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Three-time defending champion Federer, seeking his 12th Grand Slam singles crown, opened the year's final Grand Slam event on Monday with a 6-3, 6-2, 6-4 victory over 320th-ranked American Scoville Jenkins in 92 minutes.
"I'm happy to have won my first one. I'm happy I'm through," Federer said. "It always feels like a bit of a relief. Now I can relax a little maybe."
Henin, still battling a sore shoulder, took 62 minutes to beat 145th-ranked German Julia Goerges 6-0, 6-3.
She won the first set in 24 minutes, aided by Goerges' six double faults and 15 unforced errors, and cruised to the finish.
Eighth seed Serena Williams, who won her eighth Grand Slam title in January at the Australian Open, showed some rust from a long layoff due to a thumb injury but defeated Germany's Angelique Kerber 6-3, 7-5.
The younger Williams sister had not played since a Wimbledon quarter-final loss to Henin.