Icahn went public with his demands after making his case with Apple CEO Tim Cook in a Monday dinner hosted at his New York apartment. The three-hour meeting was "cordial," according to the often-confrontational Icahn, although he said the situation got "testy" when Apple's chief financial officer, Peter Oppenheimer, expressed misgivings about asking the board to reconsider how much the Cupertino, Calif., company should be spending on its own stock.
Apple didn't immediately respond to requests for comment. Icahn said Apple agreed to get back in touch with him in three weeks.
Investors are evidently hoping that Icahn's crusade will prod Apple's board to do something to lift the stock. The company's shares gained $10.59, or 2.2 percent, to $487.34 in Tuesday afternoon trading. The stock has gained 4 percent since Icahn disclosed seven weeks ago that he had bought a significant stake in Apple.
Icahn told CNBC Tuesday that his fund has invested about $2 billion in Apple so far, but didn't reveal how many shares he owns. A $2 billion investment in Apple currently represents about 0.5 percent of the company's market value of about $444 billion.
The public critiques of the tenacious Icahn threatens to compound Apple's headaches as it tries to regain some of the market share that it has lost in the mobile computing market during the past year to less expensive smartphones and tablet computers that primarily run on Google Inc.'s free Android software.
The tougher competition has slowed Apple's growth and decreased the average prices for its iPhones and iPads, trimming the company's profit margins to the dismay of investors.
Apple is hoping to bounce back with the recent release of two more iPhones, a high-end model called the 5S and slightly less expensive device called the 5C. Apple said it sold more than 9 million of the new iPhones during their first weekend on the market last month, the fastest start yet for its marquee product line.
Despite the downturn in its stock price, Apple remains the world's most valuable company. Apple also has about $147 billion in cash, but the company isn't relying on much of that money to buy back its stock. Instead, the company issued bonds to take advantage of low interest rates, a borrowing tactic that Icahn believes Apple should use again to raise even more money for stock buybacks.