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Apple CEO Tim Cook noted at a conference in San Francisco this week that he wasn’t too concerned about rising competition from and its $199 Kindle Fire.

“Amazon is a different kind of competitor. They have different strengths and so forth,” Cook said in touting the success of the iPad. “And I think they will sell a lot of of units. I think they have, and they will. But the customers that we are designing our products for are not going to be satisfied with a limited function kind of product.”

Well, how is that new rivalry over tablets actually playing out?

The research firm iSuppli released a report today showing that the introduction of the Kindle Fire and the unveiling of Apple’s very own iPhone 4S did take a bite out of Apple’s leading market share in tablet computers. According to the report — despite Apple’s record sales of 15.4 million iPads during the fourth quarter — the company’s worldwide market share in tablets slipped from 64 percent to 57 percent.

iSuppli estimated that Amazon sold 3.9 million Kindle Fires during the fourth quarter, giving it a 14 percent share. Samsung came in third with a market share of eight percent, followed by Barnes & Noble and its Nook device at seven percent. (See chart below).

“Kindle Fire shipments in the fourth quarter came right in line with the IHS early December forecast of 3.9 million units, representing a respectable start for the Fire.” said IHS iSuppli’s Rhoda Alexander. “However, the long-term viability of the product will hinge on the success of Amazon’s business gamble, which depends on tablet sales driving substantial new online merchandise sales at in order to attain profitability.”

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