MELBOURNE, Australia – Serena Williams is starting to convince a lot of people of something she’s believed all along.

The seven-time Grand Slam champion has gone from a dangerous floater to a big threat for the women’s title at the Australian Open.

“Yeah, I have it in me,” she said Sunday. “I believe in my game, and more than anything I believe in me. It doesn’t matter what people say or people write. At the end of the day, I’m my biggest fan.”

Ranked No. 81 and seemingly out of condition, Williams came into the season’s first major with only two matches in a low-key warmup. And that followed an injury-plagued 2006, when she played only four events.

She had to save match points against No. 5 Nadia Petrova to make the fourth round – her first win over a top 10 player since she won the Australian title in 2005 – then beat No. 11 Jelena Jankovic 6-3, 6-2.

Jankovic was on a hot streak, winning a title at Auckland and reaching the final at the Sydney International last week. Yet it was Williams who was aggressive from the start, making an early break and not relenting.

In the second set, Williams finished off a 17-shot rally with a trademark crosscourt backhand to end the fourth game, pumping her fist as Jankovic stared and gaped at the angle.

Helping Williams’ cause were defending champion Amelie Mauresmo and No. 3 Svetlana Kuznetsova. They were beaten, and that left No. 10 Nicole Vaidisova, a 17-year-old Czech, as the highest-ranked player in Williams’ half of the draw.

Williams next meets Shahar Peer, who beat 2004 U.S. Open champion Kuznetsova 6-4, 6-2. Peer, who spends her offseasons doing compulsory service in the Israeli army, is in the quarterfinals at a major for the first time. She lost her only previous match with Williams at Miami in 2005.

Second-seeded Mauresmo fell 6-4, 6-3 to 70th-ranked Lucie Safarova, making her exit a week earlier than she did in her breakthrough major in Melbourne last year.

Mauresmo, who also won Wimbledon last year and spent most of the season at No. 1, jumped to a 4-1 lead before 19-year-old Safarova found her range and starting passing her repeatedly.

“It’s always disappointing to go out of any tournament, even more in a Grand Slam,” Mauresmo said. “But I am going to go back to work, and that’s obviously what I need.”

This was Safarova’s first time on Rod Laver Arena. She had been 1-6 at her six previous Grand Slams.

“I came out this morning and said, ‘Wow this is a big court.’ But I felt really comfortable here,” said Safarova, who next plays fellow Czech Vaidisova, a 6-3, 6-3 winner over seventh-seeded Elena Dementieva.

While top-seeded Maria Sharapova had the day off and two top women were losing, Federer stayed on track for a 10th Grand Slam with a mostly routine 6-2, 7-5, 6-3 win over 14th-seeded Novak Djokovic of Serbia.

Andy Roddick beat No. 9 Mario Ancic 6-3, 3-6, 6-1, 5-7, 6-4 to set up a quarterfinal against friend and former housemate Mardy Fish. Roddick conceded only five points in his last four service games to turn up the heat.

“I felt like at the end of fourth set he had the momentum and was really being offensive in the points,” Roddick said. “I knew that in fifth set, win or lose, I had to turn the tables on the aggression.”

The unseeded Fish, who lived with Roddick’s family for a year in 1999, made the last eight for the first time at a major, beating No. 16 David Ferrer 6-1, 7-6 (4), 2-6, 7-5.

Fish was the 2004 Olympic silver medalist and is still coming back from wrist operations in 2005. He said a match with Roddick was almost surreal, and would bring out an almost sibling rivalry.

“Just growing up, we’ve kind of dreamt about, talked about playing in the quarterfinals, finals, whatever, of Slams,” he said.

Federer was rarely threatened by Djokovic, who won the Adelaide International tournament two weeks ago and was one of the most improved players on the ATP Tour last year. It was almost as if Federer were welcoming – or warning – Djokovic about his arrival to the big-time.

Federer faced his first break point of the match – and only second overall – in the second set and smashed an overhead into the net to allow Djokovic to pull to 4-3 and get back on serve.

But the top-ranked Swiss broke again when it counted most – using a classic backhand cross-court when Djokovic was serving to stay in the second set. He delivered an ace on match point.

“It looks like he doesn’t feel any pressure, which is really strange because he’s No. 1 and everybody wants to get his spot,” Djokovic said.

For Federer, it was a matter of timing his run.

“Straight sets every time, a day off – it couldn’t be better.” He’ll be aiming for a 34th consecutive win when he meets seventh-seeded Tommy Robredo, who beat Richard Gasquet of France 6-4, 6-2, 3-6, 6-4.

Federer is still a big favorite to claim his 10th Grand Slam title, with No. 2 Rafael Nadal and No. 15 Andy Murray – the only two men to beat him last year – meeting Monday in the fourth round.

Williams faces a tougher challenge for her eighth major, with Sharapova and Kim Clijsters on the other side of the draw.

Sharapova, the U.S. Open champion, is guaranteed of taking the No. 1 ranking at the end of the month. She plays Monday, as do No. 4 Clijsters and three-time Australian champion Martina Hingis.

To get a shot at one of them, Williams knows she has to be more consistent.

She is not the same player as when she completed a Serena Slam – holding all four majors when she won the French, Wimbledon and U.S. Open in 2002 and the Australian Open in 2003. These days, she has more on her mind than painting the lines and finishing every point a.s.a.p.

“I’ve gotten way more consistent than I have been in the past,” she said. “Just not missing as many shots as I used to miss. Just blast the ball out. I’ve got over that.”

Though she still won’t back down from a physical challenge.

“When I go out to play, I like to go to the person’s stronger side,” she said. “I guess it makes me feel macho, that if I beat them going to their stronger side it makes me feel extra awesome.”