Maria Sharapova says she doesn't like Twitter because she bores herself and would not want to inflict that on others; Sam Stosur says she is a "little bit disturbed" that someone has taken the time to create and run a profile pretending to be her biceps.

The two women prefer using Facebook to let their fans know what they are up to but the other thing they have in common is that they are looking more and more like the favourites for the women's title at the French Open.

World No.1 Victoria Azarenka, a 6-4, 6-4 winner over Alexandra Wozniak of Canada, remains a real threat but No.3 seed Agnieszka Radwanska was surprisingly beaten 6-1, 6-2 by former champion Svetlana Kuznetsova of Russia.

Having been held over until yesterday because of the marathon match between Paul-Henri Mathieu and John Isner the previous night, Sharapova made up for lost time with a 6-1, 6-1 demolition of Japan's Ayumi Morita to reach round three. Due to the delay, the Russian will have to play again today against Shuai Peng of China but having won the title in Rome just under a fortnight ago, the world No.2 is looking strong. The biceps of Stosur, meanwhile, saw the Australian to an equally impressive 6-3, 6-3 win over Nadia Petrova, the Russian who has given her plenty of trouble in the past.

The shock first-round defeat of Serena Williams cleared Sharapova's potential path to the final while Stosur, in the other half of the draw, is the big threat to world No.1 Azarenka, a potential quarter-final opponent. Having lost her last seven matches with Williams, Sharapova could be forgiven if she allowed herself a little whoop of delight when the American was upset by Virginie Razzano early on.

Yet the Russian, who is bidding to complete a full set of grand slam titles, is far too experienced to either admit such feelings or, probably, even have them. "It's tough to be in the first round of a tournament and think 'this person lost, who you could have faced in the quarter-final," Sharapova said. "I think that's just a really wrong way to look at things.

"If you're thinking like that then you're at a big disadvantage because no matter who you play, if you get to the quarter-finals, you have to play at a high level. You can't worry about if it's Serena or somebody else across the net."

As unlikely as it may have seemed a few years ago – certainly when she began her comeback in 2009 after nine months out because of shoulder surgery – Sharapova has become a fine clay-court player. Her wins on the surface in Stuttgart and Rome made her the second favourite coming into Paris, behind Williams, and Sharapova admitted it would be a massive honour to join the list of players to have won all four grand slams.

"It would mean a lot to join them," she said. "But it's not something that you think of, exactly. I think it's more of a personal achievement than anything else."

Stosur probably should have won the title two years ago when she froze on the big stage and was beaten by an emotion-packed performance from Francesca Schiavone. The Australian's confidence received a huge boost when she beat Williams to win the US Open last September, and the American's absence this time is another lift.

"She was one of the favourites to go through and maybe win the tournament," said Stosur, who now plays American Sloane Stephens. "It's definitely still open. I'm very pleased with the three matches I have played. I thought today was very, very solid and a good match. Hopefully it's going to set me up for the next one."

Third seed Radwanska began the tournament with a chance to become world No.1 but was stunned by Kuznetsova, who played her best match since reaching the last eight here 12 months ago. "I have not played so well in long time," the Russian said. "So I'm really happy."