Radwanska, Li safely through at Flushing Meadows

Radwanska, who has not been beyond the fourth round at the hard-court grand slam, swept aside her opponent 6-1 6-2 in an hour and four minutes.

The Polish world number four – who fell in the semi-final stage at Wimbledon – was quickly into her stride, and her only scare came early on in the second set when she was broken in the second game by Soler-Espinosa.

However, Radwanska broke back in the third game before doing so again in the fifth and seventh to ensure a convincing victory.

World number one Serena Williams trounced 2010 French Open champion Francesca Schiavone 6-0 6-1 in only 62 minutes.

The lopsided encounter saw Williams win 52 points to Schiavone’s 23 in a powerful display in which she did not concede a break point to her Italian opponent.

Williams’ sister and two-time US Open champion, Venus, reached the second round for the third successive year thanks to a 6-1 6-2 win over Kirsten Flipkens of Belgium.

The world number 60 broke Flipkens six times in the match as she progressed in 1:26.

Number five seed Li Na is also safely through having rarely been tested by Belarusian Olga Govortsova in a 6-2 6-2 win.

Li was dominant, winning 79 per cent of points on her first serve and 87 per cent on her second.

Govortsova was unable to break the Chinese in the match as Li, a quarter-finalist at Flushing Meadows in 2009, booked her place in the second round after 65 minutes on court.

American youngster Sloane Stephens blasted 38 winners as she overcame Luxembourg’s Mandy Minella 4-6 6-3 7-5(7-5).

It was a powerful display by Stephens as she went for her shots consistently but managed to accumulate 55 unforced errors, which accounted for nearly 50 per cent of Minella’s total points tally for the match.

Serbia’s Jelena Jankovic enjoyed a comfortable 6-3 6-4 triumph over America’s Madison Keys.

The ninth seed broke her opponent’s first service game of each set and did not give up her own delivery once as she cruised to victory.

Eighth-seeded German Angelique Kerber belted Czech Republic’s Lucie Hradecka 6-1 6-1 in only 55 minutes.

Britain’s Laura Robson, seeded 30th, found form in the second set as she thrashed Spain’s Lourdes Dominguez Lino 7-5 6-0, America’s 23rd seed Jamie Hampton also won in straight sets 6-4 6-2 against Spain’s Lara Arruabarrena and Russia’s Vera Dushevina was knocked out by 16th-seeded German Sabine Lisicki 6-2 7-6.

Other winners were Russian trio Ekaterina Makarova , Alisa Kleybanova and Anastasia Pavlyuchenkova, Spain’s Carla Suarez Navarro and Maria-Teresa Torro-Flor, Argentina’s Paula Ormaechea and Canada’s Eugenie Bouchard.

Also to advance to the second round were France’s Kristina Mladenovic and Caroline Garcia, Estonia’s Kaia Kanepi, Romania’s Sorana Cirstea, American duo Coco Vandeweghe and Bethanie Mattek-Sands, Slovakia’s Anna Schmiedlova, Japan’s Kurumi Nara, Austria’s Patricia Mayr-Achleitner, Poland’s Urszula Radwanska, Sweden’s Sofia Arvidsson, China’s Zheng Jie, Kazakhstan’s Yaroslava Shvedova and Galina Voskoboeva, plus Australia’s Ashleigh Barty.

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