Markit said the composite output PMI was at 47.7 in May, up from 46.9 in April.
The average of the Markit and NBS series declined to 54.4 from 54.7 in April.
The Markit Economics services index accelerated to a far-stronger 57.1 after 52.6 in February.
Markit's purchasing managers index for the euro zone was unchanged at 46.5 in April.
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Markit said the growth may have helped the UK economy to avoid falling back into recession.
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Markit's eurozone manufacturing Purchasing Managers' Index rose to 46.3 in December from 46.2 in November.
Markit said that eurozone confidence remained lower than at any time since before the 2008-09 financial crisis.
The Markit eurozone Purchasing Managers' Output Index was little changed from October, up to 45.8 from 45.7.
In August new export orders for manufacturers were at their weakest since March 2009, according to Markit.
"Consumer spending remained disappointingly weak in December, " said Chris Williamson, chief economist at Markit, which compiled the index.
India Manufacturing Purchasing Managers' Index, prepared by Markit, was at 54.2 in February, compared with 53.2 in January.
Still, the data company Markit said Euro zone economic activity has bottomed out and is on the upswing.
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For the Markit Flash PMI, easing a tick to 53 for this month will not surprise too many.
The Markit eurozone services purchasing managers' index (PMI) rose to 49.2 last month, up from 48.8 in February.
In economic headlines, Markit Economics' composite purchasing managers index for the U.S. showed a reading of 52 for April.
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The PMI survey from Markit fell another 0.3 points in April to 46.5.
An interesting article in the WSJ discussed the Markit Flash PMI, which looks at all of US manufacturing.
Sentiment among services businesses fell to the lowest since March 2009, said Markit.
The Bank of Scotland monthly report, carried out by the Markit financial information firm, found orders down and costs rising.
In Germany, the services sector continued to expand in March, but at a lower pace than in February, said Markit.
Outside of the eurozone, Markit's services PMI reading for the UK rose to 55.3 in March, from 53.8 in February.
The Justice Department wants to know if the firms benefited from the way prices of credit indexes were posted on Markit.com.
Markit's preliminary index measuring service and manufacturing activity for the currency bloc fell to 47.3 in February, below expectations of 48.5.
The data comes a day after a separate study by Markit said the manufacturing contraction in the eurozone worsened in March.
"Markit has been informed of an investigation by the Department of Justice into credit derivatives and related markets, " says spokeswoman Teresa Chick.
The composite survey of thousands of firms by Markit showed a continued contraction - but at a slower rate than in November.
Markit's preliminary reading on U.S. manufacturing rose to 56.1 in January from 54.0 the previous month, the strongest report since March 2011.
The manufacturing figures came in a survey from financial data firm Markit.
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Markit's February composite purchasing managers index for the region was revised higher.
Separate data also from Markit, and released before US stocks opened, showed US manufacturing activity fell for the second straight month in May.
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