In the battle of smartphone operating systems, it's safe to say the market is a firm duopoly. On one hand, you have market leader Google (NASDAQ:GOOG) (NASDAQ:GOOGL), whose Android OS has been propagated by various hardware manufacturers. On the other you have Apple's (NASDAQ:AAPL) iOS mobile ecosystem, for which the sole compatible handset is the iPhone.

For those following the worldwide competition between these two tech giants, research firm Kantar Worldpanel provides sales share numbers for the major smartphone markets. In Kantar's latest report, covering the three months ended May 2015, the picture is decidedly mixed for both Apple and Google. Here's how both fared on a year-over-year basis, according to Kantar's data.

Strong iOS markets became decidedly less so
For Apple investors, this report brought both positives and negatives. On the latter, Apple appears to have lost market share in its stronger markets. In the highly contested U.S. market, Apple's sales share fell from 32.5% in the three months ended May 2014 to 30.9% in this year's corresponding time frame. Still, it's normal for iPhone sales to slow from September-December's torrid pace.

Meanwhile, Android increased its U.S. market share by 3 percentage points, from 61.9% to 64.9%. Premium Android vendor Samsung released the newest version of its high-end Galaxy S line during this period. While Kantar credited the Galaxy S6 with increasing Android sales, it was actually outsold by the prior-generation Galaxy S5

Last year, Apple had majority share in one market: Japan. This year, there are no countries in which Apple has majority market share. After holding on to a thin majority of smartphone sales (50.7%) in Japan during the three months ended May 2014, Apple reported a year-over-year loss of 6.1 percentage points to finish with a 44.6% sales share there in the three months ended May 2015.

It appears that Android only captured 3.6 percentage points of market share in Japan during this period. The catch-all designation of "other vendors" accounted for the other 2.5 percentage points of market share that Apple lost.

Android had the same issue
If you think this report shows a further migration to Android, you'd be wrong. In Android's strongest markets, there was a move toward Apple's ecosystem.

This phenomenon was most apparent in urban China, where Apple grew from a 14.7% sales share in the three months ended May 2014 to 21.5% in this year's corresponding period. Android, on the other hand, fell from 82.7% to 76.8% during this period. In each of the last two quarters, Apple has grown total China revenue by 70% year over year, and this report confirms Apple's strong performance in the Middle Kingdom.

Android's performance wasn't as poor in the EU5 -- Europe's five top smartphone markets -- where Android reported a year-over-year market share decrease of 2.9 percentage points, from 73.7% to 70.8%, as Apple picked up market share. It appears that competition in Europe's big five is increasing in scope. Overall, the report is mixed for both Apple and Google, with no clear winner on a year-on-year basis.

This article represents the opinion of the writer, who may disagree with the “official” recommendation position of a Motley Fool premium advisory service. We’re motley! Questioning an investing thesis -- even one of our own -- helps us all think critically about investing and make decisions that help us become smarter, happier, and richer.