
Besides elaborating on the reasons why he believes it's a ''no-brainer'' for Apple to buy more of its own stock, Icahn took the unusual step of pointing out new markets where he believes the company could easily make even more money than it already does with its line of iPhones, iPads, iPods and Mac computers.
Icahn made it clear he wants to see an Apple television set, a device that has been a subject of company speculation for the past three years. Steve Jobs, Apple's late CEO and co-founder, suggested Apple was working on a TV set in interviews with his biographer before he died in October 2011.
In the letter, Icahn suggested that Apple should make a $1,600 television set with an ''ultra'' high-definition screen designed to show so-called ''4K' programming that will be coming from Internet video service Netflix Inc. and other sources later this year. Icahn thinks Apple could sell at least 25 million of the ultra-HD sets backed by its brand and software, generating about $40 billion in revenue.
Icahn also touted the possibility of Apple making a smartwatch, an idea that company CEO Tim Cook has hinted at in public appearances. Icahn also thinks the company is ideally positioned to roll out a system for handling digital payments.
The letter also jabbed at a sore point at Apple by referring to the perceived lack of innovation at the company since Jobs died. Icahn described recent upgrades to the iPhone and iPad as ''evolutionary (not revolutionary).''
Apple didn't immediately respond to requests for comment.
As has been the case through most of his public campaign, Icahn praised Cook in his letter for doing an ''excellent job.''
Icahn's main beef has been with Apple's board, which consists of Cook and seven other members. In an interview with the Fox Business network, Icahn said it's ''almost criminal'' for the board to resist increasing the company's stock buyback program and questioned the directors' intelligence about financial matters.
Icahn wants Apple to spend $50 billion buying back its stock by the end of its fiscal year ending in September. The company began the year with $37 billion left to invest in a $60 billion buyback program that doesn't expire until September 2015.