Carl Icahn thinks Apple Inc. (AAPL) is worth twice its current market value, and is pressing his good pal Tim Cook to buy back shares even more aggressively to seize on this “undervaluation” and help the stock reach his lofty target.

In a long letter to the Apple CEO published on Icahn’s Yahoo Finance Contributors Network Tumblr page, the often coarse but sometimes cuddly activist investor lays out an audaciously optimistic outlook for Apple’s new-product sales and profit growth far brighter than Wall Street’s current expectations. And he’s urging Cook to accelerate the already-rapid buyback pace to use excess cash for the benefit of long-term shareholders. Some choice excerpts:

Icahn writes: “Given the persistently excessive liquidity of $133 billion net cash on Apple’s balance sheet, we ask you to present to the rest of the Board our request for the company to make a tender offer, which would meaningfully accelerate and increase the magnitude of share repurchases.

“We thank you for being receptive to us the last time we requested an increase in share repurchases, and we thank you in advance now for any influence you may choose to have communicating to the rest of the Board the degree to which a tender offer would have a positive impact on an EPS basis for all shareholders,” he politely suggests.

Icahn pledges that his firm would not sell shares in any future tender offer. Icahn says he owns 53 million Apple shares, equivalent to about 0.9% of the company and worth more than $5.3 billion at Apple’s Wednesday closing price of $100.80. On Twitter yesterday, Icahn noted that Apple shares were up more than 50% since he first disclosed a position in the stock about a year ago.

In the letter, Icahn positively gushes over Apple’s new-product slate and its growth potential over the next few years. Icahn forecasts that in the fiscal year that just began this month, Apple’s revenue will jump 25% and earnings will surge 44% to $9.61 a share. The consensus among the dozens of analysts who follow Apple is for a much more modest 12.2% revenue gain in fiscal 2015 and 15% profit growth to $7.29 a share.

He goes on to say earnings will rise a further 30% in each fiscal 2016 and 2017. Applying a price/earnings multiple of 19 to 2015 earnings and adding the net cash he arrives at a fair-value estimate of $203 for the stock. This would make Apple, which now has a market value just above $600 billion, the first $1 trillion company in history.

Apple from early 2013 through June had already bought back a whopping $51 billion in stock, with a plan for that to get up to $90 billion by the end of calendar 2015. Icahn doesn’t specify an amount of additional repurchases he’d like to see accomplished through the tender offer he’s proposing.

Icahn lavishes praise on Cook's reign: “To be totally clear, this letter is in no way intended as a criticism of you as CEO, nor is it intended to be critical of anything you or your team are doing from an operational perspective at Apple. Quite to the contrary, we could not be more supportive of you and your team, and of the excellent work being done at Apple, a company that continues to change the world through technological innovation.

“The intention of this letter is to communicate two things to you: (1) given the earnings growth we forecast for Apple, we continue to think that the market misunderstands and dramatically undervalues Apple and (2) the excess liquidity the company continues to hold on its balance sheet affords the company an amazing opportunity to take further advantage of this valuation disconnect by accelerating share repurchases.”

On first look, Icahn’s bold letter raises a few key questions:

-Could Icahn and his team possibly be right that investors and analysts are so vastly underestimating the imminent growth surge by the biggest and most widely followed company on the market?

-Is he puffing up the bull case to make his idea for a heftier stock buyback seem like an utter no-brainer?

-Does Cook need to listen to a holder of less than 1% of the company now that the stock has had a great run?

-And will the market respond to Icahn’s loud articulation of the super-bull case yet again?

 Let’s watch and see.