CHICAGO (AP) _ Fifth-ranked Lindsay Davenport advanced to the quarterfinals of the $450,000 Ameritech Cup tennis tournament with a 6-2, 6-4 victory over South African's Joanette Kruger.

Davenport, the third seed, was disappointed in her serving, but knew that wouldn't put her in too much trouble against Kruger. Davenported watched Kruger play earlier in the week and noticed the South African wasn't hitting her second serve very hard.

``I missed a lot of first serves, and so that took away a lot of free points for myself. Maybe a couple of points could have been shorter,'' Davenport said.

In another second-round match earlier in the day, French Open champion Iva Majoli, the fourth seed, beat Belgium's Laurence Courtois 7-5, 6-1.

Majoli, who has failed to reach the quarterfinals in five of the eight tournaments since winning at Roland Garros, took control late in the first set with her baseline ground strokes.

Mary Pierce beat Romanian qualifier Catalina Cristea 6-3, 6-3 in a first-round match. Pierce, seeded fifth, used a big serve to overcome her erratic play from the baseline in her first action since the Fed Cup early last month.

``I'm not worried so much about how I do. Wins and losses are not on my mind. I need to get matches and some training,'' Pierce said, looking ahead to the season-ending Chase Championships in two weeks.

In other first-round matches Tuesday, seventh-seeded Brenda Schultz-McCarthy ousted Chanda Rubin 6-4, 7-5; Poland's Magdalena Grzybowska won the final four games and eliminated 1996 Ameritech Cup finalist Jennifer Capriati 6-3, 7-5; and eighth-seeded Lisa Raymond, showing a wide array of shots from both the baseline and net, beat veteran Helena Sukova 6-2, 6-2.