The letter, issued in conjunction with Icahn's new website "Shareholders Square Table," comes after Icahn urged Cook to use Apple's $150 billion in surplus cash to buy back company shares.
In the letter, Icahn praises Cook's leadership, but says the discussed share buyback needs to be substantial.
"We want to be very clear that we could not be more supportive of you, the existing management team, the culture at Apple and the innovative spirit it engenders. The criticism we have as shareholders has nothing to do with your management leadership or operational strategy. Our criticism relates to one thing only: the size and timeframe of Apple's buyback program. It is obvious to us that it should be much bigger and immediate," the letter says.
A copy of the letter was obtained by Street Insider.
(Read more: Apple now hordes 10% of US corporate cash )
Apple shares were down slightly in early trading.
Icahn's stake in the technology giant is now 4.7 million shares, up from 4 million, making his total stake roughly 2.46 billion, the source told CNBC.
(Read more: Ex-Apple CEO Scully: Cook should ignore Icahn )
In an earlier interview with CNBC, Icahn said he would not be dropping this issue.
"I feel very strongly about this," Icahn told CNBC after his meeting with Cook. "I can't promise you the stock will go up and I can't promise you they will do the buyback. But I can promise you that I'm not going away until they hear a lot more from me concerning this."
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