SEOUL -- Samsung Electronics Co.'s flagship Galaxy S6 smartphone has been modestly outselling its predecessor, but is struggling to keep up with Apple Inc.'s strong-selling iPhone 6, according to new figures from data tracker Counterpoint Technology Market Research.

In the three weeks after the launch of the Galaxy S6 and its curved-screen companion, the Galaxy S6 Edge, on April 10, Samsung has shipped 10 million devices to its distributors and retailers, with about six million of them sold to end users, according to Hong Kong-based Counterpoint.

That was enough to top global sales of last year's Galaxy S5 over the same period, the data tracker said, but not enough to match the April performance of the iPhone 6 and 6 Plus. Counterpoint didn't specify how many handsets it estimates Apple sold in April.

A Samsung spokeswoman said that the two flagship devices "are well accepted in the market and sustain the high demand since their market launches." She added that the company was ramping up production of the Galaxy S6 Edge, which "will support further growth" but declined to give specific figures.

Samsung doesn't typically release sales figures for individual smartphones.

Apple's large-screen smartphones, which went on sale last fall, have remained popular even eight months after their initial launch. Analysts attribute the resilience to the Cupertino, Calif.-based company's strong brand. The iPhone 6 and 6 Plus were also Apple's first foray into the large-screen handsets that have been especially popular among Asian consumers.

A spokeswoman for Apple didn't respond to a request for comment.

Samsung told investors in the spring that it expected to sell roughly three Galaxy S6 devices for each Galaxy S6 Edge device that it sells. Instead, demand for the Galaxy S6 Edge has exceeded demand, forcing Samsung mobile chief J.K. Shin to warn in April that it would take several months to ramp up manufacturing to meet demand.

While Samsung has boasted of the technology that went into the Galaxy S6 Edge's curved screen, the premium device has proven challenging to mass produce. That has hamstrung Samsung's ability to convert excitement for the device -- it received glowing reviews after it was unveiled in March -- into actual sales.

Peter Richardson, a research director at Counterpoint, said that the Samsung flagship phones could still ring up strong sales in the coming months, but warned that "the window of opportunity to do so is short as consumers don't have much patience in a market with many alternatives."

Samsung is counting heavily on the Galaxy S6 and the S6 Edge to lift sales and prop up its mobile division after one of its most bruising years in recent memory.

Last year, the South Korean technology giant's Galaxy S5 fell flat with consumers and sold just 12 million units in the three months after its April launch, compared with about 16 million units for the preceding flagship phone, according to a person familiar with the matter.

Already, the Galaxy S6 and S6 Edge are on track to fare better, despite launching in just 20 countries, compared to the Galaxy S5's initial 125-country rollout. Counterpoint said Tuesday that it still expects the Galaxy S6 and S6 Edge to sell about 50 million units, in line with analysts' expectations of between 45 million and 60 million units.

While the Galaxy S6 sales figures offer a measure of good news for Samsung, the increased sales come largely at the expense of other Samsung smartphones -- not Apple's, says Counterpoint.

The Korean company's Galaxy Note 4 and its midtier Galaxy A series of smartphones have fared particularly badly, the firm says, with sales falling by between 30% and 40%. As a result, it said, "Samsung's overall market share is not growing despite the relatively successful launch of the Galaxy S6."

The Samsung spokeswoman declined to comment on sales of its other smartphones.

Write to Jonathan Cheng at jonathan.cheng@wsj.com

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