(Updates with U.S. manufacturing data)
* U.S. factory activity expands in May
* Euro zone private business growth slows
* Chinese factory activity improves but still contracting
* Japan shows similar trend, but export orders weaken
By Angela Moon and Jonathan Cable
NEW YORK/LONDON, May 22 (Reuters) - U.S. manufacturing growth picked up to a three-month high in May, while private business activity in the euro zone grew at just under its fastest pace in three years, surveys showed on Thursday.
In Asia, China's factory sector turned in its best performance this year in May but still contracted for the fifth straight month, while Japanese factory activity cooled slightly in May but at a slower pace than in April.
Financial data firm Markit said its preliminary or "flash" U.S. Manufacturing Purchasing Managers Index rose to 56.2 in May from 55.4 in April, while factory output growth hit its fastest pace since February 2011, rising to 59.6 from 58.2.
"The U.S. manufacturing is certainly in an expansion mode, and that's a good indication going forward, that it will remain on the bullish side of the equation for sometime," said Peter Cardillo, chief market economist at Rockwell Global Capital in New York.
World stock indexes edged higher on Thursday as the data boosted investors' appetite for risky assets.
In Europe, an unexpected pickup in the service industry was offset by lackluster factory activity, but was enough to show the euro zone's fragile recovery has some traction.
"Overall, the data is a bit of a relief after disappointing first-quarter figures," said Christian Schulz, senior economist at Berenberg Bank.
"In China, the authorities are doing something and the economy is responding. (But) the European Central Bank is worried because it looks like inflation will stay low for a long period of time."
Markit's Composite Purchasing Managers' Index for the currency bloc, seen as a good indicator of growth, edged down to 53.9 from a near three-year high of 54.0 in April, matching the consensus forecast in a Reuters poll of analysts.
Readings above 50 indicate expansion, and Markit said the index pointed to second-quarter economic growth of 0.5 percent, which would be the strongest in three years and better than the 0.3 percent predicted in a Reuters poll on Wednesday.
But the data also revealed the biggest split between the euro zone's two largest economies since February. France contracted while Germany bounded ahead, although led by services rather than manufacturing.
Decent overall growth is good news for the ECB, but with official inflation at 0.7 percent, well below the central bank's 2 percent ceiling, it will be concerned that companies slashed prices for the 26th month running to drum up trade.
The ECB is expected to cut what little it has left of its main interest rate and push the deposit rate below zero next month, a Reuters poll predicted this week.
"Today's data are a reminder of the fragility and unevenness of the recovery. With inflation low and a sustained euro zone recovery not yet assured, further ECB easing in June looks all but certain," said Martin van Vliet at ING.
The economic picture was also mixed for the world's second-largest economy, China.
HSBC/Markit's Flash China Manufacturing PMI rose to 49.7 in May from April's 48.1, reaching its highest since December and beating the median forecast of 48.1.
Sub-indexes measuring output as well as domestic and foreign demand all recovered sharply. But factory jobs were shed for the 13th month and the overall index still pointed to contraction.
Premier Li Keqiang said in March it was acceptable for GDP growth to be slightly below the 7.5 percent target this year as long as the job market held up. Previously, he has said growth of at least 7.2 percent was needed to create sufficient jobs.
But just as China and the ECB look set to ease policy the Bank of England is becoming the favorite to be the first major central bank to hike interest rates - probably in the second quarter of next year.
Federal Reserve policymakers last month also began laying groundwork for an eventual retreat from easy monetary policy with a discussion of how to best control interest rates as they remove trillions of dollars from the financial system.
Meanwhile a survey showed Japanese factory activity contracted slightly in May but at a slower pace than in April, suggesting some recovery from the impact of a sales tax increase last month.
The Markit/JMMA flash Japan Manufacturing PMI rose to a seasonally adjusted 49.9 in May from April's 49.4.
Editing by Meredith Mazzilli